MrHphysics

AP Physics 1: ENERGY, Work, Power Enduring Understandings: Enduring Understanding 1.A: The internal structure of a system determines many properties of the system. Enduring Understanding 3.D: A force exerted on an object can change the momentum of the object. Enduring Understanding 3.E: A force exerted on an object can change the kinetic energy of the object. Enduring Understanding 4.B: Interactions with other objects or systems can change the total linear momentum of a system Enduring Understanding 4.C: Interactions with other objects or systems can change the total energy of a system Enduring Understanding 5.A: Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems. Enduring Understanding 5.B: The energy of a system is conserved. Enduring Understanding 5.D: The linear momentum of a system is conserved.

Essential Knowledge: Essential Knowledge 1.A.1: A system is an object or a collection of objects. Objects are treated as having no internal structure. Essential Knowledge 1.A.5: Systems have properties determined by the properties and interactions of their constituent atomic and molecular substructures. In AP Physics, when the properties of the constituent parts are not important in modeling the behavior of the macroscopic system, the system itself may be referred to as an object. Essential Knowledge 2.A.1: A vector field gives, as a function of position (and perhaps time), the value of a physical quantity that is described by a vector Essential Knowledge 3.B.2: Free–body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation. Essential Knowledge 3.B.3: Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples should include gravitational force exerted by the Earth on a simple pendulum, mass–spring oscillator Essential Knowledge 3.D.1: The change in momentum of an object is a vector in the direction of the net force exerted on the object. Essential Knowledge 3.D.2: The change in momentum of an object occurs over a time interval.

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Essential Knowledge 3.E.1: The change in the kinetic energy of an object depends on the force exerted on the object and on the displacement of the object during the interval that the force is exerted. Essential Knowledge 4.B.1: The change in linear momentum for a constant–mass system is the product of the mass of the system and the change in velocity of the center of mass. Essential Knowledge 4.B.2: The change in linear momentum of the system is given by the product of the average force on that system and the time interval during which the force is exerted. Essential Knowledge 4.C.1: The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy. Essential Knowledge 4.C.2: Mechanical energy (the sum of kinetic and potential energy) is transferred into or out of a system when an external force is exerted on a system such that a component of the force is parallel to its displacement. The process through which the energy is transferred is called work. Essential Knowledge 5.A.1: A system is an object or a collection of objects. The objects are treated as having no internal structure. Essential Knowledge 5.A.2: For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved. For an isolated or a closed system, conserved quantities are constant. An open system is one that exchanges any conserved quantity with its surroundings. Essential Knowledge 5.A.3: An interaction can be either a force exerted by objects outside the system or the transfer of some quantity with objects outside the system. Essential Knowledge 5.A.4: The boundary between a system and its environment is a decision made by the person considering the situation in order to simplify or otherwise assist in analysis. Essential Knowledge 5.B.1: Classically, an object can only have kinetic energy since potential energy requires an interaction between two or more objects. Essential Knowledge 5.B.2: A system with internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy. [Physics 1: includes mass–spring oscillators and simple pendulums. Physics 2: charged object in electric fields and examining changes in internal energy with changes in configuration.] Essential Knowledge 5.B.3: A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces. Essential Knowledge 5.B.4: The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the configuration of the objects that make up the system. Essential Knowledge 5.B.5: Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is called work. Energy transfer in mechanical or electrical systems may occur at different rates. Power is defined as the rate of energy transfer into, out of, or within a system. Essential Knowledge 5.D.1: In a collision between objects, linear momentum is conserved. In an elastic collision, kinetic energy is the same before and after. Essential Knowledge 5.D.2: In a collision between objects, linear momentum is conserved. In an inelastic collision, kinetic energy is not the same before and after the collision. Essential Knowledge 5.D.3: The velocity of the center of mass of the system cannot be changed by an interaction within the system. [Physics 1: includes no calculations of centers of mass; the equation is not provided until Physics 2. However, without doing calculations, Physics 1 students are expected to be able to locate the center of mass of highly symmetric mass distributions, such as a uniform rod or cube of uniform density, or two spheres of equal mass.]

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I kind of know what this is, but could not test well

I have a moderate grasp of this concept

I know what this is and could test well

I have a thorough understanding and could teach this to another

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(RATE YOUR UNDERSTANDING OF THE OBJECTIVES) *For signatures, if the student attempts to explain a concept to you, but not thoroughly and not to a point where you feel they understand it, initial NY for not yet. A NY will help a student identify areas of focus. Initial when they describe the concept well enough for you to understand it as well.

SIGNATURES/Initials

I have no idea 1.

AP PHYSICS 1 CHART OF UNDERSTANDING UNIT: ENERGY

Learning Objective (5.B.3.1): Describe and make qualitative and/or quantitative predictions about everyday examples of systems with internal potential energy. [SP 2.2, SP 6.4, SP 7.2] Learning Objective (5.B.4.1): Describe and make predictions about the internal energy of everyday systems. [SP 6.4, SP 7.2] Learning Objective (1.A.5.1): Model verbally or visually the properties of a system based on its substructure and relate this to changes in the system properties over time as external variables are changed. [SP 1.1, SP 7.1] Learning Objective (5.B.3.2): Make quantitative calculations of the internal potential energy of a system from a description or diagram of that system. [SP 1.4, SP 2.2] Learning Objective (5.B.3.3): Apply mathematical reasoning to create a description of the internal potential energy of a system from a description or diagram of the objects and interactions in that system. [SP 1.4, SP 2.2] Learning Objective (4.C.1.1): Calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy. [SP 1.4, SP 2.1, SP 2.2] Learning Objective (4.C.1.2): Predict changes in the total energy of a system due to changes in position and speed of objects or frictional interactions within the system. [SP 6.4] Learning Objective (4.B.1.1): Set up a representation or model showing that a single object can only have kinetic energy and use information about that object to calculate its kinetic energy. [SP 1.4, SP 2.2] Learning Objective (5.B.1.2): Translate between a representation of a single object, which can only have kinetic energy, and a system that includes the object, which may have both kinetic and potential energies. [SP 1.5] Learning Objective (5.B.4.2): Calculate changes in kinetic energy and potential energy of a system, using information from representations of that system. [SP 1.4, SP 2.1, SP 2.2] Learning Objective (3.E.1.1): Make predictions about the changes in kinetic energy of an object based on considerations of the direction of the net force on the object as the object moves. [SP 6.4, SP 7.2]

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Learning Objective (3.E.1.3): Use force and velocity vectors to determine qualitatively or quantitatively the net force exerted on an object and qualitatively whether kinetic energy of that object would increase, decrease, or remain unchanged. [SP 1.4, SP 2.2] Learning Objective (3.E.1.4): Apply mathematical routines to determine the change in kinetic energy of an object given the forces on the object and the displacement of the object. [SP 2.2] Learning Objective (5.B.5.1): Design an experiment and analyze data to examine how a force exerted on an object or system does work on the object or system as it moves through a distance. [SP 4.2, SP 5.1] Learning Objective (5.B.5.2): Design an experiment and analyze graphical data in which interpretations of the area under a force-distance curve are needed to determine the work done on or by the object or system. [SP 4.2, SP 5.1] Learning Objective (5.B.5.3): Predict and calculate from graphical data the energy transfer to or work done on an object or system from information about a force exerted on the object or system through a distance. [SP 1.4, SP 2.2, SP 6.4] Learning Objective (5.A.2.1): Define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations. [SP 6.4, SP 7.2] Learning Objective (5.B.5.4): Make claims about the interaction between a system and its environment in which the environment exerts a force on the system, thus doing work on the system and changing the energy of the system (kinetic energy plus potential energy). [SP 6.4, SP 7.2] Learning Objective (5.B.5.5): Predict and calculate the energy transfer to (i.e., the work done on) an object or system from information about a force exerted on the object or system through a distance. [SP 2.2, SP 6.4] Learning Objective (4.C.2.1): Make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the displacement of the center of mass. [SP 6.4] Learning Objective (4.C.2.2): Apply the concepts of conservation of energy and the work-energy theorem to determine qualitatively and/or quantitatively that work done on a two-object system in linear motion will change the kinetic energy of the center of mass of the system, the potential energy of the systems, and/or the internal energy of the system. [SP 1.4, SP 2.2, SP 7.2] Learning Objective (5.B.2.1): Calculate the expected behavior of a system using the object model (i.e., by ignoring changes in internal structure) to analyze a situation. Then, when the model fails, justify the use of conservation of energy principles to calculate the change in internal energy due to changes in internal structure because the object is actually a system. [SP 1.4, SP 2.1]

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ENERGY PREVIEW This unit is the study of interactions of particles and systems to exchange, store and release energy in various forms. The universe as we know it has a fixed amount of total energy. We use this principle (Law of Conservation of Energy) to evaluate exchanges of energy quantities.

Vocabulary Terms conservative force a force which does work on an object which is independent of the path taken by the object between its starting point and its ending point energy the non-material quantity which is the ability to do work on a system joule the unit for energy equal to one Newton-meter kinetic energy the energy of an inertial mass has due its motion law of conservation of energy the total energy of a system remains constant during a process mechanical energy the sum of the potential and kinetic energies in a system potential energy the energy an object has because of its position power the rate at which work is done or energy is dissipated spring constant a value that represents the stiffness of a spring ; the force per unit displacement from equilibrium system (closed) an object or collection of objects that only interacts with its members system (open) an object or collection of objects that may exchange forces and energy with its surroundings watt the SI unit for power equal to one joule of energy per second work the scalar product of force and displacement

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Work Key Concept/Idea(s):

Conceptual Example 1: A 1kg block is pushed across a table at 4m/s. What can you determine about the work done by friction? A) Friction does no work B) Friction does positive work C) Fricton does negative work D) Friction only does work if the block slows down Conceptual Example 2: If the work done by a force on an object is not zero then the force is said to be a. Conservative b. Non-conservative c. zero d. none of the above

Problem-Solving Example 1: A 1000-N bag rests on the floor. How much work is required to move it at constant speed A) 4.0 m along the floor against a friction force of 200 N B) 4.0 m vertically? C) What tension force is required to move the bag if being pulled at 100 to the horizontal?

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Problem-Solving Example 2: A sleeping hippo of mass 2000kg is pulled horizontally by a force across a floor giving it an acceleration of 1.0m/s2 for 6.0s. The coefficient of kinetic friction between the hippo and the floor is 0.2. Find: A) the net work done on the hippo. B) the work done by friction

Problem-Solving Example 3: A 300 kg piano is pushed up a 30º incline by a human at constant speed for 4m. The coefficient of kinetic friction for the piano on the platform is 0.30. Determine: A) the force exerted by the man B) the work done by the human on the piano C) the work done by friction on the piano D) the net work done on the piano

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Graph Examples: Each graph represents a force acting on a body and its location. For Each graph, determine work done by the force over the entire time interval

Use the graphs below to sketch work done by forces in application examples 1A and 2. Be sure to add a key.

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Reasoning Example 1: Student 1 and student 2 decide to displace a wagon 2m. Student 1 displaces the wagon 2m horizontally by rolling it across the ground. Student 2 lifts the wagon 2m vertically by picking it up and raising it over their head. Student 1 says “We did the same amount of work because we moved the same wagon the same distance” Student 2 says “I did more work because I applied more force to the wagon than you did over 1m” A) B) C) D)

Evaluate student 1’s statement “We did the same work” for correctness. Justify your answer Evaluate student 2’s statement “I did more work” for correctness. Justify your answer Identify any aspects of student 1’s reasoning that is correct. Justify your answer Identify any aspects of student 2’s reasoning that is correct. Justify your answer

Reasoning Example 2: Which of the following are true concepts about work? Justify your choice. a. work describes the position of an object as a function of time. b. the work done on an object is always independent of the path. c. the work done on an object depends on its path. d. work is a force e. work provides a link between force and energy.

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Kinetic Energy Key Concept(s):

Conceptual Example 1: A cat moves with speed of 5 m/s then speeds up to achieve a speed of 10 m/s. How many times greater is the new kinetic energy of the cat? a. b. c. d. e.

Half as much The same √2 times as great Twice as great Four times as great

Conceptual Example 2: Car #1 has twice the mass of car #2, but they both have the same kinetic energy. How do their speeds compare? a) b) c) d) e)

2 v 1 = v2 2 v1 = v2 4 v1 = v2 v1 = v2 8 v1 = v2

Problem-Solving Example 1: A 1500kg car travels at 20m/s before coming to a stop. Determine A) the change in kinetic energy required to stop the car. B) When going from 20m/s to 10m/s do the brakes need to do the same amount of work as they do from 10m/s to 0m/s? Justify your answer.

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Problem-Solving Example 2: A 50kg stuntman is fired from a cannon which exerts an average force of 10,000 N on the stuntman over a distance of 1.5m. What is the speed of the stuntman as he leaves the cannon?

Graph Example: Sketch a graph of Kinetic energy vs. speed. Does your graph go below zero on the KE axis? Why or why not?

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Reasoning Example: Three people argue over how to save fuel while driving. Their argument is based on using one car and whether loading it with more or less inertial mass is good for fuel economy. The transcript of their argument is below: Artimus: When driving through the city it helps fuel economy to have less inertial mass in the car because all of the changes in speed cost less energy if the car is lighter. Gretchen: But when driving on the highway, it is better to have more inertial mass because wind resistance would slow the car less for a more massive car by nature of its momentum. Overcoming these smaller speed reductions would cost less energy for the more massive car. Charlie: There is no distinct advantage to a massive or less massive car because whatever energy is consumed by the fuel is converted to roughly the same amount of kinetic energy in the car. For each person: 1-determine the correctness of their assertion 2-evaluate the validity of their reasoning

Explain your reasoning.

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Gravitational Potential Energy Key Concept(s):

Conceptual Example 1: A 5kg block is raised 3m off the ground in each of the paths shown. Compare the work input and change in gravitational potential energy of each case.

Problem-Solving Example 1: By how much does the gravitational potential energy of a 50kg high-jumper change if her center of mass rises by 1.6m during the jump?

Problem-Solving Example 2: A 1100kg car starts from rest at the bottom of a large hill and drives up until it arrives atop a 500m peak. (a) What is the car’s change in gravitational potential energy? (b) Why is it impossible for the car to accomplish this without using more than (answer to A) Joules of energy?

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Graphing Example: Ex.1: A 5kg bucket of water is pulled up from a 20m deep well at a constant speed of 1m/s. When the bucket reaches the top of the well, 1m above ground, it immediately is dropped all the way back down. On the graph provided, plot Gravitational Potential Energy vs. Position, define the ground where y = 0

For the exact same event (beginning from when the bucket is at the bottom of the well being pulled up) sketch a graph of: Kinetic Energy vs. time Gravitational Potential Energy vs. time Total Mechanical Energy vs. time

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Elastic Potential Energy Key Concept(s):

Problem-Solving Example 1: A spring has a spring stiffness constant, k = 400N/m. How much must this spring be stretched to store 100J of potential energy?

Problem-Solving Example 2: A 1400-kg taxi-cab travels on a flat surface with a speed of 30m/s when it strikes a horizontal coiled spring and is brought to rest in a distance of 4.0m. Determine the effective spring constant, k.

Problem-Solving Example 3: A 1kg mass is hung from a spring with constant k = 500N/m. It stretches the spring to an equilibrium length. A second 1kg mass is then fixed to the first one. Determine the displacement between the two equilibrium points.

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Transfers of Energy Key Concept/Idea(s):

Conceptual Example 1: A student falls down a flight of stairs as shown in the figure. They reach the bottom of the stairs and come to rest upon colliding with the floor. How do you account for their loss of gravitational potential energy?

Conceptual Example 2: A 50kg tricycle operator applies a 120N force to increase her speed over 20m. Assuming no energy is lost to nonconservative forces, the amount of Kinetic Energy gained by the system is: a. b. c. d.

Equal to the work done by the tricyclist Half the work done by the tricyclist Double the work done by the tricyclist Equal to mass times acceleration

Problem-Solving Example 1: A 10kg block, initially moving 2m/s is accelerated by a force across a floor at 2.5m/s2 for 5s. Determine the net work done on the box.

Problem-Solving Example 2: If 40,000J of work is done against gravity pushing a 1000kg car up a 100 frictionless incline for 10m of distance, determine the final speed of the car after 10m.

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Problem-Solving Example 3: How high will a 3.5-kg rock go if thrown straight up by someone who does 80.0 J of work on it? Neglect air resistance.

Problem-Solving Example 4: If it takes 4,000J of to accelerate an object from 0m/s to 10m/s over 10m a) How much more energy is required to change it from 10m/s to 20m/s? b) Assuming the same force is applied as previously, how much distance would be covered between 10m/s and 20m/s?

Problem-Solving Example 5: If the speed of a car is increased by 50%, by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the driver’s reaction time.

Problem-Solving Example 6: A 2,000kg car travelling at 30m/s must come to a complete stop over 90m. What is the minimum frictional force required for the car to stop? .

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Reasoning Example 1: Two golf balls (ball A and ball B) are thrown into identical blocks of jello in identical conditions. Each enters exactly perpendicular to the surface, embeds to a depth and comes to a stop inside the jello block. Ball A enters the jello at a speed of 20m/s, ball B enters the jello at 30m/s. Assume any friction/resistant force from jello are equal in both cases. 1. Compare depth of ball A to depth of ball B by completing the statement: Ball A sinks to a depth of ___ times the depth of ball B Explain your reasoning.

Reasoning Example 2: A 1kg block of ice (block A) slides down a steep, frictionless incline with height 1m above the floor. Another identical iceblock (block B) slides down a longer, gradual frictionless incline with height 1m. If both blocks are released at the same time… 2. Which block gets to the ground first? ___Block A ___Block B ___Neither, they reach the floor at the same time Explain your reasoning.

3. Which block has greatest speed when it reaches the floor? ___Block A ___Block B ___Neither, they reach the ground at with the same speed Explain your reasoning.

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Graphing Examples: Ex.1: A 1kg mass is dropped vertically downward 100m. Sketch sketch trend lines for Kinetic Energy, Gravitational Potential Energy and Total Mechanical Energy of the marble. Ignore any effects of air resistance.

Ex.2: A 0.1kg marble is thrown at 10m/s horizontally off the edge of a 500m cliff. On the graph below, sketch trend lines for Kinetic Energy, Gravitational Potential Energy and Total Mechanical Energy of the marble. Ignore any effects of air resistance.

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Law of Conservation of Energy Key Concept/Idea(s):

. Problem-Solving Example 1: In the high jump, Fran’s kinetic energy is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must Fran leave the ground in order to lift her center of mass 2.10 m and cross the bar with a speed of 0.70 m s ?

Problem-Solving Example 2: A 50-kg trampoline artist jumps vertically upward from the top of a platform with a speed of 5.0 m s . (a) How fast is he going as he lands on the trampoline, 5.0 m below? (b) If the trampoline behaves like a spring of constant k = 5,000N/m, how far downward will the surface displace at its lowest point?

Problem-Solving Example 3: Determine the ratio between radius R and height h such that a toy-car can safely make the loop de loop?

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Problem-Solving Example 4: A 4 kg block slides a distance (d) of 5m along a slope of inclination angle of 30o. The coefficient of kinetic friction is μk = 0.20. What will be the speed of the block at the end of the slope?

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(Energy Conservation continued with Atwood’s-like systems) Problem-Solving Example 1: M1 and M2, both of mass 1kg are attached to a string and pulley about a 300 frictionless incline as shown. Determine the total kinetic energy of the system after 3s.

Problem-Solving Example 2: Initially at rest, a 1kg mass (M2) is raised 1.0m during the dropping of a 2kg mass (M2) via a pulley system as shown. Find the velocity of M2 just before it hits the floor.

Problem-Solving Example 3: In the previous example, when M2 hits the floor, the 1kg mass (M1) continues upwards and is released from the hook. a) Determine the maximum height of M2 b) How much mechanical energy is lost to non-mechanical forms from t = 0s until M1 reaches its maximum height?

Problem-Solving Example 4: In the pulley system illustrated, M2 drops 30cm from rest. Determine the speeds of M1 and M2 (M1 hangs from 6-pulleys, M2 is suspended from 2 pulleys)

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Power(Giancoli questions) Key Concept/Idea(s):

Problem-Solving Example 1: How long will it take a 1750-W motor to lift a 315-kg piano to a sixth-story window 16.0 m above?

Problem-Solving Example 2: A driver notices that her 1480-kg car slows down from 30m/s to 20m/s in about 10.0 s on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 25m/s?

Problem-Solving Example 3: A pump is to lift 18.0 kg of water per minute through a height of 3.60 m. What output rating (watts) should the pump motor have?

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Open and Closed Systems Key Concept/Idea(s):

Example 1: A plate falls from a table and shatters on the ground. Check the following items for which, if removed, would compromise the total energy (could you say “Energy initial equals energy final” if we don’t account for the item?) a) The Earth b) An identical plate dropping event on the other side of the country c) The surrounding air d) Any listening ears to receive sound (i.e. tree-falling-in-the-woods) e) Viewers watching the event on live television from several hundred kilometers away Example 2: If the Earth and moon, suddenly lost all motion (applying the space-brakes) then gravitationally fell into each other, colliding and becoming one body… a) What would happen to their mutual center of mass?

b) How much energy would be transferred to heat of the system?

c) Where did the “heat energy” come from (inside or outside of the system)?

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Example 3: A car is moving at 20m/s when it slams on its brakes and skids to a stop over 40m a) Identify the change in total mechanical energy b) Identify the total change in energy c) Describe the “work” done as positive or negative. Justify your answer d) Regarding the work done, was this from an internal force or an external force?

Example 4: Explain why (in terms of both forces and energy) one particle or object MUST interact with another object in order to change its motion

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(Additional Multiple-Choice Practice from njctl.org)

PSI AP Physics I Work and Energy Multiple-Choice questions 1. A driver in a 2000 kg Porsche wishes to pass a slow moving school bus on a 4 lane road. What is the average power in watts required to accelerate the sports car from 30 m/s to 60 m/s in 9 seconds? A. 1,800 B. 5,000 C. 100,000 D. 300,000 2. A force F is at an angle θ above the horizontal and is used to pull a heavy suitcase of weight mg a distance d along a level floor at constant velocity. The coefficient of friction between the floor and the suitcase is μ. The work done by the force F is: A. Fdcos θ - μ mgd B. Fdcos θ C. -μ mgd D. 2Fdsin θ - μ mgd 3. A force of 20 N compresses a spring with a spring constant 50 N/m. How much energy is stored in the spring? A. 2 J B. 4 J C. 5 J D. 6 J 4. A stone is dropped from the edge of a cliff. Which of the following graphs best represents the stone's kinetic energy KE as a function of time t?

A. 5.

B.

C.

D.

A 4 kg ball is attached to a 1.5 m long string and whirled in a horizontal circle at a constant speed 5 m/s. How much work is done on the ball during one period? A. 9 J B. 4.5 J C. 2 J D. 0 J

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6. A student pushes a box across a horizontal surface at a constant speed of 0.6 m/s. The box has a mass of 40 kg, and the coefficient of kinetic friction is 0.5. The power supplied to the box by the person is: A. 40 W B. 60 W C. 120 W D. 150 W 7. You need to move three identical couches from the first to the second floor of an apartment building. The first time, you and a friend make a mistake and carry a couch up to the third floor and then back down to the second floor. The second couch is carried directly from the first to the second floor. On your third trip, you decide to put a ramp over the staircase and you both push the couch up the ramp to the second floor. During which trip did you perform the most work on the couch? A. The first trip B. The second trip C. The third trip D. The same work was performed for each trip.

8. A force F is applied in horizontal to a 10 kg block. The block moves at a constant speed of 2 m/s across a horizontal surface. The coefficient of kinetic friction between the block and the surface is 0.5. The work done by the force F in 1.5 minutes is: A. 9000 J B. 5000 J C. 3000 J D. 2000 J

Questions 9-10: A ball swings from point 1 to point 3 in the diagram to the right. Assume that the ball is in Simple Harmonic Motion and point 3 is 2 m above point 2 (the lowest point).

9. What happens to the kinetic energy of the ball when it moves from point 1 to point 2? A. increases B. decreases C. remains the same D. more information is required 10. What is the velocity of the ball at point 2? A. 2.2 m/s B. 3.5 m/s C. 5.1 m/s D. 6.3 m/s 27 https://sites.google.com/site/mrhphysics/

11. As shown above, a block with a mass of m slides at a constant velocity V0 on a horizontal frictionless surface. The block collides with a spring and comes to rest when the spring is compressed to the maximum value. If the spring constant is K, what is the maximum compression in the spring? A. V0 (m/K)1/2 B. KmV0 C. V0K/m D. V0 (K/m)1/2

Questions 12-13: A 2 kg block is released from rest from the top of an inclined plane, as shown in the diagram to the right. There is no friction between the block and the surface.

12. How much work is done by the gravitational force on the block? A. 80 J B. 60 J C. 40 J D. 20 J 13. What is the speed of the block when it reaches the horizontal surface? A. 3.2 m/s B. 4.3 m/s C. 5.8 m/s D. 7.7 m/s 14. A crane lifts a 300 kg load at a constant speed to the top of a building 60 m high in 15 s. The average power expended by the crane to overcome gravity is: A. 10,000 W 28 https://sites.google.com/site/mrhphysics/

B. 12,000 W C. 15,000 W D. 30,000 W

15. A satellite with a mass m revolves around Earth in a circular orbit with a constant radius R. What is the kinetic energy of the satellite if Earth’s mass is M? A. ½ mv2 B. mgh C. ½GMm/R2 D. ½ GMm/R Questions 16-17: An apple of mass m is thrown horizontally from the edge of a cliff of height H, as shown to the right.

16. What is the total mechanical energy of the apple with respect to the ground when it is at the edge of the cliff? A. ½ mv02 B. mgH C. mgH + ½ mv02 D. ½ mv02- mgH 17. What is the kinetic energy of the apple just before it hits the ground? A. ½ mv02 + mgH B. ½ mv02 - mgH C. mgH D. ½ mv02 Questions 18-19: A 500 kg roller coaster car starts from rest at point A and moves down the curved track, as shown to the right. Assume the track is frictionless.

18. Find the speed of the car at the lowest point B. A. 10 m/s B. 20 m/s C. 30 m/s D. 40 m/s 29 https://sites.google.com/site/mrhphysics/

19. Find the speed of the car when it reaches point C. A. 10 m/s B. 20 m/s C. 30 m/s D. 40 m/s 20. Two projectiles A and B are launched from the ground with velocities of 50 m/s at 60 ̊ (projectile A) and 50 m/s at 30 ̊ (projectile B) with respect to the horizontal. Assuming there is no air resistance involved, which projectile has greater kinetic energy when it reaches the highest point? A. projectile A B. projectile B C. they both have the same non-zero kinetic energy D. they both have zero kinetic energy Questions 21-22: An object with a mass of 2.0 kg is initially at rest at a position x = 0. A non constant force F is applied to the object over a displacement of 6.0 m, as shown in the graph to the right.

21. What is the total work done on the object at the end of 6.0 m? A. 200 J B. 190 J C. 170 J D. 150 J 22. What is the velocity of the object at x = 6.0 m? A. 300 m/s B. 150 m/s C. 25 m/s D. 12 m/s 23. A metal ball is held stationary at a height h0 above the floor and then thrown downward. Assuming the collision with the floor is elastic, which graph best shows the relationship between the net energy E of the metal ball and its height h with respect to the floor?

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24. A toy car travels with speed V0 at point X. Point Y is a height H below point X. Assuming there is no frictional losses and no work is done by the motor, what is the speed at point Y? A. (2gH+ ½ V02)1/2 B. V0-2gH C. (2gH + V02)1/2 D. 2gH+ (½V02)1/2 25. A rocket is launched from the surface of a planet with mass M and radius R. What is the minimum velocity the rocket must be given to completely escape from the planet’s gravitational field? A. (2GM/R)1/2 B. (2GM/R)3 C. (GM/R)1/2 D. 2GM/R 26. A block of mass m is placed on the frictionless inclined plane with an incline angle θ. The block is just in a contact with a free end on an unstretched spring with a spring constant k. If the block is released from rest, what is the maximum compression in the spring: A. kmg sinθ B. kmg cosθ C. 2mg sinθ /k D. mg/k

Questions 27-29: In a physics lab, a student uses three light, frictionless wheeled carts as shown to the right. Each cart is loaded with blocks of equal mass.

27. The same force F is applied to each cart and they move equal distances d. In which one of these three cases is more work done by force F? A. cart I B. cart II C. cart III D. the same work is done on each cart 28. The same force F is applied to each cart and they move equal distances d. Which cart will have more kinetic energy at the end of distance d? A. cart I B. cart II 31 https://sites.google.com/site/mrhphysics/

C. cart III D. all three will have the same kinetic energy 29. The same force F is applied to each cart and they move equal distances d. Which cart will move faster at the end of distance d? A. cart I B. cart II C. cart III D. all three will move with the same velocity 30. A box of mass M begins at rest with point 1 at a height of 6R, where 2R is the radius of the circular part of the track. The box slides down the frictionless track and around the loop. What is the ratio between the normal force on the box at point 2 to the box’s weight?

A. B. C. D.

1 2 3 4

31. A ball of mass m is fastened to a string. The ball swings in a vertical circle of radius r with the other end of the string held fixed. Assuming that the ball moves at a constant speed, the difference between the string’s tension at the bottom of the circle and at the top of the circle is: A. mg B. 2mg C. 3mg D. 6mg

Directions: For each of the following, two of the suggested answers will be correct. Select the best two choices to earn credit. No partial credit will be earned if only one correct choice is selected. 32. The following are characteristics of energy: A. The amount of energy in an isolated system can be changed by an external force performing work on it. B. Thermal energy can never be changed into mechanical energy. 32 https://sites.google.com/site/mrhphysics/

C. Mechanical energy can be changed into thermal energy. D. Energy is only present in an object when it is moving.

33. When analyzing work and energy problems, it is important to define the system and the environment in such a way as to make the problems easier to solve. The following are characteristics of a system: A. Forces internal to the system can change its total mechanical energy. B. A system can be one object or a collection of objects. C. The system is external to the system boundary. D. Objects cannot pass through the system boundary. 34. A constant force, F, is applied to a block sitting on a bench. There could be other forces acting on the block at the same time. In which of the following cases is no work done on the block by F? A. The force is applied to the block, and it moves in the same direction as the force. B. The force is applied to the block, and the block moves in the opposite direction of the force. C. The block does not move. D. The force is applied perpendicular to the block’s motion. 35. A student swings a ball around his head in a perfectly circular, horizontal orbit with a constant Tension. Which of the following is true concerning the ball’s orbit? A. No work is done by the Tension force since the force is perpendicular to the ball’s motion. B. The ball maintains a constant speed. C. The ball maintains a constant velocity. D. The ball’s speed increases due to the work done by the Tension force.

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MULTIPLE CHOICE ANSWER KEY: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

D B C B D C D A A D A B D B D C A C B B D D B C A C D D A A B A, C B, D C, D A, B

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AP Physics Released Free Response – Work Power Energy

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C2007M3.

The apparatus above is used to study conservation of mechanical energy. A spring of force constant 40 N/m is held horizontal over a horizontal air track, with one end attached to the air track. A light string is attached to the other end of the spring and connects it to a glider of mass m. The glider is pulled to stretch the spring an amount x from equilibrium and then released. Before reaching the photogate, the glider attains its maximum speed and the string becomes slack. The photogate measures the time t that it takes the small block on top of the glider to pass through. Information about the distance x and the speed v of the glider as it passes through the photogate are given below.

(a) Assuming no energy is lost, write the equation for conservation of mechanical energy that would apply to this situation. (b) On the grid below, plot v2 versus x2 . Label the axes, including units and scale.

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(c) i. Draw a best-fit straight line through the data. ii. Use the best-fit line to obtain the mass m of the glider. (d) The track is now tilted at an angle θ as shown below. When the spring is unstretched, the center of the glider is a height h above the photogate. The experiment is repeated with a variety of values of x.

Assuming no energy is lost, write the new equation for conservation of mechanical energy that would apply to this situation starting from position A and ending at position B

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C2008M3

In an experiment to determine the spring constant of an elastic cord of length 0.60 m, a student hangs the cord from a rod as represented above and then attaches a variety of weights to the cord. For each weight, the student allows the weight to hang in equilibrium and then measures the entire length of the cord. The data are recorded in the table below:

(a) Use the data to plot a graph of weight versus length on the axes below. Sketch a best-fit straight line through the data.

(b) Use the best-fit line you sketched in part (a) to determine an experimental value for the spring constant k of the cord.

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The student now attaches an object of unknown mass m to the cord and holds the object adjacent to the point at which the top of the cord is tied to the rod, as shown. When the object is released from rest, it falls 1.5 m before stopping and turning around. Assume that air resistance is negligible. (c) Calculate the value of the unknown mass m of the object. (d) i. Determine the magnitude of the force in the cord when the when the mass reaches the equilibrium position. ii. Determine the amount the cord has stretched when the mass reaches the equilibrium position.

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2008B2

Block A of mass 2.0 kg and block B of mass 8.0 kg are connected as shown above by a spring of spring constant 80 N/m and negligible mass. The system is being pulled to the right across a horizontal frictionless surface by a horizontal force of 4.0 N, as shown, with both blocks experiencing equal constant acceleration. (a) Calculate the force that the spring exerts on the 2.0 kg block. (b) Calculate the extension of the spring. The system is now pulled to the left, as shown below, with both blocks again experiencing equal constant acceleration.

(c) Is the magnitude of the acceleration greater than, less than, or the same as before? ____ Greater ____ Less ____ The same Justify your answer. (d) Is the amount the spring has stretched greater than, less than, or the same as before? ____ Greater ____ Less ____ The same Justify your answer. (e) In a new situation, the blocks and spring are moving together at a constant speed of 0.50 m s to the left. Block A then hits and sticks to a wall. Calculate the maximum compression of the spring.

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2009B1

In an experiment, students are to calculate the spring constant k of a vertical spring in a small jumping toy that initially rests on a table. When the spring in the toy is compressed a distance x from its uncompressed length L0 and the toy is released, the top of the toy rises to a maximum height h above the point of maximum compression. The students repeat the experiment several times, measuring h with objects of various masses taped to the top of the toy so that the combined mass of the toy and added objects is m. The bottom of the toy and the spring each have negligible mass compared to the top of the toy and the objects taped to it. (a) Derive an expression for the height h in terms of m, x, k, and fundamental constants. With the spring compressed a distance x = 0.020 m in each trial, the students obtained the following data for different values of m.

(b) i. What quantities should be graphed so that the slope of a best-fit straight line through the data points can be used to calculate the spring constant k ? ii. Fill in one or both of the blank columns in the table with calculated values of your quantities, including units. (c) On the axes below, plot your data and draw a best-fit straight line. Label the axes and indicate the scale.

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(d) Using your best-fit line, calculate the numerical value of the spring constant. (e) Describe a procedure for measuring the height h in the experiment, given that the toy is only momentarily at that maximum height.

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1974B1. A pendulum consisting of a small heavy ball of mass m at the end of a string of length L is released from a horizontal position. When the ball is at point P, the string forms an angle of θ with the horizontal as shown.

θ

(a) In the space below, draw a force diagram showing all of the forces acting on the ball at P. Identify each force clearly.

(b) Determine the speed of the ball at P. (c) Determine the tension in the string when the ball is at P.

1974B7. A ski lift carries skiers along a 600 meter slope inclined at 30°. To lift a single rider, it is necessary to move 70 kg of mass to the top of the lift. Under maximum load conditions, six riders per minute arrive at the top. If 60 percent of the energy supplied by the motor goes to overcoming friction, what average power must the motor supply?

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1975B1. A 2-kilogram block is released from rest at the top of a curved incline in the shape of a quarter of a circle of radius R. The block then slides onto a horizontal plane where it finally comes to rest 8 meters from the beginning of the plane. The curved incline is frictionless, but there is an 8-newton force of friction on the block while it slides horizontally. Assume g = 10 meters per second2. a. Determine the magnitude of the acceleration of the block while it slides along the horizontal plane. b. How much time elapses while the block is sliding horizontally? c. Calculate the radius of the incline in meters.

1975B7. A pendulum consists of a small object of mass m fastened to the end of an inextensible cord of length L. Initially, the pendulum is drawn aside through an angle of 60° with the vertical and held by a horizontal string as shown in the diagram above. This string is burned so that the pendulum is released to swing to and fro. a. In the space below draw a force diagram identifying all of the forces acting on the object while it is held by the string.

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b. Determine the tension in the cord before the string is burned. c. Show that the cord, strong enough to support the object before the string is burned, is also strong enough to support the object as it passes through the bottom of its swing.

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1977 B1. A block of mass 4 kilograms, which has an initial speed of 6 meters per second at time t = 0, slides on a horizontal surface. a. Calculate the work W that must be done on the block to bring it to rest.

If a constant friction force of magnitude 8 newtons is exerted on the block by the surface, determine the following: b. The speed v of the block as a function of the time t. c. The distance x that the block slides as it comes to rest

1978B1. A 0.5 kilogram object rotates freely in a vertical circle at the end of a string of length 2 meters as shown above. As the object passes through point P at the top of the circular path, the tension in the string is 20 newtons. Assume g = 10 meters per second squared.

(a) On the following diagram of the object, draw and clearly label all significant forces on the object when it is at the point P.

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(b) Calculate the speed of the object at point P. (c) Calculate the increase in kinetic energy of the object as it moves from point P to point Q. (d) Calculate the tension in the string as the object passes through point Q.

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1979B1. From the top of a cliff 80 meters high, a ball of mass 0.4 kilogram is launched horizontally with a velocity of 30 meters per second at time t = 0 as shown above. The potential energy of the ball is zero at the bottom of the cliff. Use g = 10 meters per second squared.

a. Calculate the potential, kinetic, and total energies of the ball at time t = 0. b.

On the axes below, sketch and label graphs of the potential, kinetic, and total energies of the ball as functions of the distance fallen from the top of the cliff

Top of cliff

Distance Fallen (m)

Bottom of cliff

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c.

On the axes below sketch and label the kinetic and potential energies of the ball as functions of time until the ball hits

1981B1. A 10-kilogram block is pushed along a rough horizontal surface by a constant horizontal force F as shown above. At time t = 0, the velocity v of the block is 6.0 meters per second in the same direction as the force. The coefficient of sliding friction is 0.2. Assume g = 10 meters per second squared. a. Calculate the force F necessary to keep the velocity constant.

The force is now changed to a larger constant value F'. The block accelerates so that its kinetic energy increases by 60 joules while it slides a distance of 4.0 meters. b.

Calculate the force F'.

c.

Calculate the acceleration of the block.

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1981B2. A massless spring is between a 1-kilogram mass and a 3-kilogram mass as shown above, but is not attached to either mass. Both masses are on a horizontal frictionless table. In an experiment, the 1-kilogram mass is held in place and the spring is compressed by pushing on the 3-kilogram mass. The 3-kilogram mass is then released and moves off with a speed of 10 meters per second. Determine the minimum work needed to compress the spring in this experiment.

1982B3. A child of mass M holds onto a rope and steps off a platform. Assume that the initial speed of the child is zero. The rope has length R and negligible mass. The initial angle of the rope with the vertical is o, as shown in the drawing above. a. Using the principle of conservation of energy, develop an expression for the speed of the child at the lowest point in the swing in terms of g, R, and cos o b. The tension in the rope at the lowest point is 1.5 times the weight of the child. Determine the value of cos

o.

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1985B2. Two 10-kilogram boxes are connected by a massless string that passes over a massless frictionless pulley as shown above. The boxes remain at rest, with the one on the right hanging vertically and the one on the left 2.0 meters from the bottom of an inclined plane that makes an angle of 60° with the horizontal. The coefficients of kinetic friction and static friction between the left-hand box and the plane are 0.15 and 0.30, respectively. You may use g = 10 m/s2, sin 60° = 0.87, and cos 60° = 0.50. a. What is the tension T in the string? b. On the diagram below, draw and label all the forces acting on the box that is on the plane.

c. Determine the magnitude of the frictional force acting on the box on the plane.

The string is then cut and the left-hand box slides down the inclined plane. d. Determine the amount of mechanical energy that is converted into thermal energy during the slide to the bottom. e. Determine the kinetic energy of the left-hand box when it reaches the bottom of the plane.

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1986B2. One end of a spring is attached to a solid wall while the other end just reaches to the edge of a horizontal, frictionless tabletop, which is a distance h above the floor. A block of mass M is placed against the end of the spring and pushed toward the wall until the spring has been compressed a distance X, as shown above. The block is released, follows the trajectory shown, and strikes the floor a horizontal distance D from the edge of the table. Air resistance is negligible.

Determine expressions for the following quantities in terms of M, X, D, h, and g. Note that these symbols do not include the spring constant.

a. The time elapsed from the instant the block leaves the table to the instant it strikes the floor b. The horizontal component of the velocity of the block just before it hits the floor c. The work done on the block by the spring d. The spring constant

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1991B1. A 5.0-kilogram monkey hangs initially at rest from two vines, A and B. as shown above. Each of the vines has length 10 meters and negligible mass. a. On the figure below, draw and label all of the forces acting on the monkey. (Do not resolve the forces into components, but do indicate their directions.)

b. Determine the tension in vine B while the monkey is at rest.

The monkey releases vine A and swings on vine B. Neglect air resistance. c. Determine the speed of the monkey as it passes through the lowest point of its first swing. d. Determine the tension in vine B as the monkey passes through the lowest point of its first swing.

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1992B1. A 0.10-kilogram solid rubber ball is attached to the end of an 0.80 meter length of light thread. The ball is swung in a vertical circle, as shown in the diagram above. Point P, the lowest point of the circle, is 0.20 meter above the floor. The speed of the ball at the top of the circle is 6.0 meters per second, and the total energy of the ball is kept constant. a.

Determine the total energy of the ball, using the floor as the zero point for gravitational potential energy.

b.

Determine the speed of the ball at point P, the lowest point of the circle.

c.

Determine the tension in the thread at i. the top of the circle; ii. the bottom of the circle.

The ball only reaches the top of the circle once before the thread breaks when the ball is at the lowest point of the circle. d.

Determine the horizontal distance that the ball travels before hitting the floor.

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1996B2 (15 points) A spring that can be assumed to be ideal hangs from a stand, as shown above. a.

You wish to determine experimentally the spring constant k of the spring. i.

What additional, commonly available equipment would you need?

ii.

What measurements would you make?

iii. How would k be determined from these measurements? b.

Suppose that the spring is now used in a spring scale that is limited to a maximum value of 25 N, but you would like to weigh an object of mass M that weighs more than 25 N. You must use commonly available equipment and the spring scale to determine the weight of the object without breaking the scale. i.

Draw a clear diagram that shows one way that the equipment you choose could be used with the spring scale to determine the weight of the object,

ii.

Explain how you would make the determination.

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1997B1. A 0.20 kg object moves along a straight line. The net force acting on the object varies with the object's displacement as shown in the graph above. The object starts from rest at displacement x = 0 and time t = 0 and is displaced a distance of 20 m. Determine each of the following. a. The acceleration of the particle when its displacement x is 6 m. b. The time taken for the object to be displaced the first 12 m. c. The amount of work done by the net force in displacing the object the first 12 m. d. The speed of the object at displacement x = 12 m. e. The final speed of the object at displacement x = 20 m.

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2002B2. A 3.0 kg object subject to a restoring force F is undergoing simple harmonic motion with a small amplitude. The potential energy U of the object as a function of distance x from its equilibrium position is shown above. This particular object has a total energy E: of 0.4 J. (a) What is the object's potential energy when its displacement is +4 cm from its equilibrium position? (b) What is the farthest the object moves along the x axis in the positive direction? Explain your reasoning. (c) Determine the object's kinetic energy when its displacement is – 7 cm. (d) What is the object's speed at x = 0 ?

(e) Suppose the object undergoes this motion because it is the bob of a simple pendulum as shown above. If the object breaks loose from the string at the instant the pendulum reaches its lowest point and hits the ground at point P shown, what is the horizontal distance d that it travels?

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2004B1. A roller coaster ride at an amusement park lifts a car of mass 700 kg to point A at a height of 90 m above the lowest point on the track, as shown above. The car starts from rest at point A, rolls with negligible friction down the incline and follows the track around a loop of radius 20 m. Point B, the highest point on the loop, is at a height of 50 m above the lowest point on the track.

(a) i. Indicate on the figure the point P at which the maximum speed of the car is attained. ii. Calculate the value vmsx of this maximum speed.

(b) Calculate the speed vB of the car at point B.

(c) i. On the figure of the car below, draw and label vectors to represent the forces acting on the car when it is upside down at point B.

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ii. Calculate the magnitude of all the forces identified in (c)

(d) Now suppose that friction is not negligible. How could the loop be modified to maintain the same speed at the top of the loop as found in (b)? Justify your answer.

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B2004B1. A designer is working on a new roller coaster, and she begins by making a scale model. On this model, a car of total mass 0.50 kg moves with negligible friction along the track shown in the figure above. The car is given an initial speed vo = 1.5 m/s at the top of the first hill of height 2.0 m. Point A is located at a height of 1.9 m at the top of the second hill, the upper part of which is a circular arc of radius 0.95 m.

(a) Calculate the speed of the car at point A.

(b) On the figure of the car below, draw and label vectors to represent the forces on the car at point A.

(c) Calculate the magnitude of the force of the track on the car at point A.

(d) In order to stop the car at point A, some friction must be introduced. Calculate the work that must be done by the friction force in order to stop the car at point A.

(e) Explain how to modify the track design to cause the car to lose contact with the track at point A before descending down the track. Justify your answer.

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B2005B2

A simple pendulum consists of a bob of mass 0.085 kg attached to a string of length 1.5 m. The pendulum is raised to point Q, which is 0.08 m above its lowest position, and released so that it oscillates with small amplitude θ between the points P and Q as shown below.

(a) On the figures below, draw free-body diagrams showing and labeling the forces acting on the bob in each of the situations described.

i. When it is at point P

ii. When it is in motion at its lowest position

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(b) Calculate the speed v of the bob at its lowest position. (c) Calculate the tension in the string when the bob is passing through its lowest position.

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2005B2.

2. (10 points) A simple pendulum consists of a bob of mass 1.8 kg attached to a string of length 2.3 m. The pendulum is held at an angle of 30° from the vertical by a light horizontal string attached to a wall, as shown above. (a) On the figure below, draw a free-body diagram showing and labeling the forces on the bob in the position shown above.

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(b) Calculate the tension in the horizontal string. (c) The horizontal string is now cut close to the bob, and the pendulum swings down. Calculate the speed of the bob at its lowest position.

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B2006B2

A small block of mass M is released from rest at the top of the curved frictionless ramp shown above. The block slides down the ramp and is moving with a speed 3.5vo when it collides with a larger block of mass 1.5M at rest at the bottom of the incline. The larger block moves to the right at a speed 2vo immediately after the collision.

Express your answers to the following questions in terms of the given quantities and fundamental constants. (a) Determine the height h of the ramp from which the small block was released. (b) The larger block slides a distance D before coming to rest. Determine the value of the coefficient of kinetic friction µ between the larger block and the surface on which it slides.

2006B1

An ideal spring of unstretched length 0.20 m is placed horizontally on a frictionless table as shown above. One end of the spring is fixed and the other end is attached to a block of mass M = 8.0 kg. The 8.0 kg block is also attached

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to a massless string that passes over a small frictionless pulley. A block of mass m = 4.0 kg hangs from the other end of the string. When this spring-and-blocks system is in equilibrium, the length of the spring is 0.25 m and the 4.0 kg block is 0.70 m above the floor. (a) On the figures below, draw free-body diagrams showing and labeling the forces on each block when the system is in equilibrium. M = 8.0 kg

m = 4.0 kg

(b) Calculate the tension in the string. (c) Calculate the force constant of the spring.

The string is now cut at point P. (d) Calculate the time taken by the 4.0 kg block to hit the floor. (e) Calculate the maximum speed attained by the 8.0 kg block as it oscillates back and forth

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B2008B2

A 4700 kg truck carrying a 900 kg crate is traveling at 25 m/s to the right along a straight, level highway, as shown above. The truck driver then applies the brakes, and as it slows down, the truck travels 55 m in the next 3.0 s. The crate does not slide on the back of the truck.

(a) Calculate the magnitude of the acceleration of the truck, assuming it is constant. (b) On the diagram below, draw and label all the forces acting on the crate during braking.

(c) i. Calculate the minimum coefficient of friction between the crate and truck that prevents the crate from sliding.

ii. Indicate whether this friction is static or kinetic. ____ Static ____Kinetic

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Now assume the bed of the truck is frictionless, but there is a spring of spring constant 9200 N/m attaching the crate to the truck, as shown below. The truck is initially at rest.

(d) If the truck and crate have the same acceleration, calculate the extension of the spring as the truck accelerates from rest to 25 m s in 10 s. (e) At some later time, the truck is moving at a constant speed of 25 m/s and the crate is in equilibrium. Indicate whether the extension of the spring is greater than, less than, or the same as in part (d) when the truck was accelerating. ___ Greater ___ Less ___ The same Explain your reasoning.

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C1973M2. A 30-gram bullet is fired with a speed of 500 meters per second into a wall. a. b.

If the deceleration of the bullet is constant and it penetrates 12 centimeters into the wall, calculate the force on the bullet while it is stopping. If the deceleration of the bullet is constant and it penetrates 12 centimeters into the wall, how much time is required for the bullet to stop?

C1982M1. A 20 kg mass, released from rest, slides 6 meters down a frictionless plane inclined at an angle of 30º with the horizontal and strikes a spring of unknown spring constant as shown in the diagram above. Assume that the spring is ideal, that the mass of the spring is negligible, and that mechanical energy is conserved. a. Determine the speed of the block just before it hits the spring. b. Determine the spring constant given that the distance the spring compresses along the incline is 3m when the block comes to rest. c. Is the speed of the block a maximum at the instant the block strikes the spring? Justify your answer.

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C1983M3. A particle of mass m slides down a fixed, frictionless sphere of radius R. starting from rest at the top. a. In terms of m, g, R. and , determine each of the following for the particle while it is sliding on the sphere. i. The kinetic energy of the particle ii. The centripetal acceleration of the mass

C1985M2. An apparatus to determine coefficients of friction is shown above. At the angle shown with the horizontal, the block of mass m just starts to slide. The box then continues to slide a distance d at which point it hits the spring of force constant k, and compresses the spring a distance x before coming to rest. In terms of the given quantities and fundamental constants, derive an expression for each of the following. a. s the coefficient of static friction. b.

c.

E, the loss in total mechanical energy of the block-spring system from the start of the block down the incline to the moment at which it comes to rest on the compressed spring. k,

the coefficient of kinetic friction.

C1987M1. An adult exerts a horizontal force on a swing that is suspended by a rope of length L, holding it at an angle with the vertical. The child in the swing has a weight W and dimensions that are negligible compared to L. The weights of the rope and of the seat are negligible. In terms of W and , determine

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a) The tension in the rope b) The horizontal force exerted by the adult. c) The adult releases the swing from rest. In terms of W and passes through its lowest point

determine the tension in the rope as the swing

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C1988M2.

A 5-kilogram object initially slides with speed vo in a hollow frictionless pipe. The end of the pipe contains a spring as shown. The object makes contact with the spring at point A and moves 0.1 meter before coming to rest at point B. The graph shows the magnitude of the force exerted on the object by the spring as a function of the objects distance from point A. a. Calculate the spring constant for the spring. b. Calculate the decrease in kinetic energy of the object as it moves from point A to point B. c. Calculate the initial speed vo of the object

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C1989M1. A 0.1 kilogram block is released from rest at point A as shown above, a vertical distance h above the ground. It slides down an inclined track, around a circular loop of radius 0.5 meter, then up another incline that forms an angle of 30° with the horizontal. The block slides off the track with a speed of 4 m/s at point C, which is a height of 0.5 meter above the ground. Assume the entire track to be frictionless and air resistance to be negligible. a. Determine the height h . b. On the figure below, draw and label all the forces acting on the block when it is at point B. which is 0.5 meter above the ground.

c. d. e. f.

Determine the magnitude of the velocity of the block when it is at point B. Determine the magnitude of the force exerted by the track on the block when it is at point B. Determine the maximum height above the ground attained by the block after it leaves the track. Another track that has the same configuration, but is NOT frictionless, is used. With this track it is found that if the block is to reach point C with a speed of 4 m/s, the height h must be 2 meters. Determine the work done by the frictional force.

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C1989M3. A 2-kilogram block is dropped from a height of 0.45 meter above an uncompressed spring, as shown above. The spring has an elastic constant of 200 newtons per meter and negligible mass. The block strikes the end of the spring and sticks to it. a. Determine the speed of the block at the instant it hits the end of the spring. b. Determine the force in the spring when the block reaches the equilibrium position c. Determine the distance that the spring is compressed at the equilibrium position d. Determine the speed of the block at the equilibrium position e.

Is the speed of the block a maximum at the equilibrium position, explain.

C1990M2. A block of mass m slides up the incline shown above with an initial speed v o in the position shown.

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a. If the incline is frictionless, determine the maximum height H to which the block will rise, in terms of the given quantities and appropriate constants. b. If the incline is rough with coefficient of sliding friction , the box slides a distance d = h2 / sin θ along the length of the ramp as it reaches a new maximum height h2. Determine the new maximum height h2 in terms of the given quantities.

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3m

C1991M1. A small block of mass 3m moving at speed v0 / 3 enters the bottom of the circular, vertical loop-the-loop shown above, which has a radius r. The surface contact between the block and the loop is frictionless. Determine each of the following in terms of m, vo , r, and g. a. The kinetic energy of the block and bullet when they reach point P on the loop b. The speed vmin of the block at the top of the loop to remain in contact with track at all times c The new required entry speed vo΄ at the bottom of the loop such that the conditions in part b apply.

C1993M1. A massless spring with force constant k = 400 newtons per meter is fastened at its left end to a vertical wall, as shown in Figure 1. Initially, block C (mass mc = 4.0 kilograms) and block D (mass mD = 2.0 kilograms) rest on a rough horizontal surface with block C in contact with the spring (but not compressing it) and with block D in contact with block C. Block C is then moved to the left, compressing the spring a distance of 0.50 meter, and held in place while block D remains at rest as shown in Figure 11. (Use g = 10 m/s2.) a. Determine the elastic energy stored in the compressed spring. Block C is then released and accelerates to the right, toward block D. The surface is rough and the coefficient of friction between each block and the surface is = 0.4. The two blocks collide instantaneously, stick together, and move to the right at 3 m/s. Remember that the spring is not attached to block C. Determine each of the following. b. The speed vc of block C just before it collides with block D c. The horizontal distance the combined blocks move after leaving the spring before coming to rest

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C2002M2. The cart shown above has a mass 2m. The cart is released from rest and slides from the top of an inclined frictionless plane of height h. Express all algebraic answers in terms of the given quantities and fundamental constants. a. Determine the speed of the cart when it reaches the bottom of the incline. b. After sliding down the incline and across the frictionless horizontal surface, the cart collides with a bumper of negligible mass attached to an ideal spring, which has a spring constant k. Determine the distance x m the spring is compressed before the cart and bumper come to rest.

C2004M1

A rope of length L is attached to a support at point C. A person of mass m1 sits on a ledge at position A holding the other end of the rope so that it is horizontal and taut, as shown. The person then drops off the ledge and swings down on the rope toward position B on a lower ledge where an object of mass m2 is at rest. At position B the person grabs hold of the object and simultaneously lets go of the rope. The person and object then land together in the lake at point D, which is a vertical distance L below position B. Air resistance and the mass of the rope are negligible. Derive expressions for each of the following in terms of m1, m2, L, and g.

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(a) The speed of the person just before the collision with the object (b) The tension in the rope just before the collision with the object (c) After the person hits and grabs the rock, the speed of the combined masses is determined to be v’. In terms of v’ and the given quantities, determine the total horizontal displacement x of the person from position A until the person and object land in the water at point D.

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Supplemental

One end of a spring of spring constant k is attached to a wall, and the other end is attached to a block of mass M, as shown above. The block is pulled to the right, stretching the spring from its equilibrium position, and is then held in place by a taut cord, the other end of which is attached to the opposite wall. The spring and the cord have negligible mass, and the tension in the cord is FT . Friction between the block and the surface is negligible. Express all algebraic answers in terms of M, k, FT , and fundamental constants.

(a) On the dot below that represents the block, draw and label a free-body diagram for the block.

(b) Calculate the distance that the spring has been stretched from its equilibrium position.

The cord suddenly breaks so that the block initially moves to the left and then oscillates back and forth. (c) Calculate the speed of the block when it has moved half the distance from its release point to its equilibrium position. (d) Suppose instead that friction is not negligible and that the coefficient of kinetic friction between the block and the surface is µk . After the cord breaks, the block again initially moves to the left. Calculate the initial acceleration of the block just after the cord breaks.

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