Learning Objectives – AP Physics C Mechanics (Semester 1) Unit 1: Kinematics (1D, 2D, & Rotational) [~20% of Exam]
1. Motion in OneDimension (1D) a. I understand the general relationships among position, velocity, and acceleration ( x, v, a ) for the motion of a particle along a straight line, so that: i. Given a GRAPH of xt , vt , or at , I can recognize in what time intervals the other two are positive, negative, or zero and can identify or sketch a graph of each as a function of time. ii. Given an EXPRESSION for one of the kinematic quantities, position, velocity or acceleration, as a function of time, I can use algebra (or calculus ) to determine the other two as a function of time, and find when these quantities are zero or achieve their maximum and minimum values. b. I understand the special case of motion with constant acceleration , so I can: i. Write down EXPRESSIONS for velocity and position as functions of time, and identify or sketch graphs of these quantities. ii. Use the 4 kinematic equations to solve problems involving motion in a straight line with a constant acceleration. 2. Motion in TwoDimensions (2D); including “Projectiles” a. I am able to add, subtract, & resolve vectors (precalculus), so I can: i. Determine the horizontal & vertical components of an angled vector. ii. Determine the net displacement of a particle or the location of a particle relative to another. iii. Determine the change in velocity of a particle or the velocity of one particle relative to another. b. Given functions x ( t ) and y ( t ) that describe 2D motion, I can determine the horiz. & vert. components, and magnitude and direction of the particle's velocity and acceleration as functions of time. c. I understand the motion of projectiles in a uniform gravitational field, so I can: i. Recognize when an object is in freefall and experiences an acceleration due to gravity of 9.8 2 m/s , (which can always be rounded to 10) ii. Write down expressions for the horizontal and vertical components of velocity and position as functions of time, and sketch or identify graphs of these components. iii. Use these expressions in analyzing the motion of a projectile that is projected with an arbitrary initial velocity. 3. Rotational Kinematics a. I understand the analogy between translational (or linear) and rotational (or angular) kinematics so I can: i. Write and apply the 4 kinematic equations, but using angular acceleration, angular velocity, and angular displacement (𝛼, 𝜔, 𝛳) of an object that rotates about a fixed axis with constant angular acceleration. ii. Determine tangential and angular variables using: x = rθ, v = rω, a = rα for spinning objects. b. I am able to use the righthand rule to determine the vector direction of rotational variables (instead of saying “clockwise” and “counterclockwise”).
c. I understand ROLLING motion, so I can: i. Write down, justify, and apply the relation between linear and angular velocity, or between linear and angular acceleration, for an object of circular crosssection that rolls without slipping along a fixed plane, and determine the velocity and acceleration of an arbitrary point on such an object. ii. Apply the equations of translational and rotational motion simultaneously in analyzing rolling with slipping.
Unit 2: Forces & Torques [~20% of Exam]
1. (Newton’s 1st Law) “Static Equilibrium.” Situations in which an object IS NOT accelerating must have balanced forces and torques. a. I understand that all horizontal forces must add to zero if there is no horizontal acceleration. b. I understand that all vertical forces must add to zero if there is no vertical acceleration. c. I understand that all torques must add to zero if there is no angular acceleration. i. Apply these conditions in analyzing the equilibrium of a rigid object under the combined influence of a number of forces applied at different locations on the object. 2. (Newton’s 2nd Law) “Dynamics.” Situations in which an object IS accelerating must have unbalanced forces and torques. a. I understand how Newton's Second Law, F , applies to an object subject to forces such as net = ma gravity, the pull of strings, or contact forces, so I can: i. Draw a welllabeled, freebody diagram showing all forces that act on the object. ii. Write down the vector equation that results from applying Newton's Second Law to the object, and take components of this equation along appropriate axes. This is called “sum of forces.” b. I am able to analyze situations in which an object moves with specified acceleration under the influence of one or more forces so I can determine the magnitude and direction of the net force, or of one of the forces that makes up the net force, along either the vertical or horizontal directions. c. I can determine the angular acceleration with which a rigid object is accelerated about a fixed axis when subjected to a specified external torque. d. I understand the significance of the coefficient of friction, so I can: i. Write down the relationship between the normal and frictional forces on a surface. ii. Analyze situations in which an object moves along a rough inclined plane or horizontal surface. iii. Analyze under what circumstances an object will start to slip, or to calculate the magnitude of the force of static friction. 3. (Newton’s 3rd Law) “Systems.” Situations in which there are multiple objects, there are always equal and opposite forces between any 2 objects. a. I can identify the force pairs and the objects on which they act, and state the magnitude and direction of each force. b. I recognize that the equal but opposite forces of contact act between two objects that accelerate together along a horizontal or vertical line, or between two surfaces that slide across one another.
c. I know that tension is the same on both ends of light string with no objects in between. That is to say, I know that the tension is constant in a light string that passes over a massless pulley and should be able to use this fact to analyze the motion of a system of two objects joined by a string. d. I am able to solve problems in which application of Newton's laws leads to two or three simultaneous linear equations involving unknown forces or accelerations. e. I am able to analyze problems involving strings and massless or massive pulleys. 4. Torque a. I can calculate the magnitude and direction of the torque associated with a given force. b. I can calculate the torque on a rigid object due to gravity. 5. Rotational Inertia a. I have QUALITATIVE understanding of rotational inertia, so I can: i. Determine by inspection which of a set of symmetrical objects of equal mass has the greatest rotational inertia. ii. Determine by what factor an object's rotational inertia changes if all its dimensions are increased by the same factor. b. I have QUANTITATIVE understanding of rotational inertia, so I can calculate the rotational inertia of: i. A collection of point masses lying in a plane about an axis perpendicular to the plane. ii. A thin rod of uniform density, about an arbitrary axis perpendicular to the rod. iii. A cylinder about its axis. iv. A sphere about its axis. v. For any rotating object whose axis of rotation is not at its center of mass, I able to state and apply the parallelaxis theorem.
Unit 3: Circular Motion [~3% of Exam] 1. I understand the uniform circular motion of a particle, so I can: a. Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal acceleration, or mathematically written as: a = v 2/r . b. Describe the direction of the particle's velocity and acceleration at any instant during the motion. c. Determine the components of the velocity and acceleration vectors at any instant, and sketch or identify graphs of these quantities. i. Motion in a horizontal circle (e.g., mass on a rotating merrygoround, or car rounding a banked curve). ii. Motion in a vertical circle (e.g., mass swinging on the end of a string, cart rolling down a curved track, rider on a Ferris wheel). iii. Determine the radial and tangential acceleration of a point on a rigid object.
Unit 4: Impulse & Center of Mass [~3% of Exam]
1. Center of mass. a. I understand the technique for finding center of mass, so I can: i. Visually identify by inspection the center of mass of a symmetrical object. ii. Calculate the center of mass of a system consisting of two such objects. iii. Use integration to find the center of mass of a thin rod of nonuniform density. b. I am able to understand and apply the relation between centerofmass’ velocity and linear momentum, and between centerofmass acceleration and net external force for a system of particles. 2. I can apply the Impulse Equation: Fnet ⠂ , so I can: Δt = mv mvo a. Mathematically use the impulse equation to solve for unknown variables i. Given an expression for net force as a function of time , use calculus (integrate) to determine impulse. ii. Use calculus to calculate the change in momentum of an object given a function F (t) for the net force acting on the object as a function of time. iii. Conceptually use the impulse equation to explain the relationship between the force of impact and time of impact for different situations (e.g., “cradling” an object as you catch it, airbags saving lives, etc.) 3. I can analyze GRAPHS of net force as a function of time , so that I can: a. Determine the AREA under a force versus time graph to determine the impulse delivered to an object. b. Use the graph with the impulse equation to determine numerical information including initial and final velocities, average/maximum impact force, and time of impact.
Unit 5: Conservation of Momentum (linear & angular) [~10% of Exam] 1. Linear Collisions (along “one dimension” or along a straight line). a. Total momentum of all objects is ALWAYS the same before and after any collision. This law is called “conservation of total momentum,” and is an extension from Newton's 3rd law i. Use this law to solve for speeds before or after collisions between two objects ii. This law can be applied to each direction (horizontal and vertical, separately; and using righttriangle trigonometry iii. I can state and apply the relations between linear momentum and centerofmass motion for a system of particles. b. Recognize that this law can only be applied to situations in which both objects are FREE TO MOVE ALONG A LINE (no forces other than the one between them) 2. Collisions involving rotation (about a fixed axis): a. Definition of angular momentum for differentshaped objects, and for a point mass moving in a line b. Total anular. momentum of all objects is ALWAYS the same before and after any collision. This law is called “conservation of total momentum,” and is an extension from Newton's 3rd law i. Use this law to solve for rotational velocities, ω, before or after collisions c. Recognize that this law can only be applied to situations in which both objects are FREE TO ROTATE (no torques other than the one related to the collision)
Unit 6: Work & Conservation of Energy [~25% of Exam] 1. Work is change in energy, which occurs when any force acts along a distance. a. I understand the definition of work, including when it is positive, negative, or zero. i. (+)W → lost; ()W → gained b. Work, W (or change in energy) for a certain form of energy is i. Algebraically defined as: W = F ∙ x or W = Fx cos ϴ ii. Graphically defined as AREA under a Fx graph iii. Calculus—given an expression for F(x), integrate to calculate work done by that force . c. “Power” is time rate of change in energy. i. Calculate the power required to maintain the motion of an object with constant acceleration (e.g., to move an object along a level surface, to raise an object at a constant rate, or to overcome friction for an object that is moving at a constant speed). ii. Calculate the work performed by a force that supplies constant power, or the average power supplied by a force that performs a specified amount of work. d. “Workenergy principle,” i. Work in relation to kinetic energy 1. Mathematically defined as: WFnet = Fnet ∙x … Fnet (x) = ∆K 2. Calculus—given an expression for Fnet (x), integrate to calculate ∆K ii. Work in relation to potential energy 1. Mathematically defined as: W Fg = ∆U g or W Felastic = ∆U el 2. Graphically, given a Ux graph, determine the force...force is ()slope [or negative slope] 3. Calculus a. given an expression for F(x), integrate to calculate () ∆U b. given an expression for U(x), differentiate to calculate the ()force 2. Identify the forms of energy utilized in any particular situation : a. “Potential” energies are all dependent upon position of the object. b. “Elastic potential energy” is dependent upon a spring’s compression or stretching distance , and is 2 mathematically defined as U el = ½kx c. “Gravitational potential energy” is dependent upon height/altitude above its lowest point , and is mathematically defined as U g = mgh i. I am able to define center of gravity and use this concept to express the gravitational potential energy of a rigid object in terms of the position of its center of mass. d. “Kinetic energy” is dependent upon a velocity and rotational velocity , and is mathematically 2 2 defined as K , and K 𝜔 (rotationally). lin = ½mv ,(linearly) rot = ½I 3. Conservation of Energy a. Total energy of a system at any moment is always the same amount. Total energy (“before”) must EQUAL the total energy (“after”) b. Must include ALL connected objects; (cannot look at only one object if that object is part of a system)
c. Mathematically defined as: ±W + U + K + U = U + K + U (W is work done by an F g el g el F “external” force) i. Implement “work done by an external force” as a term in expressions of conservation of energy ii. I am able to apply conservation of energy in analyzing the motion of systems of connected objects, such as an Atwood's machine (pulley system). iii. I am able to recognize and solve problems that call for application both of conservation of energy and forces/torques. (e.g., pulley systems, objects rolling down inclines) 4. Energy lost during collisions a. E (for collisions) is unpredictable; and therefore, not defined by an equation diss b. Conservation of energy before/after collision is only way to determine E during collision diss i. Mathematically determine if a collision is “elastic” (E = 0), or “inelastic” (E ≠ 0) diss diss 1. “Perfectly inelastic” collisions occur when both objects end up sticking together.
Unit 7: Oscillation [~10% of Exam] 1. Simple harmonic motion (force and energy relationships). I understand simple harmonic motion, so they can: a. Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period and frequency of the motion. b. Write down an appropriate expression for displacement of the form A sin wt or A cos wt to describe the motion. c. Find an expression for velocity as a function of time. d. State the relations between acceleration, velocity and displacement, and identify points in the motion where these quantities are zero or achieve their greatest positive and negative values e. State and apply the relation between frequency and period. 2 2 f. Recognize that a system that obeys a differential equation of the form d2 x/dt =𝜔 A must execute simple harmonic motion, and determine the frequency and period of such motion. g. State how the total energy of an oscillating system depends on the amplitude of the motion, sketch, or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic. h. Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic and potential energy is constant. i. Calculate the maximum displacement or velocity of a particle that moves in simple harmonic motion with specified initial position and velocity. j. Develop a qualitative understanding of resonance so they can identify situations in which a system will resonate in response to a sinusoidal external force. 2. Massspring systems. I am able to apply their knowledge of simple harmonic motion to the case of a mass on a spring, so I can: a. Derive the expression for the period of oscillation of a mass on a spring. b. Apply the expression for the period of oscillation of a mass on a spring. c. Analyze problems in which a mass hangs from a spring and oscillates vertically.
d. Analyze problems in which a mass attached to a spring oscillates horizontally. e. Determine the period of oscillation for systems involving series or parallel combinations of identical springs, or springs of differing lengths. 3. Pendulum and other oscillations. I am able to apply their knowledge of simple harmonic motion to the case of a pendulum, so I can: a. Derive the expression for the period of a simple pendulum. b. Apply the expression for the period of a simple pendulum. c. State what approximation must be made in deriving the period. d. Analyze the motion of a torsional pendulum or physical pendulum in order to determine the period of small oscillations.
Unit 8: Universal Gravitation [~10% of Exam] 1. Newton's law of gravity.
I know Newton's Law of Universal Gravitation, so I can: a. Determine the force that one spherically symmetrical mass exerts on another. b. Determine the strength of the gravitational field at a specified point outside a spherically symmetrical mass. c. Describe the gravitational force inside and outside a uniform sphere, and calculate how the field at the surface depends on the radius and density of the sphere. 2. Orbits of planets and satellites I understand the motion of an object in orbit under the influence of gravitational forces, so I can: a. For a circular orbit: i. Recognize that the motion does not depend on the object's mass; describe qualitatively how the velocity, period of revolution, and centripetal acceleration depend upon the radius of the orbit; and derive expressions for the velocity and period of revolution in such an orbit. ii. Derive Kepler's Third Law for the case of circular orbits. iii. Derive and apply the relations among kinetic energy, potential energy, and total energy for such an orbit b. For an elliptical orbit: i. State Kepler's three laws of planetary motion and use them to describe in qualitative terms the motion of an object in an elliptical orbit. ii. Apply conservation of angular momentum to determine the velocity and radial distance at any point in the orbit iii. Apply angular momentum conservation and energy conservation to relate the speeds of an object at the two extremes of an elliptical orbit iv. Apply energy conservation in analyzing the motion of an object that is projected straight up from a planet's surface or that is projected directly toward the planet from far above the surface. After Electricity and Magnetism, but before exam...
RESISTIVE FORCE Students should understand the effect of drag forces on the motion of an object, so they can: (1) Given a situation in which acceleration changes as a function of velocity and time, I can write an appropriate DIFFERENTIAL EQUATION, and also solve it for v(t) using calculus, incorporating correctly a given initial value for v (and using an algorithm as an o alternate method). (2) Find the terminal velocity of an object moving vertically under the influence of a retarding force dependent on velocity. (3) Describe qualitatively, with the aid of graphs, the acceleration, velocity, and displacement of such a particle when it is released from rest or is projected vertically with specified initial velocity. (4) Use Newton's Second Law to write a differential equation for the velocity of the object as a function of time. (5) Use the method of separation of variables to derive the equation for the velocity as a function of time from the differential equation that follows from Newton's Second Law. (6) Derive an expression for the acceleration as a function of time for an object falling under the influence of drag forces.
