Q1. (30 pts) Flyball governer: Old steam engines used a governer to stabilize the rotational speed ω of the shaft coming out of the engine (see the figures below). Use the schematic figure on the left for your calculations below. The action principle of the governer is simple: as the shaft rotates faster, the balls (m) move outwards and the collar (A) moves up to uncover a steam hole and slow down the engine. Assume no friction and that everything other than the balls is massless.
A real example with a slightly different geometry.
(a) (10 pts) Using the figure on the left, express the gravitational potential energy and the kinetic energy of the system in terms of the opening angle (γ) of the arms as the single generalized coordinate and construct the Lagrangian. (b) (10 pts) By considering the effective potential of the one dimensional system or constructing the equation of motion, find the equilibrium value(s) for γ. (c) (10 pts) Investigate the stability of the equilibrium solution(s) in part (b).
Q2. (35 pts) Particle on a sliding wedge (Homework problem): A particle with mass m slides without friction on a wedge of angle θ and mass M which itself can move without friction on a horizontal surface. We want to treat the constraint of the particle on the wedge by the method of Lagrange multipliers. Let the coordinates of the particle be (x1 , y1 ) and the lowest point on the inclined surface of the wedge be (x2 , 0). (a) (15 pts) Write down the Lagrangian and the constraint equation. (b) (10 pts) Obtain the equations of motion. (c) (10 pts) Find the constraint force.
Q3. (35 pts) A different central force: A particle with mass m is in a circular orbit under the influence of the central force f~(r) = −kˆ r/r3 . (a) (10 pts) Find the angular momentum l0 . (b) (5 pts) Plot the effective potential for l > l0 and l < l0 . When does the system have bound orbits? (c) (10 pts) When the angular momentum is l0 , the particle receives a radial impulse, so that it has nonzero radial speed vr at t = 0. Find r(t) for t > 0. (d) (10 pts) Find θ(t) for the case in part (c).
Dec 4, 2017 - Q1. (30 pts) Flyball governer: Old steam engines used a governer to stabilize the rotational speed Ï of the shaft coming out of the engine (see ...
