Homework 3

Due Wednesday March 2nd at 5pm

Q1 Satellite

NASA has launched a satellite into an circular orbit around the earth and wants to increase the radius slightly while maintaining a circular orbit. NASA scientist propose to fire the engines briefly, applying a small impulse to the satellite. One scientist says that it doesn't matter if the impulse is applied in a direction tangential to the satellite motion or perpendicular to the motion, arguing that both approaches will simply fine tune the total energy of the system. Another scientist disagrees and argues that the one of the options would definitely not work. Which scientist would you side with, and why?

Q2 Ice Rink

Consider the motion of a hockey puck of mass m on a perfectly circular bowl-shaped ice rink with radius a. You may neglect the e ffects of friction. The sides of ice bowl are of height h. The central region of the bowl (r < 0.8a) is perfectly flat.

a) Draw a sketch of the potential for this system. Set the zero of potential energy at the top of the walls.

b) Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?

c) Situation 2: a stationary puck, at a distance a/2 from the center of the rink, is hit in such a way that it's initial velocity v_0 (vector) is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?

d) In situation 2, what is the angular momentum of the puck immediately after it is struck?

e) Draw a sketch of the e ective potential for situation 2.

f) In situation 2, for what minimum value of v_0 (vector) does the puck just escape the rink?

Q3: Binary Stars

The fi gure below shows the orbit traced out by the position vector r(t) of a “fictitious” reduced mass under the influence of a gravitational central force that points toward the origin.

The vector r(t) also describes the separation vector for a binary star system. The binary stars have mass m_1 and m_2 respectively.

a) Assuming that m_2 = m_1, sketch the orbits of m_1 and m_2 (we already did this exercise in class).

b) Repeat part a) for m_2 > m_1