By Pavel Panchekha, Jeffrey Prouty

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Problem Set 7 (Given 12/04/11)

Level 1

1. In a standard mass-spring system, a spring of constant $$k$$ is stretched a distance $$A$$ from its equilibrium position and released. At any given time, what is the total energy of the mass-spring system? By what angle would you have to displace a simple pendulum of length $$L$$ and mass $$m$$ to obtain the same total energy?
2. Suppose I launch a projectile from the surface of the moon. Neglecting gravitational attraction to any bodies besides the moon and the earth, what is the minimum velocity with which I must fire the projectile such that it will never return to either earth or moon?
3. In lecture we discussed the difference between the final speed of a block, a hoop, and a cylinder rolling down a ramp. We assumed no friction between the block and the ramp. For an arbitrary coefficient of friction between the block and the ramp, what is the new the final speed of the block? What would the coefficient of friction have to be for the block to end up with the same speed as the hoop, assuming the hoop loses no energy to friction since it is rolling?

Level 2

1. Solve the AP Physics C 2002 Mechanics, question 2
2. AP Physics C 2006 Mechanics, question 3
3. Derive the equation for simple harmonic motion of a simple pendulum using the energy method discussed in class (you should already know what answer to expect!).

Level 3

1. A binary star system consists of two stars of mass $$M$$ orbiting each other in circular motion. The distance between the center of mass of the two stars is $$R$$. Suppose a small mass $$m$$ is placed in the center between the two stars. The mass is given a small kick in the direction perpendicular to the plane of the orbit of the two stars. Find the period of oscillation for the subsequent motion of the mass.