**Unformatted text preview: **Homework 4, PHY 5200, Fall 2016 (due on 10/05/2016)
1. A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose
other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle
of radius r0 with angular velocity ω0 , but I now pull the string down through the hole until a length r remains
between the hole and the particle. (a) What is the particle’s angular velocity now? (b) Assuming that I pull the
string so slowly that we can approximate the particle’s path by a circle of slowly shrinking radius, calculate the
work I did pulling the string. (c) Compare your answer to part (b) with the particle’s gain in kinetic energy.
2. (extra 2 pts)
Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the puck is given a tiny
nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface
of the sphere?
3. Prove that the direction of ∇f (r) at any point r is the direction in which f increases fastest as we move away
from r.
4. Which of the following forces is conservative? (a) F~ = k(x, 2y, 3z) where k is a constant. (b) F~ = k(y, x, 0). (c)
F~ = k(y, x, 0). For those which are conservative, find the corresponding potential energy U . 156 5. A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod. A massless string (length
l) attached to the ball runs over a massless, frictionless pulley (at a distance b from the rod) and supports
a block of mass M , as shown in Figure below. The positions of the two masses can be specified by the one
angle θ. (a) Write down the potential energy U (θ). (Assume that the pulley and ball have negligible size.) (b)
Chapter 4 Energy
By differentiating U (θ) find whether the system has an equilibrium position, and for what values of m and M
equilibrium can occur. Discuss the stability of any equilibrium positions. Figure 4.27 Problem 4.36 (The subscript "o" is to emphasize that this is the period for small oscillations.) 4.35 ** Consider the Atwood machine of Figure 4.15, but suppose that the pulley has radius R and
moment of inertia I. (a) Write down the total energy of the two masses and the pulley in terms of the
coordinate x and I. (Remember that the kinetic energy of a spinning wheel is i ho2 .) (b) Show (what
is true for any conservative one-dimensional system) that you can obtain the equation of motion for the
coordinate x by differentiating the equation E = const. Check that the equation of motion is the same ...

View
Full Document

- Fall '16
- Sergei Voloshin
- mechanics, Friction, Kinetic Energy, Mass, Work, 2 pts, frictionless horizontal table, θ.