Assignment 3

Problem 1:

A cylindrical shell rolls without slipping down an incline as shown below.  If it starts from rest, how far must it roll along the incline to obtain a speed v?

 

Problem 2:

An inclined plane of length L makes an angle θ > 0o with respect to the horizontal.  A very thin ring with radius R rolls down this plane.  It reaches the bottom in time Δtring.
(a)  If a uniform density disk or sphere with the same radius R roll down this plane, at what time do they reach the bottom?  Find Δtdisk/Δtring and Δtsphere/Δtring.
(b)  If all three object should reach the bottom at the same time Δtring, one needs to vary the angle θ for the disk and the sphere.  Find sinθdisk/sinθ and sinθsphere/sinθ.

Problem 3:

A rope rests on two platforms which are both inclined at angle θ (which you are free to pick) as shown.  The rope has uniform density and its coefficient of friction with the platform is 1.  The system has left-right symmetry.  What is the largest possible fraction of the rope that does not touch the platform?  What angle θ allows this maximum value?

Problem 4:

A uniform disk of radius R and mass M is spinning about its diameter with angular velocity ω, as shown below.  Located on the rim of the disk, at an angle θ from the spin axis, is point P, and P is moving with speed vp.  Point P is now suddenly fixed.  Show that the subsequent linear speed vc of the center of the disk is vc = vp/5.

Problem 5:

A solid sphere toy globe of mass M and radius R rotates freely without friction with an initial angular velocity ω0.  A bug of mass m starts at one pole N and travels with constant speed v to the other pole S along a meridian in time T.  The axis of rotation of the globe is held fixed.  Show that during the time the bug is traveling the globe rotates through an angle
Δθ = (πω0R/v) (2M/(2M + 5m))1/2.

Useful integral: ∫0dx/(a + b cosx) = 2π/(a2 - b2)1/2,  (a2 > b2).