**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 > 0^{o} with respect
to the horizontal. A very thin ring with radius R rolls down this plane. It
reaches the bottom in time Δt_{ring}.

(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 Δt_{disk}/Δt_{ring}
and Δt_{sphere}/Δt_{ring}.

(b) If all three object should reach the bottom at the same time Δt_{ring},
one needs to vary the angle θ for the disk and the sphere. Find sinθ_{disk}/sinθ
and sinθ_{sphere}/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 v_{p}.
Point P is now suddenly fixed. Show that the subsequent linear speed v_{c}
of the center of the disk is v_{c }= v_{p}/5.

**Problem 5:**

A solid sphere toy globe of mass M and radius R rotates freely without
friction with an initial angular velocity **ω**_{0}. 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 = (πω_{0}R/v) (2M/(2M + 5m))^{1/2}.

Useful integral: ∫_{0}^{2π}dx/(a + b cosx) = 2π/(a^{2}
- b^{2})^{1/2}, (a^{2} > b^{2}).