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?
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.
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?
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.
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: ∫02πdx/(a + b cosx) = 2π/(a2 - b2)1/2, (a2 > b2).