A hoop of radius R and mass M is hung from a nail and displaced from equilibrium

Question

A hoop of radius R and mass M is hung from a nail and displaced from equilibrium by a small angle theta. Starting from Newton's Laws, find the period of the motion. An object of mass m1 is sliding on a frictionless, horizontal surface. It is attached to a spring of a spring of force constant k. A second block of mass m2 is on top of the first block. If the amplitude of the oscillation is A, what is the minimum value of the coefficient of static friction mu such that the top block will not slip on the bottom block? Two pendulums have the same length L and total mass m. Pendulum A is a very small ball swinging at the end of a massless bar. Pendulum B has half the mass in the uniform bar and half the mass in the ball. Which has the larger period? A small particle of mass m slides without friction is a spherical bowl of radius r. (a) Show that the motion of the particle is the same as if it were attached to a string of length r. The figure shows a particle of mass m1 that is displaced by a small distance s1, where s1, is much smaller than r. A second particle of mass m2 is displaced in the opposite direction a distance of s2 = 3s1, where s2 is also much smaller than r. If the particles are released at the same time, where do they meet? The figure shows a solid cylinder of mass M and radius R attached to a horizontal massless spring. The cylinder can roll without slipping along the horizontal surface. The force constant of the spring is k. The system is released from rest at a position where the spring is stretched by amount d. Find the translation kinetic energy and the rotational kinetic energy of the cylinder as it passes through the equilibrium position. Show that the center of mass of the cylinder executes simple harmonic motion and find the period.

Explanation / Answer

(t) = 0 cos((2/T) t)

S = S0 cos((2/T) t)

S1 + S2 = (s1 + s2) cos((2/T)t) = s1 + s2

therefore they meet each other at t=T/2 or in other words at the bottom of the bowl.

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