A meter stick of total length l is pivoted a distance d from one end on a friction-less bearing. The stick is suspended so that it becomes a pendulum. This is called a "physical" pendulum because the mass is distributed over the body of the stick. Assume the total mass is m and the mass density of the stick is uniform. The acceleration of gravity is g. Find T and V as functions of generalized coordinate θ and velocity ˙θ. Do this by considering the stick to be divided into infinitesimal parts of length dl and integrating to find the total kinetic and potential energy. Set up the Lagrangian and find the equation of motion.

Two pendula are shown in the figure. Each consists of a solid ball with uniform density and has a mass M. They are each suspended from the ceiling with massless rod as shown in the figure. The ball on the left pendulum is very small. The ball of the right pendulum has radius 1/2 L.   L = 1.6 m Find the period T of the left pendulum for small displacements in s.  Find the period T of the right pendulum for small displacements in s.

Pr5. A pointlike body of mass m made of lead is fixed inside a homogeneous solid sphere of radius R and mass m at distance R/2 from the center of the sphere. This body is placed on a horizontal rough surface. Find the period of small oscillations of the sphere around its equi- librium position. (The sphere rolls without slip- ping on the surface. The moment of inertia of a homogeneous sphere of mass m and radius R is 2mR* /5.) R/2

(a) An oscillating object repeats its motion every 3.3 seconds. (i) What is the period of this oscillation? (ii) What is its frequency? (iii) What is its phase rate (i.e. angular frequency)? (b) A magnesium atom (mass of 24 proton masses) in a crystal is measured to oscillate with a frequency of roughly 10l3 Hz. What is the effective spring constant of the forces holding that atom in the crystal? (c) Sitting on a trampoline, a person with mass m sinks a distance Az below the trampoline's normal level surface. (i) If the person gently bounces on the trampoline (without leaving its surface), what would be the person's period of oscillation T? (You should not need the person's mass, but if you think you do, assume and state a value.) (ii) Find T for Az = 45 cm. Check: For Az = 20 cm you should find T = 0.90 s.

A mass m is attached to a spring of force constant 76.0 N/m and allowed to oscillate. The figure (Figure 1) shows a graph of its velocity vx as a function of time t. Find the period, frequency, and angular frequency of this motion. What is the amplitude (in cm)? At what times does the mass reach the position x=±A in the interval between t=0s and t=1.8s including the endpoints? Find the maximum acceleration of the mass. Find the times at which the maximum acceleration occurs in the interval between t=0s and t=1.8s including the endpoints. What is the mass m?

Note: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)

A horizontal spring mass system oscillates on a frictionless plane.  At time t=0, it is moving left at position x=9 cm.  It has velocity v=0, at positions x=0 cm and 12 cm, and completes one full cycle in 2 seconds.  Write the position and velocity kinematic equations for this oscillating system, including the phase constant.

MärkxsfH) Problem 1: A mass of 20 grams stretches a spring by 10/169 meters. (a) Find the spring constant k, the angular frequency w, as well as the period T and frequency f of free undamped motion for this spring-mass system. (b) Find the general solution x(t) of the DE for the free spring-mass system. (c) Suppose that an exterior force of F(t) = 2.5 sin(12t) in Newtons acts on the spring-mass system. Find the equation of motion (the solution x(t)) of the system if the mass initially is at rest in its equilibrium position.