A solid cylinder (mass M and radius R) on a table has a mechanism that attaches a spring (of constant k) to its central axis. The other end of the spring is fixed, so that the spring is horizontal. The cylinder is able to roll on the table without slipping. Let's pull the mass to one side and release it so that it rolls back and forth. Show that the motion of the center of mass is simple harmonic, and find the frequency of ocillation, w. HINT: There are two ways (at least) to do this. Each involves comparing information about this system to the simple harmonic oscillator we know, the mass on a spring. If you can get the force/torque equation to look like -kx = ma. or you can get the energy budget to look like /2mv? + />kx? = ETOTAL, then the motion is simple harmonic. Comparison with w = (k/m}1/2 should then give you the frequency of oscillation.

deal with a mass-spring-dashpot sys- tem having position function x(t) satisfying Eq. (4). We write xo = x (0) and vo = x'(0) and recall that p = c/(2m), w k/m, and wi = wổ – p². The system is critically damped, overdamped, or underdamped, as specified in each problem. Probler 24. (Critically damped) Show in this case that x(1) = (xo + voi + pxot)e¬P!.

Consider a solid uniform ball of mass m and radius R rolling without slipping near the bottom of a frictionless hemispherical fishbowl of radius r as shown. The ball oscillates about the center of the fishbowl. Set the gravitatio nal potential energy to be zero at the bottom of the bowl. r(1- cos 0) A. Express the total energy of the ball as a function of the ball's distance from the center x and its velocity. Hint 1: Since the ball is rolling without slipping, its angular velocity can be expressed in terms of its linear velocity, and it has translational and rotational energy. Hint 2: In writing gravitational potential U, use the approximation cos 0 ~ 1 –, then use SOH-CAH-TOA to rewrite 0 in terms of x and r, then use sin 0 ~ 0 to eliminate the variable 0. `Find the oscillation frequency of the ball. Hint: Since the ball is undergoing simple harmonic motion, x(t) (and hence v(t)) follows a certain form. Also, since the energy of the system is conserved, then dE/dt = 0. B.

Question 1. Find the steady state solution of the forced Mass-Spring-Damper with the following parameters undergoing forcing function F(t). 3 kg, c = 22 Ns/m, k = 493 N/m, F(t) = 21 cos(6t) in the form of (t) = A cos(6t – 8) Enter your answers for A and to four decimal places in the appropriate boxes below: m = A: d:

A cylindrical disc with a mass of 0.619 kg and radius of 0.575 m, is positioned such that it will oscillate as a physical pendulum as shown below. If the period of the small angle oscillations is to be 0.343 s, at what distance from the center of the disc should the axis of rotation be fixed? Assume that the position of the fixed axis is on the actual disc. The moment of inertia of a disc about its center is 1 = 0.5 M R²...Hint: Use the parallel axis theorem.

A particle moves in a force field described by the Yukowa potentialV (r) = −k/r exp(−r/a)where k and a are positive. Write the equations of motion and reduce them to the equivalentone-dimensional problem. Use the effective potential to discuss the qualitative nature of the orbits for different values of the energy and the angular momentum.

A uniform annular ring of mass m and inner and outer radil a and b, respectively, is pivoted around an axis perpendicular to the plane of the ng at point P (see figurm bet). Detemine ts penod sf cscillation. (Use the following as necessary: m, a, b, and g for acceleration due to gravity) T-

Quartic oscillations Consider a point particle of mass m (e.g., marble whose radius is insignificant com- pared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function: x4 y(x) = A ¹ Bx² + B² B²), (1) Where x represents the distance along the horizontal axis and y the height in the vertical direction. The direction of Earth's constant gravitational field in this system of coordinates is g = −gŷ, with ŷ a unit vector along the y direction. This is just a precise way to say with math that gravity points downwards and greater values of y point upwards. A, B > 0. (a) Find the local extrema of y(x). Which ones are minima and which ones are maxima? (b) Sketch the function y(x). (c) What are the units of A and B? Provide the answer either in terms of L(ength) or in SI units. (d) If we put the point particle at any of the stationary points found in (a) and we displace it by a small quantity³. Which stationary locations…

Let B be the solid bounded by the surfaces z = x^2+2y^2, x^2+y^2=16, and the xy-plane(distances in cm). If B has a constant mass density of 5g/cm^3, find the moment of inertia of B about the axis through(4,3,2) that is perpendicular to the yz-plane

A particle of mass m is suspended from a support by a light string of length which passes through a small hole below the support (see diagram below). The particle moves in a vertical plane with the string taut. The support moves vertically and its upward displacement (measured from the ring) is given by a function z = h(t). The effect of this motion is that the string-particle system behaves like a simple pendulum whose length varies in time. I b) [Expect to a few lines to wer these questions.] a) Write down the Lagrangian of the system. Derive the Euler-Lagrange equations. z=h(t) Compute the Hamiltonian. Is it conserved?