Question

#2. [Swinging Disk] A uniform circular disk of mass M and radius R is set swinging side-to-side about a frictionless pivot P at its edge (a) What is the disk's moment of inertia about the pivot? (b) Write an expression for the net torque acting on the disk about the pivot when the disk is displaced to the right by angle θ CM (c) Write Newton's 2nd Law for Rotation for the disk when it is displaced as shown. Be careful to give torque and angular acceleration the correct signs. Take the angle θ and angular acceleration α to be (+) in the CCW (U) direction (d) Derive a differential equation for θ(t) by writing the angular acceleration as d2θ/df in your equation from part (c), and then solve for d2e(t)/dt2. Show that the resulting equation is not the differential equation for Simple Harmonic Motion (SHM). Why isn't it? Now use the small angle approximation sinf θ (in rad) to show that your equation from part (d) approximates the differential equation for SHM when the angle θ is small (« 1 rad) (f) Using your equation from part (e), read off an expression for the natural angular frequency Oo of the small angle SHM oscillations of the disk. From this, write expressions for the natural frequency and period of the disk's small angle oscillations. Show that your results agree with the corresponding results derived for the physical pendulum in lecture and the textbook. (g) What is the length of a simple pendulum that has the same oscillation frequency as this disk? Please give your reasoning

Answer 1