CALC The balance wheel of a watch vibrates with an angular amplitude ?, angular frequency ?, and phase angle ? = 0. (a) Find expressions for the angular velocity d?/dt and angular acceleration d2?/dt2 as functions of time. (b) Find the balance wheel’s angular velocity and angular acceleration when its angular displacement is ?, and when its angular displacement is ?/2 and ? is decreasing. (?Hint:? Sketch a graph of ? versus t.)

Solution 44E The angular displacement of a SHM is given by, = cos(t + )…..(1), where is angular amplitude, is angular frequency and is phase angle. Given that, = 0 Therefore, from equation (1), = cost…..(2) (a) Angular velocity can be calculated by differentiating equation (2) with respect to time d = ()sint dt d dt = sint…..(3) This is the angular velocity as a function of time. Angular frequency can be calculated by differentiating equation (3) with respect to time, 2 2 dt2 = cost…..(4) This is the angular acceleration as a function of time. (b) When angular displacement is , We get from...