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 equation (2), = cost cost = 1 Therefore, sint = 0 Using this value in equation (3), we can get angular velocity, d dt = 0 Therefore, angular velocity at displacement is zero. Angular acceleration can be calculated from equation (4) using cost = 1 d 2 dt2 = Therefore, a ngular acceleration a t displacement is .2 When the displacement is 2, From equation (2), 2 = cost cost = 1/2 3 Then, sint = 2 From equation (3), d 3 Angular velocity dt = 2 d = 3 dt 2 3 The required angular velocity is 2 From equation (4), 2 Angular acceleration d 2 = /2 dt This is the required angular acceleration.