The motion of an under-damped oscillator is described by Eq. (14.42). Let the phase angle be

The motion of an under-damped oscillator is described by Eq. (14.42). Let the phase angle ɸ be zero.

(a) According to this equation, what is the value of x at t = 0?

(b) What are the magnitude and direction of the velocity at t = 0? What does the result tell you about the slope of the graph of x versus t near t = 0?

(c) Obtain an expression for the acceleration α_x at t = 0. For what value or range of values of the damping constant b (in terms of k and m) is the acceleration at negative, zero, and positive? Discuss each case in terms of the shape of the graph of x versus t near t = 0.