A mass-and-spring system oscillates with amplitude \(A\) and angular frequency \(\omega\) (a) What is the average speed during one complete cycle of oscillation? (b) What is the maximum speed? (c) Find the ratio of the average speed to the maximum speed. (d) Sketch a graph of \(v_{x}(t),\) and refer to it to explain why this ratio is greater than \(\frac{1}{2}\).

In this problem, we have a mass-and-spring system oscillating with amplitude A and angular frequency ω. The velocity formula for the system is given by \(v_x(t) = -A\omega\sin(\omega t + \phi)\). The maximum speed is obtained when the sine function has its maximum value, which is 1. Thus, the maximum speed is given by \(v_\text{max} = A\omega\). The average speed of one complete cycle is the total distance covered over the total time taken, which can be calculated as \(v_\text{avg} = \frac{4A\omega}{2\pi}\). The ratio of average speed to maximum speed is \(\frac{v_\text{avg}}{v_\text{max}} = \frac{2}{\pi}\). The graph of \(v_x(t)\) represents a sine wave with a maximum value of \(A\omega\) and a minimum value of \(-A\omega\), oscillating about the t-axis. The average speed is greater than half of the maximum speed because it takes into account the symmetric slopes of the sine wave in both the positive and negative half cycles.