Chapter 4, Problem 25CT

To determine

To find:theequation of the simple hormonic motion of the ball.

Answer to Problem 25CT

The equation of the simple hormonic motion of the ball is $d=-6\cos \pi t$ .

Explanation of Solution

Given:

A ball on a spring that starts at its lowest point of 6 inches below the equilibrium.

The maximum height of the point is 6 inches above the equilibrium and period 2s.

Definition used:

A point that moves on a coordinate line is in simple hormonic motion when its distance d from the origin at time t is given by either $d=a\sin \omega t\text{ or }d=a\cos \omega t$ where a and $\omega$ are real numbers such that $\omega >0$ . The motion has amplitude $\left|a\right|$ and period $\frac{2\pi }{\omega }$ .

Calculation:

Here, the lowest point of the ball is 6 inches below the equilibrium and maximum height of the point is 6 inches above the equilibrium.

This implies that the amplitude $\left|a\right|=6$ .

The period of the ball is 2 seconds.

  $\begin{array}{l}\frac{2\pi }{\omega }=2\\ \omega =\pi \end{array}$

Thus, the value of $\omega =\pi$ .

The ball starts from lowest point to maximum point, the amplitude $a=-6$ .

Substitute $a=-6$ and $\omega =\pi$ in $d=a\cos \omega t$ ,

  $d=-6\cos \pi t$

Therefore, the equation of the simple hormonic motion of the ball is $d=-6\cos \pi t$ .