Chapter 8, Problem 67RE

(a)

To determine

To describe:The motion of the object.

Explanation of Solution

Given: $d=-2\cos \left(\pi t\right)$

The equation of simple harmonic motion is given by,

  $y=a\cos \omega t$

And the given equation is,

  $d=-2\cos \left(\pi t\right)$

Therefore, the motion of the object is simple harmonic motion.

(b)

To determine

To find: The maximum displacement.

Explanation of Solution

Given: $d=-2\cos \left(\pi t\right)$

The equation of simple harmonic motion is given by,

  $y=a\cos \omega t$

And the given equation is,

  $d=-2\cos \left(\pi t\right)$

The amplitude is,

  $\begin{array}{c}\left|a\right|=\left|-2\right|\\ =2\end{array}$

Therefore, the maximum displacement is $2$ .

(c)

To determine

To find: The time required for one oscillation.

Explanation of Solution

Given: $d=-2\cos \left(\pi t\right)$

The equation of simple harmonic motion is given by,

  $y=a\cos \omega t$

And the given equation is,

  $d=-2\cos \left(\pi t\right)$

The time period,

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

The time required for one oscillation is $2$ .

(d)

To determine

To find: The time required for one oscillation.

Explanation of Solution

Given: $d=-2\cos \left(\pi t\right)$

The equation of simple harmonic motion is given by,

  $y=a\cos \omega t$

And the given equation is,

  $d=-2\cos \left(\pi t\right)$

The frequency is calculated as,

  $\begin{array}{c}f=\frac{1}{T}\\ =\frac{1}{2}\end{array}$

Therefore, the frequency is $\frac{1}{2}$ .