Question

The displacement d (in meters) of an object at time t (in seconds) is given $d=5\cos \left(\frac{\pi }{2}t\right)$

(a) Describe the motion of the object.

(b) What is the maximum displacement from its resting position?

(c) What is the time required for one oscillation?

(d) What is the frequency?

Short answer

a) It is a simple harmonic motion

b) The amplitude is 5

c) The period is 4

d) The frequency is $\frac{2}{\pi }$

Step-by-step solution

Given information

Our given equation is$d=5\cos \left(\frac{\pi }{2}t\right)$

Part (a) Step 1: Explanation

As the object is modeled by cosine function the given wave is simple harmonic.

Part (b) Step 1: Part b) Step 1: The maximum displacement of object from resting position is the amplitude

Comparing the equation with standard wave equation we get$a=5$

Part (c) Step 1:  Find period 

We get,

$$\begin{gathered} \omega =\frac{2\pi }{T} \\ \frac{\pi }{2}=\frac{2\pi }{T} \\ T=4 \end{gathered}$$

Part (d) Step 1: Find frequency

We have,

$$\begin{gathered} f=\frac{1}{\omega } \\ f=\frac{2}{\pi } \end{gathered}$$

Conclusion

a) It is a simple harmonic motion

b) The amplitude is 5

c) The period is 4

d) The frequency is$\frac{2}{\pi }$