Question

an object moves in simple harmonic motion described by the equation d = - 5 sin(\frac{\pi}{4}t) where t is measured in seconds and d in inches. find the maximum displacement, the frequency, and the time required for one cycle.
a. find the maximum displacement.
in.
(type an integer or a fraction.)

Explanation:

Step1: Recall sine - function property

The general form of simple harmonic motion is $d = A\sin(\omega t)$. For the function $y = \sin x$, the range of $y$ is $[- 1,1]$. In the equation $d=-5\sin(\frac{\pi}{4}t)$, the amplitude $A$ determines the maximum displacement.

Step2: Calculate maximum displacement

Since the amplitude $A = |-5|=5$, the maximum displacement of the object is the absolute - value of the coefficient of the sine function. So the maximum displacement is 5 inches.

Step3: Find the angular frequency $\omega$

For the equation $d=-5\sin(\frac{\pi}{4}t)$, we have $\omega=\frac{\pi}{4}$.

Step4: Calculate the period $T$

The formula for the period of a simple - harmonic motion is $T=\frac{2\pi}{\omega}$. Substituting $\omega = \frac{\pi}{4}$ into the formula, we get $T=\frac{2\pi}{\frac{\pi}{4}}=8$ seconds. The period $T$ is the time required for one cycle.

Step5: Calculate the frequency $f$

The frequency $f$ and the period $T$ are related by the formula $f=\frac{1}{T}$. Since $T = 8$ seconds, then $f=\frac{1}{8}$ Hz.

Answer:

a. 5
b. $\frac{1}{8}$ Hz
c. 8 seconds