Find a model for simple harmonic motion satisfying the specified conditions.
Displacement (t = 0): 0
Amplitude: 5 meters
Period: 10 seconds

A) $d = 5 \sin \left( \frac { \pi t } { 5 } \right)$
B) $d = 5 \sin ( 2 \pi t )$
C) $d = 5 \sin \left( \frac { \pi t } { 12 } \right)$
D) $d = 10 \sin ( \pi t )$
E) $d = 5 \cos ( 2 \pi t )$

Question

Find a model for simple harmonic motion satisfying the specified conditions.  
Displacement (t = 0): 0
Amplitude: 5 meters
Period: 10 seconds
 
A) $d = 5 \sin \left( \frac { \pi t } { 5 } \right)$ 
B) $d = 5 \sin ( 2 \pi t )$  
C)  $d = 5 \sin \left( \frac { \pi t } { 12 } \right)$  
D) $d = 10 \sin ( \pi t )$  
E) $d = 5 \cos ( 2 \pi t )$

Quizplus · Accepted Answer

The period $T$ of a simple harmonic motion described by $d = A \sin(\omega t + \phi)$ or $d = A \cos(\omega t + \phi)$ is given by $T = \frac{2\pi}{\omega}$. Given that the amplitude is 5 meters and the period is 10 seconds, and the displacement at $t = 0$ is 0 (which matches a sine function starting at 0), we find the angular frequency $\omega$ using the period: $\omega = \frac{2\pi}{T} = \frac{2\pi}{10} = \frac{\pi}{5}$. Thus, the model is $d = 5 \sin \left( \frac { \pi t } { 5 } \right)$.

Find a Model for Simple Harmonic Motion Satisfying the Specified d=5sin⁡(πt5)d = 5 \sin \left( \frac { \pi t } { 5 } \right)d=5sin(5πt​)

Find a Model for Simple Harmonic Motion Satisfying the Specified $d = 5 \sin \left( \frac { \pi t } { 5 } \right)$