Question

A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and s is in inches: s(t)=−3cos(πt+π4). Determine the position of the spring at t=1.5 s. Find the velocity of the spring at t=1.5 s.

Answer 1

Answer:

The position and velocity of the spring is −2.12 inches and -6.66 ft/s.

Explanation:

Given that,

The position function,

$s(t)=-3\cos(\pi t+\dfrac{\pi}{4})$

We need to calculate the position of the spring at t = 1.5 s

Using position function

$s(t)=-3\cos(\pi t+\dfrac{\pi}{4})$

Put the value of t in the function

$s(1.5)=-3\cos(\pi\times1.5+\dfrac{\pi}{4})$

$s(1.5)=-2.12\ inches$

We need to calculate the velocity of the spring

Using position function

$s(t)=-3\cos(\pi t+\dfrac{\pi}{4})$

On differentiating

$\dfrac{ds}{dt}=3\pi\sin(\pi t+\dfrac{\pi}{4})$

$v(t)=3\pi\sin(\pi t+\dfrac{\pi}{4})$

Put the value into the formula

$v(1.5)=3\pi\sin(1.5\pi+\dfrac{\pi}{4})$

$v(1.5)=-6.66\ ft/sec$

Hence, The position and velocity of the spring is −2.12 inches and -6.66 ft/s.