Question

An object's motion is described by the equation d = 4 sin(pi t ) . The displacement, d , is measured in meters. The time, t , is measured in seconds. Answer the following questions:
(a) What is the object's position at t = 0 ?
(b) What is the object's maximum displacement from its resting position?
(c) How much time is required for one oscillation?
(d) What is the frequency?

Answer 1

The equation of the motion is
d = 4 sin (π t)

a) When t = 0
d = 4 sin 0
d = 0

b) We take the derivative
4π cos πt = 0
t = 1/2
d = 4 sin π/2 = 4 meters

c) 1 second is required

d) The frequency is 1Hz

Answer 2

Answer:

Step-by-step explanation:

The given equation is:

$d=4sin({\pi}t)$, where d is the displacement.             (1)

(A) The position of the object at t=0 is:

$d=4sin({\pi}(0))$

⇒$d=4sin(0)$

⇒$d=0$

The object will be at rest at t=0.

(B) In order to find the maximum displacement from the resting position, we will differentiate the given displacement.

Thus,  Differentiating with respect to t, we have

$d'=4cos({\pi}t)({\pi})$

⇒$d'=4{\pi}cos({\pi}t)$

Now, d'=0

⇒$4{\pi}cos({\pi}t)=0$

⇒$cos{\pi}t=0$

⇒${\pi}t=\frac{{\pi}}{2}$

⇒$t=\frac{1}{2}$

Substitute the value of t in equation (1), we get

⇒$d=4sin({\pi}(\frac{1}{2}))$

⇒$d=4(1)$

⇒$d=4m$

Thus, the maximum displacement from its resting position will be 4m.

(C) Time required for one oscillation is equal to the period which is equal to=$\frac{2\pi}{|b|}=\frac{2\pi}{\pi}=2$

thus,  time  required for one oscillation will be 2 seconds.

(D) Frequency is nothing but the reciprocal of period that is $\frac{|b|}{2\pi}=\frac{\pi}{2\pi}=\frac{1}{2}$

Thus, the frequency will be equal to $\frac{1}{2}$.