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 \(\frac{1}{2}\).
