Question

The displacement of an object attached to a spring and executing simple harmonic motion is given by \[x=2\pi \times {{10}^{-2}}\cos \pi t\text{ meters}\]. The time at which the maximum speed first occur is:
A). 0.5 s
B). 0.75 s
C). 0.125 s
D). 0.25 s

Answer 1

Hint: Simple harmonic motion is a sinusoidal motion executed by an object. Here we have attached this object with a spring. So, it will execute the linear simple harmonic motion. From the displacement equation, we can find out the velocity of the motion.

Formula used: \[v=\dfrac{dx}{dt}\], where v is the velocity and x is the displacement.

Complete step by step answer:

In this question, the displacement of an object attached to the spring is given.
\[x=2\pi \times {{10}^{-2}}\cos \pi t\text{ meters}\]
From this, we can find out the velocity, since the velocity is the rate of change of displacement.
\[v=\dfrac{dx}{dt}\]
\[v=\dfrac{d(2\pi \times {{10}^{-2}}\cos \pi t)}{dt}\]
\[v= -2\pi \times {{10}^{-2}}\pi \sin \pi t\]
\[|v|=2\pi \times {{10}^{-2}}\pi \sin \pi t\]
Here we have to find the time at which the first maximum speed occurs that’s why we consider only magnitude. Since it is a simple harmonic motion, it will occur in repeated motions. We have to find the time for the first maximum speed.
Since it is a sine-based equation, the maxima and minima depend upon that.
The maximum value of the sine is 1. So we can equate \[\sin \pi t\] to 1.
\[\sin \pi t=1\]
This is only possible if \[\sin \pi t=\sin \dfrac{\pi }{2}\]
\[\dfrac{\pi }{2}=\pi t\], or \[t=\dfrac{1}{2}\]
The time required for the first maximum speed is 0.5 seconds. Therefore, the correct answer is option A.

Additional information:
Simple harmonic motion can be defined as the repeating motion with a sinusoidal function of time.
\[x(t)=A\cos (\omega t+\phi )\]
Amplitude is the maximum displacement occurring during the propagation in either direction. The simple harmonic motion will obey Hooke’s law.
\[F=-kx\], where k is the spring constant and x is the displacement.
The period of the simple harmonic motion is the time required to complete one oscillation by an oscillator. So after every period, the motion will get repeated.

Note: During the simple harmonic motion, the velocity will be zero at extreme positions. While the velocity will be maximum at equilibrium positions. Do not think that, \[\sin \pi t=0\] since \[\sin \pi =0\]. The simple harmonic functions are dependent on time. So that, the time (t) has a great impact on that function.