Question

The Maximum velocity of a simple harmonic motion represented by $y = 3\sin \left( {100t + \dfrac{\pi }{6}} \right)$ is given by
A. $300\,units$
B. $\dfrac{{3\pi }}{6}\,units$
C. $100\,units$
D. $\dfrac{\pi }{6}\,units$

Answer 1

Simple harmonic motion (abbreviated SHM) is a type of periodic motion in mechanics and physics in which the restoring force on a moving item is directly proportional to the magnitude of the item's displacement and acts towards the item's equilibrium position. It causes an oscillation that, if not interrupted by friction or other energy dissipation, can last indefinitely.

Formula used:
${v_{\max }} = a\omega$
Where, $a =$Amplitude of wave and $\omega =$Angular velocity of wave.

Complete step by step answer:
Simple harmonic motion can be used to mimic a number of motions, but it is best exemplified by the oscillation of a mass on a spring when it is subjected to Hooke's law's linear elastic restoring force. The motion has a single resonance frequency and is sinusoidal in time.

Other phenomena, such as the motion of a simple pendulum, can be modelled using basic harmonic motion, albeit the net force on the object at the end of the pendulum must be proportional to the displacement for it to be an accurate model. Molecular vibration can also be modelled using simple harmonic motion.

Given: $y = 3\sin \left( {100t + \dfrac{\pi }{6}} \right)$
From this equation of the wave we can say that:
Amplitude of the wave is
$a = 3$
And Angular velocity of the wave is:
$\omega = \,100\,rad/\sec$
As we know that the velocity can be obtained as:
${v_{\max }} = a\omega$
$\Rightarrow {v_{\max }} = 3 \times 100 \\\ \therefore {v_{\max }} = 300\,units$

Hence, the correct answer is option A.

Note: Because any mass subject to a force in stable equilibrium functions as a harmonic oscillator for small vibrations, the harmonic oscillator concept is particularly essential in physics. Harmonic oscillators are found in abundance in nature and are used in a variety of man-made systems, including clocks and radio circuits.