Question

: A particle is subjected to two simple harmonic motions along $x$ and $y$ directions according to $$$$ $x = 3\sin 100\pi t$ $y = 4\sin 100\pi t$
A. Motion of the particle will be an ellipse traveling in the clockwise direction.
B. Motion of the particle is in a straight line with a slope $\dfrac{4}{3}$ .
C. Motion will be simple harmonic motion with amplitude $5$.
D. Phase difference between two motions is $\dfrac{\pi }{2}$ .

Answer 1

Simple harmonic motion is a unique type of periodic motion where the restoring force of the moving body is directly proportional to the displacement and directed towards the body’s equilibrium position. For example, the motion of the pendulum and spring motion. Every simple harmonic motion is periodic but every simple harmonic motion is not periodic like the motion of the planet around the sun it is periodic but not simple harmonic.
Formula used:
The slope of a straight line = $\dfrac{y}{x}$

Complete step by step answer:
A particle is subjected to two simple harmonic motions along $x$ and $y$ directions,
The slope of a straight line = $\dfrac{y}{x}$ ………………….. (1)
Given that $x = 3\sin 100\pi t$………. (2)
And $y = 4\sin 100\pi t$ ……….. (3)
Divide equation (3) by equation (4)
We get $\dfrac{y}{x} = \dfrac{4}{3}$ ………… (5)
From equation (1) and (5)
We get, $slope = \dfrac{4}{3}$

So, the correct answer is “Option B”.

Note:
The overall energy of the particle exhibiting simple harmonic motion is conserved. Many physical systems show simple harmonic motion like the electron in a wire carrying alternating current, the vibrating particles of the medium in sound waves, etc.
The main distinction between simple harmonic motion and periodic motion is that periodic motion refers to any type of repeated motion where simple harmonic motion refers to a particular type of periodic motion in which the restoring force is proportional to the displacement.
Simple harmonic motions are rare in nature since frictional forces are not negligible and bodies that move in an oscillatory manner lessen their amplitude in their interaction with the air that surrounds them. Simple harmonic movement is characterized by having constant amplitude.