Question

The displacement of a particle varies according to the relation $x = 4\left( {\cos \pi t + \sin \pi t} \right)$ . the amplitude of the particle is:
(A) $ - 4$
(B) $4$
(C) $4\sqrt 2 $
(D) $8$

Answer 1

Hint: - In the case of simple harmonic motion there will be a mean position and two extreme positions. The pendulum performs the SHM with respect to the mean position and within the range of two extreme positions. Generally, simple harmonic motions are denoted with the sinusoidal or cosecant functions.
Formula used:
The equation of SHM is
$x = A\sin \left( {\omega t + \phi } \right)$ ,
where $A$ is the amplitude,
$x$ is the displacement,
$\omega $ is the angular frequency,
$t$ is the time,
$\phi $ is the phase difference.

Complete step-by-step solution:
We have to calculate the amplitude of the particle from the given expression. The given expression is $x = 4\left( {\cos \pi t + \sin \pi t} \right)$ . This gives the relation between displacement, amplitude, and the angular velocity.
It is given that,
$x = 4\left( {\cos \pi t + \sin \pi t} \right)$
Divide and multiply the equation by $\sqrt 2 $ we get,
$x = 4\sqrt 2 \left( {\dfrac{1}{{\sqrt 2 }}\cos \pi t + \dfrac{1}{{\sqrt 2 }}\sin \pi t} \right)$
$ \Rightarrow x = 4\sqrt 2 \left( {\sin 45\cos \pi t + \cos 45\sin \pi t} \right)$
Applying the trigonometric equation,
$x = 4\sqrt 2 \sin \left( {\pi t + 45} \right)$ .............. $\left( 1 \right)$
Comparing the equation $\left( 1 \right)$ with,
$x = A\sin \left( {\pi t + \phi } \right)$
Therefore, the amplitude of the particle is $A = 4\sqrt 2 $

Hence, the correct answer is option (C) $4\sqrt 2 $ .

Additional information: It is the periodic motion of a point along a straight line such that its acceleration is always towards a fixed point and it is directly proportional to its distance from that point. It is essentially repetitive movement back and forth of an object through an equilibrium or central position such that maximum displacement on one side of its position is equal to maximum displacement on the other side.
The time period is the time taken for an object to complete one cycle of its periodic motion, such as time taken by a pendulum to create a full back and forth swing. The total of the average acceleration is the half time period for equilibrium position in a simple harmonic motion.

Note: The simple harmonic motion is a special type of periodic motion where the restoring force applied on the object is directly proportional to the magnitude of displacement and acts towards the object’s equilibrium position. Moreover, all simple harmonic motions are periodic motions but all periodic motions are not simple harmonic motions.