Question

The equation $x = a\sin 2t + b\cos 2t$ will represent an SHM.
A. True
B. False

Answer 1

Recall the condition for a motion of an object to be simple harmonic motion. This condition is that the acceleration of the particle performing simple harmonic motion is directly proportional to displacement of the particle. Derive the relation for acceleration from the given equation of motion and check whether that equation represents simple harmonic motion or not.

Formula used:
The acceleration $a$ of an object performing simple harmonic motion is given by
$a = - {\omega ^2}x$ …… (1)
Here, $\omega$ is the angular speed of the object and $x$ is the displacement of the object.

Complete step by step answer:
We have given an equation of displacement of an object as
$x = a\sin 2t + b\cos 2t$ …… (2)
From equation (1), we can conclude that for an object performing simple harmonic motion, the acceleration $a$ of object is directly proportional to displacement $x$ of that object and the angular velocity $\omega$ of the object is constant.
$a \propto x$

The velocity of an object is the rate of change of displacement with time.
$v = \dfrac{{dx}}{{dt}}$
Substitute $a\sin 2t + b\cos 2t$ for $x$ in the above equation.
$v = \dfrac{{d\left( {a\sin 2t + b\cos 2t} \right)}}{{dt}}$
$\Rightarrow v = 2a\cos 2t - 2b\sin 2t$

The acceleration of an object is the rate of change of velocity with time.
$a = \dfrac{{dv}}{{dt}}$
Substitute $2a\cos 2t - 2b\sin 2t$ for $v$ in the above equation.
$a = \dfrac{{d\left( {2a\cos 2t - 2b\sin 2t} \right)}}{{dt}}$
$\Rightarrow a = - 4a\sin 2t - 4b\cos 2t$
$\Rightarrow a = - 4\left( {a\sin 2t + b\cos 2t} \right)$
Substitute $x$ for $a\sin 2t + b\cos 2t$ in the above equation.
$\therefore a = - 4x$

From the above equation, it is clear that for the given equation of displacement, the acceleration of the object is directly proportional to displacement of the object.Therefore, the given equation represents an equation of simple harmonic motion.

Hence, the correct option is A.

Note: The students should be careful while taking the derivatives of the equations because if these derivatives are not taken correctly, the final derivation for acceleration of the object will be incorrect and we will not be able to check whether the given equation represents simple harmonic motion or not.