Question

Two simple harmonic motions of angular frequency 100 and 1000 rad/S have the same displacement amplitude. The ratio of their maximum acceleration is:
A. $1:{{10}^{3}}$
B. $1:{{10}^{4}}$
C. $1:10$
D. $1:{{10}^{2}}$

Answer 1

Hint: In this problem we just need to take a ratio of acceleration of simple harmonic motion one to simple harmonic motion two, in which displacement A(amplitude) is the same for both the motion and the acceleration is in the opposite direction of motion.

Formula used:
 $a=-{{\omega }^{2}}A$

Complete Step by step solution:
Given,
${{\omega }_{1}}$=100 rad/s (angular frequency of simple harmonic motion one)
${{\omega }_{2}}$=1000rad/s (angular frequency of simple harmonic motion two)
Now, from formula $a=-{{\omega }^{2}}A$, here $a$(acceleration), $\omega $(angular frequency) and $A$(amplitude).

We can take ratio of acceleration of one to two which is given by,
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{-{{\omega }_{1}}^{2}A}{-{{\omega }_{2}}^{2}A}$ ${{a}_{1}}$(Acceleration of first motion) ${{a}_{2}}$(Acceleration of second motion) and $A$(amplitude).

Now,$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{\omega }_{1}}^{2}}{{{\omega }_{2}}^{2}}$
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{10}^{2}}}{{{1000}^{2}}}=\dfrac{1}{{{10}^{2}}}$

So, the ratio of maximum acceleration of simple harmonic motion one to simple harmonic motion two is $1:{{10}^{2}}$, so option D is correct.

Additional information:
The maximum acceleration can be derived from differentiating the distance time equation of simple harmonic motion ($x=A\sin (\omega t)$two times with respect to time, in which the maximum acceleration would be equal to $a=-{{\omega }^{2}}A$ which is at the time of maximum amplitude, as we know at that time maximum force has been exerted on the body in the opposite direction.

Note:
In solving these problems carefully read the statement given to calculate maximum acceleration i.e. In the above problem statement we have given the displacement the same (amplitude), so we just need to find the ratio of square of individual angular velocities of motions. If displacement was also different then we needed to consider amplitude to calculate the ratio.