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Two simple harmonic motions of angular frequency 100 and 1000 rad s$^{-1}$ have the same displacement amplitude. The ratio of their maximum accelerations is -
A
$1 : 10^{3}$
B
$1 : 10^{4}$
C
$1 : 10$
D
$1 : 10^{2}$
Detailed Solution
Acceleration in SHM: $a = -\omega^2 x$, so the maximum acceleration is $a_{max} = \omega^2 A$.
Both motions have the same amplitude $A$.
$\dfrac{a_1}{a_2} = \dfrac{\omega_1^2 A}{\omega_2^2 A} = \dfrac{\omega_1^2}{\omega_2^2}$
$\dfrac{a_1}{a_2} = \dfrac{(100)^2}{(1000)^2} = \dfrac{10^4}{10^6}$
$\dfrac{a_1}{a_2} = \dfrac{1}{10^2}$
So the ratio of the maximum accelerations is $1 : 10^{2}$.
Both motions have the same amplitude $A$.
$\dfrac{a_1}{a_2} = \dfrac{\omega_1^2 A}{\omega_2^2 A} = \dfrac{\omega_1^2}{\omega_2^2}$
$\dfrac{a_1}{a_2} = \dfrac{(100)^2}{(1000)^2} = \dfrac{10^4}{10^6}$
$\dfrac{a_1}{a_2} = \dfrac{1}{10^2}$
So the ratio of the maximum accelerations is $1 : 10^{2}$.