Question

A body executes simple harmonic motion under the action of a force $F_1$ with a frequency $n_1$ and under another force $F_2$ with frequency $n_2$. If both forces act simultaneously in the same direction, the frequency of oscillation of the body is

• A. $n_1 + n_2$
• B. $\dfrac{n_1 + n_2}{2}$
• C. $\sqrt{n_1^2 + n_2^2}$
• D. $\sqrt{n_1 n_2}$

Hint:

When SHM forces combine in the same direction, the net frequency is the square root of the sum of squares of individual frequencies.

Answer 1

The Correct Option is C

If two simple harmonic restoring forces act in the same direction on the same body, their accelerations (and hence force constants) add.
Since frequency $n = \dfrac{1}{2\pi} \sqrt{\dfrac{k}{m}}$, the effective force constant becomes $k_{\text{eq}} = k_1 + k_2$.
Thus, $n_{\text{eq}} = \sqrt{n_1^2 + n_2^2}$