Question

When a mass \(\mathrm{m}\) is suspended from the free end of a massless spring having force constant \(\mathrm{k}\), its oscillates with frequency \(\mathrm{f}\). Now if the spring is divided into two equal parts and a mass $2 \mathrm{~m}$ is suspended from the end of anyone of them, it will oscillate with a frequency equal to......... (A) \(\mathrm{f}\) (B) \(2 \mathrm{f}\) (C) \((\mathrm{f} / \sqrt{2})\) (D) \(\sqrt{2 f}\)

Short answer

The frequency of oscillation when a mass $2\mathrm{m}$ is suspended from the end of one of the divided springs is equal to the initial frequency, $f$. So, the answer is (A) \(\mathrm{f}\).

Step-by-step solution

Expression for frequency of oscillation for mass m

We can first express the frequency of oscillation for a mass m suspended from a spring with force constant k using the formula: \(f = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\)

Force constant of the divided spring

When the spring is divided into two equal parts, its force constant changes. The force constant for each part of the spring when it is divided is given by: \(k' = 2k\) The new force constant, \(k'\), is twice the original force constant.

Expression for frequency of oscillation for mass 2m and divided spring

Now we need to find the frequency of oscillation when a mass 2m is suspended from the end of one of the divided springs having force constant \(k'\). We can use the same formula for frequency, with the new values for mass and force constant: \(f' = \dfrac{1}{2\pi}\sqrt{\dfrac{k'}{2m}}\)

Substitute the new force constant

We can now substitute the new force constant, \(k' = 2k\), into the expression for frequency: \(f' = \dfrac{1}{2\pi}\sqrt{\dfrac{2k}{(2m)}}\)

Simplify and compare with the original frequency

Simplify the expression for the new frequency: \(f' = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\) Comparing this expression with the original frequency expression, we can see that they are equal: \(f' = f\) Therefore, the correct answer is: (A) \(\mathrm{f}\)