Question

A block having mass \(\mathrm{M}\) is placed on a horizontal frictionless surface. This mass is attached to one end of a spring having force constant \(\mathrm{k}\). The other end of the spring is attached to a rigid wall. This system consisting of spring and mass \(\mathrm{M}\) is executing SHM with amplitude \(\mathrm{A}\) and frequency \(\mathrm{f}\). When the block is passing through the mid-point of its path of motion, a body of mass \(\mathrm{m}\) is placed on top of it, as a result of which its amplitude and frequency changes to \(\mathrm{A}^{\prime}\) and \(\mathrm{f}\). The ratio of frequencies \((\mathrm{f} / \mathrm{f})=\ldots \ldots \ldots\) (A) \(\sqrt{\\{} \mathrm{M} /(\mathrm{m}+\mathrm{M})\\}\) (B) \(\sqrt{\\{\mathrm{m} /(\mathrm{m}+\mathrm{M})\\}}\) (C) \(\sqrt{\\{\mathrm{MA} / \mathrm{mA}}\\}\) (D) \(\sqrt{[}\\{(\mathrm{M}+\mathrm{m}) \mathrm{A}\\} / \mathrm{mA}]\)

Short answer

None of the given options are correct. The ratio of the initial frequency to the final frequency is: \(\frac{f}{f'} = \sqrt{1 + \frac{m}{M}}\)

Step-by-step solution

Frequency of Simple Harmonic Motion

The frequency of a mass-spring system undergoing simple harmonic motion is given by the following formula: \(f = \frac{1}{2 \pi} \sqrt{\frac{k}{M}}\) Initially, the frequency of the block with mass M is: \(f = \frac{1}{2 \pi} \sqrt{\frac{k}{M}}\)

Frequency of the Combined System

When mass m is added to the system, the total mass becomes (M + m), and the frequency of the combined system is given by: \(f' = \frac{1}{2 \pi} \sqrt{\frac{k}{M + m}}\)

Calculate the Ratio of Frequencies

Now, we want to find the ratio of the initial frequency (f) to the final frequency (f'): \(\frac{f}{f'} = \frac{\frac{1}{2 \pi} \sqrt{\frac{k}{M}}}{\frac{1}{2 \pi} \sqrt{\frac{k}{M + m}}}\)

Simplify the Ratio

We can now simplify this expression, canceling out common terms: \(\frac{f}{f'} = \frac{\sqrt{\frac{k}{M}}}{\sqrt{\frac{k}{M + m}}}\) Dividing two square roots is the same as taking the square root of the ratio: \(\frac{f}{f'} = \sqrt{\frac{\frac{k}{M}}{\frac{k}{M + m}}}\) Cancel out k in the numerator and denominator: \(\frac{f}{f'} = \sqrt{\frac{M + m}{M}}\) Which is equivalent to: \(\frac{f}{f'} = \sqrt{1 + \frac{m}{M}}\)

Choosing the Correct Option

Comparing this to the given options, we find that none of the options exactly match our final answer. It's important to double check our working and ensure we didn't miss any steps, but assuming we followed the correct process, it looks like there could be an error in the question or the given options.