A mass M is suspended from a spring of negligible mass. The...

A mass $M$ is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period $T$. If the mass is increased by $m$, the time period becomes $\frac{5 T}{3}$. Then the radio of $\frac{m}{M}$ is [2003] (A) $\frac{3}{5}$ (B) $\frac{25}{9}$ (C) $\frac{16}{9}$ (D) $\frac{5}{3}$

A mass $M$ is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period $T$. If the mass is increased by $m$, the time period becomes $\frac{5 T}{3}$. Then the radio of $\frac{m}{M}$ is [2003]
(A) $\frac{3}{5}$
(B) $\frac{25}{9}$
(C) $\frac{16}{9}$
(D) $\frac{5}{3}$

Key Concepts

Scaling Relationships in Oscillatory Systems

Scaling relationships in oscillatory systems involve understanding how modifications to system parameters, such as mass or spring constant, influence the time period of oscillation. In the context of a mass-spring oscillator, increasing the mass results in a longer oscillation period by a factor that is proportional to the square root of the increase in mass, demonstrating a direct mathematical relationship between mass addition and the change in time period.

Time Period Dependency

For a mass-spring system executing simple harmonic motion, the time period (T) is given by the formula T = 2??(M/k), where M is the mass and k is the spring constant. This relationship highlights that the time period increases as the square root of the mass, meaning any change in mass directly affects the oscillation period.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position. It is characterized by sinusoidal oscillations and is governed by the principles of energy conservation, with kinetic and potential energy continuously interchanging during oscillation.

Mass-Spring Oscillator

A mass-spring oscillator is a classic example of SHM where a mass is attached to a spring that obeys Hooke's law. The motion of the system is described by the force equation F = -kx, where k is the spring constant and x is the displacement. The system oscillates about the equilibrium position with a time period determined by the mass and the spring constant.

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