One end of a spring of force constant k is fixed to a...

One end of a spring of force constant $k$ is fixed to a vertical wall and the other to a block of mass $m$ resting on a smooth horizontal surface. There is another wall at a distance $x_{0}$ from the block. The spring is then compressed by $2 x_{0}$ and then released. The time taken to strike the wall iv (a) $\frac{1}{6} \pi \sqrt{\frac{k}{m}}$ (b) $\sqrt{\frac{k}{m}}$ (c) $\frac{2 \pi}{3} \sqrt{\frac{m}{k}}$ (d) $\frac{\pi}{4} \sqrt{\frac{k}{m}}$

One end of a spring of force constant $k$ is fixed to a vertical wall and the other to a block of mass $m$ resting on a smooth horizontal surface. There is another wall at a distance $x_{0}$ from the block. The spring is then compressed by $2 x_{0}$ and then released. The time taken to strike the wall iv
(a) $\frac{1}{6} \pi \sqrt{\frac{k}{m}}$
(b) $\sqrt{\frac{k}{m}}$
(c) $\frac{2 \pi}{3} \sqrt{\frac{m}{k}}$
(d) $\frac{\pi}{4} \sqrt{\frac{k}{m}}$

Key Concepts

Phase of Motion in SHM

The phase of a simple harmonic oscillator describes the state of its sinusoidal motion at any given time and is determined by the initial displacement and velocity. It is key to analyzing the motion over specific intervals of a cycle, such as finding the time taken to move from one point in the oscillatory path to another.

Oscillation Period and Angular Frequency

In a mass-spring system, the angular frequency is defined as ? = ?(k/m), and the period of oscillation is then given by T = 2??(m/k). Understanding these relationships allows one to calculate the time intervals for different segments of the oscillation, crucial for determining durations such as the time taken to reach a barrier during the motion.

Simple Harmonic Motion

This is the type of motion exhibited by a mass-spring system where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. The motion is periodic and can be described mathematically with sine or cosine functions depending on the initial conditions.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position, expressed as F = -kx. This law forms the basis of analyzing the dynamics of spring-mass systems and is crucial for establishing the restoring force that leads to oscillatory motion.

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