Chittaranjan Dasgupta and Asok Kumar Das Solutions for Chapter: Oscillations, Exercise 1: EXERCISE

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$. The piston and the cylinder have equal cross-sectional area $A$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $p_0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surroundings, the piston executes a simple harmonic motion with the frequency

A particle of mass in is attached to one end of a mass-less spring of force constant $k$, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial velocity $u_0$. When the speed of the particle is $0.5 u_0$, it collides elastically with a rigid wall. After this collision

What is the phase difference between two simple harmonic motions represented by $x_1=A \sin \left(\omega t+\frac{\pi }{6}\right)$ and $x=A \cos \omega t$?

Two simple harmonic motions are given by $x_1=a \sin \omega t+a \cos \omega t$ and $x=a \sin \omega t+\frac{a}{\sqrt{3}} \cos \omega t$. The ratio of amplitudes of first and second motion and the phase difference between them are respectively $n$ is

When a particle executing shm oscillates with a frequency $v$, then the kinetic energy of the particle

The displacement of a particle in a periodic motion is given by $y=4{\cos }^2 \left(\frac{t}{2}\right)\sin 1000t$. This displacement may be considered as the result of superposition of $n$ is

A particle moves with simple harmonic motion in a straight line. In first $\tau \mathrm{s}$ after starting from rest it travels a distance $a$ and in the next $\tau \mathrm{s}$ it travels $2a$ in the same direction, then

A small mass $m$ attached to one end of a spring with a negligible mass and unstretched length $L$, executes vertical oscillations with angular frequency ${\omega }_0$. When the mass is rotated with an angular speed $\omega$ by holding the other end of the spring at a fixed point, the mass moves uniformly in a circular path in a horizontal plane. Then the increase in length of the spring during this rotation is