Spring Mass System

Two masses are connected by a spring as shown in the figure. One of the masses was given velocity $v=2k$ as shown in figure where $k$ is the spring constant. Then maximum extension in the spring will be

The coefficient of friction between block of mass $m$ and $2\mathrm{m}$ is $\mu =2\tan \left(\theta \right)$. There is no friction between a block of mass $2\mathrm{m}$ and inclined plane. The maximum amplitude of two block system for which there is no relative motion between both the blocks-

In the case of a simple pendulum executing $\mathrm{SHM},$ at $t=0$, the bob is not at the mean position. The graph drawn between the tension $\left(T\right)$ in the string and time $\left(t\right)$ is

The bob of simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob get suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

A body of mass $1 \mathrm{kg}$ is executing simple harmonic motion. Its displacement $y\left(\mathrm{cm}\right)$ at $t$ seconds is given by $y=6\sin \left(100t+\frac{\pi }{4}\right)$. Its maximum kinetic energy is

Among the following displacement-time graphs, the one that represent damped harmonic oscillation is ____

Two simple pendulums of length $5 \mathrm{m}$ and $20 \mathrm{m}$, respectively, are given small linear displacement in one direction at the same time. They will again be in the phase when the pendulum of shorter length has completed ___ oscillations.

The damped harmonic oscillator is represented by the energy-time graph given as

Four massless springs whose force constants are $2k$, $2\mathit{\text{k}}$, $\mathit{\text{k}}$ and $2\mathit{\text{k}}$, respectively, are attached to a mass $M$ kept on a frictionless plane (as shown in figure). If the mass $M$ is displaced in the horizontal direction, then the frequency of the system is, 

Find the natural frequency of oscillation of the system shown in the figure. Pulleys are massless and frictionless. Spring and strings are also massless. 

A body $P$ is suspended from an elastic spring and set into oscillation. Its frequency is found to be $n_1$. Now, after removing the body$P$, a body $Q$ is suspended from the same spring and then set into oscillations. Its frequency is found to be $n_2$. If both $P$ and $Q$ are suspended from the same spring and set into oscillations, then the frequency of the oscillations will be:

A spring mass system (mass $m,$ spring constant $k$ and natural length $l$ ) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system rotates about it's axis with an angular velocity $\omega ,\left(k>>m{\omega }^2\right)$ the relative change in the length of the spring is best given by the option:

A spring whose unstrentches length is $l$ has a force constant $k.$ The spring is cut into two pieces of unstretches lengths $l_1$ and $l_2$ where, $l_1=n{l}_2$ and $n$ is an integer. The ratio $k_1/k_2$ of the corresponding force constants, $k_1$ and $k_2$ will be:

Two light identical springs of spring constant $k$ are attached horizontally at the two ends of a uniform horizontal rod $AB$ of length $l$ and mass $m.$ The rod is pivoted at its center 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

A simple pendulum of length $L$ has mass $M$ and it oscillates freely with amplitude $A$. At the extreme position, its potential energy is ($g$ = acceleration due to gravity)

A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of ${10}^{12} {\mathrm{s}}^{-1}$ . What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver,$=108 \mathrm{g} {\mathrm{mol}}^{-1}$ and Avogadro number $=6.02\times {10}^{23}$)

A $1 \mathrm{kg}$ block attached to a spring vibrates with a frequency of $1 \mathrm{Hz}$ on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to a$8 \mathrm{kg}$block placed on the same table. So, the frequency of vibration of the $8 \mathrm{kg}$ block is 

Two discs of masses $M_1$ and $M_2$ are placed on a frictionless surface. Now, the spring is compressed by small displacement $x$. Then, the time period of the given system is,

A body is in simple harmonic motion with time period  $T=0.5\text{s}$ and amplitude $A=1\text{cm}$. Find the average velocity in the interval in which it moves from equilibrium position to half of its amplitude.