If the function $\sin ^2 \omega t$ ( $t$ is time in second) represents a periodic motion, then the period of the motion is

On a smooth inclined plane a block of mass $M$ is fixed to two rigid supports using two springs as shown in the figure. If each spring has spring constant $k$, then the period of oscillation of the block is

(Neglect the masses of the springs)

A particle is executing simple harmonic motion with amplitude $A$. The ratio of the kinetic energies of the particle when it is at displacements of $\frac{A}{4}$ and $\frac{A}{2}$ from the mean position is

A particle is executing simple harmonic motion starting from its mean position. If the time period of the particle is 1.5 s , then the minimum time at which the ratio of the kinetic and total energies of the particle becomes 3:4 is

The equations for the displacements of two particles in simple harmonic motion are $y_1=0.1 \sin \left(100 \pi t+\frac{\pi}{3}\right)$ and $y_2=0.1 \cos \pi t$ respectively. The phase difference between the velocities of the two particles at a time $t=0$ is

A spring is stretched by 0.2 m when a mass of 0.5 kg is suspended to it. The time period of the spring when 0.5 kg mass is replaced with a mass of 0.25 kg is suspended to it is

(Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A body of mass 4 kg attached to a spring of force constant $64 \mathrm{Nm}^{-1}$ executes simple harmonic motion on a frictionless horizontal surface. The time period of oscillation is

A particle is executing simple harmonic motion with amplitude $A$. At a distance ' $x$ ' from the mean position, when the particle is moving towards extreme position it receives a blow in the direction of motion which instantaneously doubles its velocity. The new amplitude of the particle is

(Frequency is constant during the motion)

If the displacement ' $x$ ' of a body in motion in terms of time ' $t$ ' is given by $x=A \sin (\omega t+\theta)$, then the minimum time at which the displacement becomes maximum is

If the maximum velocity and maximum acceleration of a particle executing simple harmonic motion are respectively $5 \mathrm{~ms}^{-1}$ and $10 \mathrm{~ms}^{-2}$, then the time period of oscillation of the particle is

A body of mass 1 kg is suspended from a spring of force constant $600 \mathrm{Nm}^{-1}$. Another body of mass 0.5 kg moving vertically upwards hits the suspended body with a velocity of $3 \mathrm{~ms}^{-1}$ and embedded in it. The amplitude of motion is

For a particle executing simple harmonic motion, the ratio of kinetic and potential energies at a point where displacement is one half of the amplitude is

When the mass attached to a spring is increased from 4 kg to 9 kg , the time period of oscillation increases by $0.2 \pi \mathrm{~s}$. Then, the spring constant of the spring is

The kinetic energy of a particle executing simple harmonic motion at a displacement of 3 cm from the mean position is 4 mJ . If the amplitude of the particle is 5 cm , then the maximum force acting on the particle is

A body of mass 1 kg is attached to the lower end of a vertically suspended spring of force constant $600 \mathrm{~N}-\mathrm{m}^{-1}$. If another body of mass 0.5 kg moving vertically upward hits the suspended body with a velocity $3 \mathrm{~ms}^{-1}$ and embedded in it, then the frequency of the oscillation is

If the displacement $y$ (in cm ) of a particle executing simple harmonic motion is given by the equation $y=5 \sin (3 \pi t)+5 \sqrt{3} \cos (3 \pi t)$, then the amplitude of the particle is

The angular frequency of a block of mass 0.1 kg oscillating with the help of a spring of force constant $2.5 \mathrm{~N}-\mathrm{m}^{-1}$ is

As shown in the figure, two blocks of masses $m_1$ and $m_2$ are connected to spring of force constant $k$. The blocks are slightly displaced in opposite directions to $x_1, x_2$ distances and released. If the system executes simple harmonic motion, then the frequency of oscillation of the system ( $\omega$ ) is

The displacement of a particle of mass 2 g executing simple harmonic motion is $x=8 \cos \left(50 t+\frac{\pi}{12}\right) \mathrm{m}$, where $t$ is time in second. The maximum kinetic energy of the particle is

Two simple harmonic motions are represented by $y_1=5[\sin 2 \pi t+\sqrt{3} \cos 2 \pi t]$ and $y_2=5 \sin \left[2 \pi t+\frac{\pi}{4}\right]$. The ratio of their amplitudes is

When a mass $m$ is connected individually to the springs $k_1$ and $k_2$, the oscillation frequencies are $v_1$ and $v_2$. If the same mass is attached to the two springs as shown in the figure, the oscillation frequency would be

One bar magnet is in simple harmonic motion with time period $T$ in an earth's magnetic field. If its mass is increased by 9 times the time period becomes

In a spring block system as shown in figure. If the spring constant $k=9 \pi^2 \mathrm{Nm}^{-1}$, then the time period of oscillation is

A 3 kg block is connected as shown in the figure. Spring constants of two springs $k_1$ and $k_2$ are $50 \mathrm{Nm}^{-1}$ and $150 \mathrm{Nm}^{-1}$ respectively. The block is released from rest with the springs unstretched. The acceleration of the block in its lowest position is $\left(g=10 \mathrm{~ms}^{-2}\right)$

A particle is executing simple harmonic motion with an instantaneous displacement $$x=A \sin ^2\left(\omega t-\frac{\pi}{4}\right)$$. The time period of oscillation of the particle is

If the amplitude of a lightly damped oscillator decreases by $$1.5 \%$$ then the mechanical energy of the oscillator lost in each cycle is

A body is executing S.H.M. At a displacement $$x$$ its potential energy is 9 J and at a displacement $$y$$ its potential energy is 16 J . The potential energy at displacement $$(x+y)$$ is

A hydrometer executes simple harmonic motion when it is pushed down vertically in a liquid of density $$\rho$$. If the mass of hydrometer is $$m$$ and the radius of the hydrometer tube is $$r$$, then the time period of oscillation is

An object undergoing simple harmonic motion takes 0.5 s to travel from one point of zero velocity to the next such point. The angular frequency of the motion is

A cone with half the density of water is floating in water as shown in figure. It is depressed down by a small distance $$\delta(\ll< H)$$ and released. The frequency of simple harmonic oscillations of the cone is

In case of a forced vibration, the resonance wave becomes very sharp when the

A particle executing simple harmonic motionalong a straight line with an amplitude A,attains maximum potential energy when itsdisplacement from mean position equals

The bob of a simple pendulum is a sphericalhollow ball filled with water. A plugged holenear the bottom of the oscillating bob getssuddenly unplugged. During observation, tillwater is coming out the time period ofoscillation would

A block of mass $$\mathrm{l} \mathrm{kg}$$ is fastened to a spring of spring constant of $$100 ~\mathrm{Nm}^{-1}$$. The block is pulled to a distance $$x=10 \mathrm{~cm}$$ from its equilibrium position $$(x=0 \mathrm{~cm})$$ on a frictionless surface, from rest at $$t=0$$. The kinetic energy and the potential energy of the block when it is $$5 \mathrm{~cm}$$ away from the mean position is

The scale of a spring balance which can measure from 0 to $$15 \mathrm{~kg}$$ is $$0.25 \mathrm{~m}$$ long. If a body suspended from this balance oscillates with a time period $$\frac{2 \pi}{5} \mathrm{~s}$$, neglecting the mass of the spring, find the mass of the body suspended.

A spring is stretched by 0.40 m when a massof 0.6 kg is suspended from it. The period ofoscillations of the spring loaded by 255 g andput to oscillations is close to (g = 10 ms$$^{-2}$$)

A heavy brass sphere is hung from a springand it executes vertical vibrations with periodT. The sphere is now immersed in anon-viscous liquid with a density (1/10 )ththat of brass. When set into verticalvibrations with the sphere remaining insideliquid all the time, the time period will be