Simple Harmonic Motion

A particle moves with simple harmonic motion in a straight line. In first \(\tau s,\) after starting from rest it travels a distance \(a,\) and in next \(\tau s\) it travels \(2a,\) in same direction, then:

The amplitude of a damped oscillator decreases to \(0.9\) times its original magnitude in \(5s\). In another \(10s\) it will decrease to \(\alpha\) times its original magnitude, where \(\alpha\) equals

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}$...

If a simple pendulum has significant amplitude (up to a factor of \(1/e\) of original ) only in the period between \(t = 0s\,\,to\,\,t = \tau \,s,\) then \(\tau \,\) may be called the average life of the pendulum When the spherical bob of t...

A mass \(M,\) attached to a horizontal spring, executes \(S.H.M.\) with amplitude \({A_1}.\) When the mass \(M\) passes through its mean position then a smaller mass \(m\) is placed over it and both of them move together with amplitude $${A...

Two particles are executing simple harmonic motion of the same amplitude \(A\) and frequency \(\omega\) along the \(x\)-axis. Their mean position is separated by distance \({X_0}\left( {{X_0} > A} \right)\). If the maximum separation betwe...

If \(x,\) \(v\) and \(a\) denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period \(T,\) then, which of the following does not change with time?

A particle of mass \(m\) executes simple harmonic motion with amplitude a and frequency \(v.\) The average kinetic energy during its motion from the position of equilibrium to the end is

A point mass oscillates along the \(x\)-axis according to the law \(x = {x_0}\,\cos \left( {\omega t - \pi /4} \right).\) If the acceleration of the particle is written as \(a = A\,\cos \left( {\omega t + \delta } \right),\) then

Two springs, of force constant \({k_1}\) and \({k_2}\) are connected to a mass \(m\) as shown. The frequency of oscillation of the mass is \(f.\) If both \({k_1}\) and \({k_2}\) are made four times their original values, the frequency of os...

The displacement of an object attached to a spring and executing simple harmonic motion is given by \(x = 2 \times {10^{ - 2}}\) \(cos\) \(\pi t\) metre. The time at which the maximum speed first occurs is

Starting from the origin a body oscillates simple harmonically with a period of \(2\) \(s.\) After what time will its kinetic energy be \(75\%\) of the total energy?

The maximum velocity of a particle, executing simple harmonic motion with an amplitude \(7\) \(mm,\) is \(4.4\) \(m/s.\) The period of oscillation is

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motoin of angular frequency \(\omega .\) The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first tim...

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

The function \({\sin ^2}\left( {\omega t} \right)\) represents

Two simple harmonic motions are represented by the equations \({y_1} = 0.1\,\sin \left( {100\pi t + {\pi \over 3}} \right)\) and \({y_2} = 0.1\,\cos \,\pi t.\) The phase difference of the velocity of particle \(1\) with respect to the velo...

If a simple harmonic motion is represented by \({{{d^2}x} \over {d{t^2}}} + \alpha x = 0.\) its time period is

A particle at the end of a spring executes \(S.H.M\) with a period \({t_1}\). While the corresponding period for another spring is \({t_2}\). If the period of oscillation with the two springs in series is \(T\) then

In forced oscillation of a particle the amplitude is maximum for a frequency \({\omega _1}\) of the force while the energy is maximum for a frequency \({\omega _2}\) of the force; then

A particle of mass \(m\) is attached to a spring (of spring constant \(k\)) and has a natural angular frequency \({\omega _0}.\) An external force \(F(t)\) proportional to \(\cos \,\omega t\left( {\omega \ne {\omega _0}} \right)\) is appli...

The bob of a simple pendulum executes simple harmonic motion in water with a period \(t,\) while the period of oscillation of the bob is \({t_0}\) in air. Neglecting frictional force of water and given that the density of the bob is $$\left...

The total energy of particle, executing simple harmonic motion is

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 $${{5T} \over 3}$...

The length of a simple pendulum executing simple harmonic motion is increased by \(21\%\). The percentage increase in the time period of the pendulum of increased length is

A body executes simple harmonic motion. The potential energy \((P.E),\) the kinetic energy \((K.E)\) and total energy \((T.E)\) are measured as a function of displacement \(x.\) Which of the following statements is true ?

Two particles \(A\) and \(B\) of equal masses are suspended from two massless springs of spring of spring constant \({k_1}\) and \({k_2}\), respectively. If the maximum velocities, during oscillation, are equal, the ratio of amplitude of $...

The displacement of particle varies according to the relation
\(x=4\)\(\left( {\cos \,\pi t + \sin \,\pi t} \right).\) The amplitude of the particle is

If a spring has time period \(T,\) and is cut into \(n\) equal parts, then the time period of each part will be

A child swinging on a swing in sitting position, stands up, then the time period of the swing will

In a simple harmonic oscillator, at the mean position