Simple Harmonic Motion

Two oscillating simple pendulums with time periods $T$ and $\dfrac{4T}{3}$ are in phase at a given time. They will be again in phase after an elapse of time

Two springs of force constants '$2k$' and '$k$' are connected to a mass '$m$' as shown. Mass is displaced slightly to one side and released. The frequency of oscillation of the two springs-mass system is

A simple pendulum of length $l_1$ has periodic time $2.4$ s. Another simple pendulum of length $l_2 < l_1$, has periodic time $1.8$ s. The periodic time of simple pendulum of length $(l_1 - l_2)$ is nearly

A particle is performing S.H.M. about $x = 0$, with an amplitude $a$ and time period $T$. The speed of the particle at $x = \dfrac{a}{3}$ will be

For a particle executing S.H.M., the potential energy is $n$ times the kinetic energy when its displacement from mean position is $\left(\dfrac{2\sqrt{2}}{3}\right)A$, where $A$ is the amplitude of S.H.M. The value of $n$ is

The displacement of a particle varies with time according to the relation$x = a\sin\omega t + b\cos\omega t$

A particle executes a simple harmonic motion with a periodic time $8$ second. At time $t = 0$, it is at a mean position. The ratio of the distance traveled by a particle in the 2nd and that in the 1st second of its motion is($\sin 45^\circ ...

A particle executing linear S.H.M. has velocities $V_1$ and $V_2$ at distance $x_1$ and $x_2$ respectively, from the mean position, its angular velocity is

A spring force constant $180$ N/m is loaded with a mass $0.2$ kg. The amplitude of oscillations is $4$ cm. When mass comes to equilibrium position, its velocity is

A particle is performing simple harmonic motion about $x = 0$ with an amplitude '$a$' and periodic time T. The speed of the particle at $x = \dfrac{a}{3}$ will be

All the springs in fig (a), (b) and (c) are identical, each one having force constant K. Mass m is attached to each system. If $T_a$, $T_b$ and $T_c$ are the periodic time of oscillations of the three systems in fig (a), (b) and (c) respect...

A particle of mass '$m$' is executing S.H.M. about the origin on x-axis with frequency $\sqrt{\dfrac{Ka}{\pi m}}$, where K is a constant and a is the amplitude of S.H.M. If '$x$' is the displacement of a particle at time '$t$', the potentia...

Two particles A and B of equal masses are suspended from two massless springs of spring constants $K_A$ and $K_B$ respectively. The maximum velocities of the particles during oscillations are equal. The ratio of the amplitude of 'B' to that...

Three masses 100g, 300g and 500g are suspended at the end of a spring as shown in figure and are in equilibrium. When the 500 g mass is removed, the system oscillates with a period of 3 second. When the 300 g mass is also removed, it will o...

A particle executing simple harmonic motion starts from mean position with amplitude '$A$' and periodic time '$T$'. At what displacement is its speed one-fourth of the maximum speed?

For a particle executing S.H.M., its potential energy is 8 times its kinetic energy at a certain displacement '$x$' from the mean position. If '$A$' is the amplitude of S.H.M., the value of '$x$' is

Two simple pendulums of lengths $L_1$ and $L_2$ have periodic time $T_1$ and $T_2$ respectively $(T_1 > T_2)$. The time period of the pendulum of length $(L_1-L_2)$ is$[(L_1-L_2) > 60\text{ cm}]$

For a particle performing linear S.H.M. of amplitude 'r', the potential energy is '$\lambda$' times its total energy. The displacement of particle is

A light spring is suspended with mass '$m_1$' at its lower end and its upper end is fixed to a rigid support. The mass is pulled down a short distance and then released. The period of oscillation is T second. When a mass '$m_2$' is added to...

The displacement of a particle performing linear S.H.M. is given by $y = A\cos[\pi(t + \phi)]$. If at $t = 0$, the displacement is $y = 2$ cm and velocity is $2\pi$ cm/s, the value of amplitude A in cm is

For a particle P performing S.H.M., when displacement is 'x', potential energy and restoring force acting on it is denoted by 'E' and 'F' respectively. The relation between x, E and F is

A rectangular block of mass 'm' and cross-sectional area 'A' floats on a liquid of density '$\varrho$'. It is given a small vertical displacement from equilibrium, it starts oscillating with frequency (g=acceleration due to gravity)

The maximum potential energy of a block executing S.H.M. is E and amplitude of oscillation is 'A'. The kinetic energy of the block at amplitude A/2 is

A mass $m_1$ performs S.H.M. with amplitude A which is connected to horizontal spring. While mass $m_1$ passing through mean position, another mass $m_2$ ($m_2 < m_1$) is placed on it so that both masses move together with amplitude $A_1$. ...

A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is $1.6$ m. The period of Oscillation of the sphere is (Acceleration due to gravity $g = 10 \text{ m/s}^2$)

A particle starts from mean position and performs S.H.M. with period $6$ second. At what time its kinetic energy is $50\%$ of total energy? ($\cos 45^\circ = 1/\sqrt{2}$)

A particle executes linear S.H.M. with amplitude $4$ cm. The magnitude of velocity and acceleration is equal when it is at $3$ cm from mean position. Time period is

A pendulum clock is running slow, In order to correct it, we should

The minimum phase difference between two simple harmonic motions is$x_1 = \dfrac{1}{\sqrt{2}} \sin \omega t + \dfrac{1}{\sqrt{2}} \cos \omega t$$x_2 = \sin \omega t + \cos \omega t$     $\left[ \sin \dfrac{\pi}{4} = \cos \dfrac{\pi}{4} = \d...

A body of mass $0.4$ kg performs simple harmonic motion. It experiences a restoring force of $0.4$ N when its displacement from the mean position is 4 cm. The force constant and magnitude of acceleration respectively are

A spring executes S.H.M. with mass 10 kg attached to it. The force constant of spring is 10 N/m. If at any instant its velocity is 40 cm/s, the displacement at that instant is(Amplitude of S.H.M. is 0.5 m)

The velocity of a particle executing SHM varies with displacement ($x$) as $4v^2 = 50 - x^2$.The time period of oscillations is $x/7$ s.The value of $x$ is (Take $\pi = 22/7$)

The amplitude of a particle executing SHM is $3$ cm. The displacement at which its kinetic energy will be $25\%$ more than the potential energy is (in cm)

A block is fastened to a horizontal spring. The block is pulled to a distance $x = 10$ cm from its equilibrium position (at $x = 0$) on a frictionless surface from rest. The kinetic energy of the block at $x = 5$ cm is $0.25$ J. The spring ...

There is a body having mass $m$ and performing SHM with amplitude '$a$'. There is a restoring force $F=-Kx$, where $x$ is the displacement. The total energy of the body depends upon which of the following?

A particle executes simple harmonic motion between $x = -A$ and $x = +A$. If time taken by particle to go from $x = 0$ to $\dfrac{A}{2}$ is $2$s, then time taken by particle in going from $x = \dfrac{A}{2}$ to A is

Two massless springs of spring constants $K_1$ and $K_2$ are connected one after the other forming a single chain, suspended vertically and a certain mass is attached to the free end. If $x_1$ and $x_2$ are their respective extensions and '...

A mass attached to a spring performs S.H.M. whose displacement is $x = 3 \times 10^{-3} \cos 2\pi t$ m. The time taken to obtain maximum speed for the first time is (in s)

A mass $0.4$ kg performs S.H.M. with a frequency $\dfrac{16}{\pi}$ Hz. At a certain displacement it has kinetic energy $2$ J and potential energy $1.2$ J. The amplitude of oscillation is

An oscillating pendulum suspended from the roof of a lift which is at rest has time period $T_1$. When lift moves up with acceleration 'a' its time period is $T_2$. When lift moves down with acceleration 'a' its time period is $T_3$. The re...

All the springs in fig. (a), (b) and (c) are identical, each having force constant K each. Mass m is attached to each system. If $\mathrm{T}_a, \mathrm{~T}_b$ and $\mathrm{T}_{\mathrm{c}}$ are the time periods of oscillations of the three s...

A point particle of mass 200 gram is executing S.H.M. of amplitude 0.2 m . When the particle passes through the mean position, its kinetic energy is $16 \times 10^{-3} \mathrm{~J}$. The equation of motion of this particle is (Initial phase ...

A simple pendulum starts oscillating simple harmonically from its mean position ( $\mathrm{x}=0$ ) with amplitude ' $a$ ' and periodic time ' $T$ '. The magnitude of velocity of pendulum at $x=\frac{a}{2}$ is

A spring executes S.H.M. with mass 1 kg attached to it. The force constant of the spring is $4 \mathrm{~N} / \mathrm{m}$. If at any instant its velocity is $20 \mathrm{~cm} / \mathrm{s}$, the displacement at that instant is (Amplitude of S....

The ratio of the frequencies of two simple pendulums is $4: 3$ at the same place. The ratio of their respective lengths is

At a place, the length of the oscillating simple pendulum is made $\frac{1}{4}$ times keeping amplitude same then the total energy will be

A mass ' $M$ ' attached to a horizontal spring executes S.H.M. of 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 $\mathrm{A}_2$....

A particle is performing S.H.M. starting from extreme position. Graphical representation shows that between displacement and acceleration, there is a phase difference of

A simple pendulum is suspended from ceiling of a lift when lift is at rest its period is ' T '. With what acceleration ' $a$ ' should lift be accelerated upward in order to reduce the period to ' $T/2$ '? (take ' g ' as acceleration due to ...

A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is 1.6 m . The period of oscillation of the sphere is (acceleration due to gravity $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )