Simple Harmonic Motion

The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If the length of the spring is cut to half, the same mass oscillates with frequency $v_2$. The value of $v_2 / v_1$ is $\_\_\_\_$ .

A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is 5 sec . The time required by it to move from $x=A$ to $x=\frac{A}{\sqrt{2}}$ is $\_\_\_\_$ sec.

A spring stretches by 2 mm when it is loaded with a mass of 200 g . From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maxmimum energy in the spring are $\_\_\_\_$ Hz...

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The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2=50-x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7} \mathrm{~s}$, the value of $x$ is $\_\_\_\_$ .

A uniform disc of radius $R$ and mass $M$ is free to oscillate about the axis $A$ as shown in the figure. For small oscillations the time period is $\_\_\_\_$ .
(g is acceleration due to gravity)

The equation of motion of a particle is given by $x = a \sin(50t + \pi/3)$ cm. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are ________.

The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t)=A \sin \omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t=T / 2 \beta$....

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses $1 \text{ kg}$ and $0.2 \text{ kg}$ with a separation more than spring natural length and are released. Assuming the horizontal surfa...

A spring of force constant $15 \mathrm{~N} / \mathrm{m}$ is cut into two pieces. If the ratio of their length is $1: 3$, then the force constant of smaller piece is $\_\_\_\_$ $\mathrm{N} / \mathrm{m}$.

A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is $\_\_\_\_$

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . The length of the string required for the pendulum to perform 40 oscillations in the same time duration is
$\_\_\_\_$ cm . [Assume that the mass of the pendulum rem...

Using a simple pendulum experiment $g$ is determind by measuring its time period $T$. Which of the following plots represent the correct relation between the pendulum length $L$ and time period $T$ ?

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/s. The frequency of this simple harmonic oscillator is ______ Hz. [ take $\pi = \frac{22}{7}$ ]

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is 200 N/m. The blo...

Two simple pendulums having lengths $l_1$ and $l_2$ with negligible string mass undergo angular displacements $\theta_1$ and $\theta_2$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then ...

Two blocks of masses $m$ and $M,(M>m)$, are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then ( $\mu=$ coeffi...

A particle is subjected to two simple harmonic motions as :

\(x_1=\sqrt{7} \sin 5 \mathrm{tcm}\)

and $x_2=2 \sqrt{7} \sin \left(5 t+\frac{\pi}{3}\right) \mathrm{cm}$
where $x$ is displacement and $t$ is time in seconds.
The maximum acce...

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.
Reason (R) ...

Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1 and k2, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maxi...

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Knowing initial position $\mathrm{x}_0$ and initial momentum $p_0$ is enough to determine the position and momentum at...

A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then $\frac{\mathrm{D}}{\mathrm{d}}$ is

A particle oscillates along the $x$-axis according to the law, $x(\mathrm{t})=x_0 \sin ^2\left(\frac{\mathrm{t}}{2}\right)$ where $x_0=1 \mathrm{~m}$. The kinetic energy $(\mathrm{K})$ of the particle as a function of $x$ is correctly repre...

A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y \pi \times 10^{-2} \mathrm{~s}$, where the value of...

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time...

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of \(4 \mathrm{~m}, 2 \mathrm{~ms}^{-1}\) and \(16 \mathrm{~ms}^{-2}\) at a certain instant. The amplitude of the motion is ...

A particle of mass \(0.50 \mathrm{~kg}\) executes simple harmonic motion under force \(F=-50(\mathrm{Nm}^{-1}) x\). The time period of oscillation is \(\frac{x}{35} s\). The value of \(x\) is _________.
(Given \(\pi=\frac{22}{7}\))

An object of mass \(0.2 \mathrm{~kg}\) executes simple harmonic motion along \(x\) axis with frequency of \(\left(\frac{25}{\pi}\right) \mathrm{Hz}\). At the position \(x=0.04 \mathrm{~m}\) the object has kinetic energy \(0.5 \mathrm{~J}\) ...

A particle is doing simple harmonic motion of amplitude \(0.06 \mathrm{~m}\) and time period \(3.14 \mathrm{~s}\). The maximum velocity of the particle is _________ \(\mathrm{cm} / \mathrm{s}\).

A simple pendulum doing small oscillations at a place \(R\) height above earth surface has time period of \(T_1=4 \mathrm{~s}\). \(\mathrm{T}_2\) would be it's time period if it is brought to a point which is at a height \(2 \mathrm{R}\) fr...

In simple harmonic motion, the total mechanical energy of given system is \(E\). If mass of oscillating particle \(P\) is doubled then the new energy of the system for same amplitude is:

The displacement of a particle executing SHM is given by \(x=10 \sin \left(w t+\frac{\pi}{3}\right) m\). The time period of motion is \(3.14 \mathrm{~s}\). The velocity of the particle at \(t=0\) is _______ \(\mathrm{m} / \mathrm{s}\).

A particle performs simple harmonic motion with amplitude \(A\). Its speed is increased to three times at an instant when its displacement is \(\frac{2 A}{3}\). The new amplitude of motion is \(\frac{n A}{3}\). The value of \(n\) is _______...

The time period of simple harmonic motion of mass \(M\) in the given figure is \(\pi \sqrt{\frac{\alpha M}{5 k}}\), where the value of \(\alpha\) is _________.

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is \(\frac{x}{8}\), where \(x=\) _________.

A simple harmonic oscillator has an amplitude \(A\) and time period \(6 \pi\) second. Assuming the oscillation starts from its mean position, the time required by it to travel from \(x=\) A to \(x=\frac{\sqrt{3}}{2}\) A will be $$\frac{\pi}...

The bob of a pendulum was released from a horizontal position. The length of the pendulum is \(10 \mathrm{~m}\). If it dissipates \(10 \%\) of its initial energy against air resistance, the speed with which the bob arrives at the lowest poi...

A particle executes simple harmonic motion with an amplitude of \(4 \mathrm{~cm}\). At the mean position, velocity of the particle is \(10 \mathrm{~cm} / \mathrm{s}\). The distance of the particle from the mean position when its speed becom...

A simple pendulum of length $1 \mathrm{~m}$ has a wooden bob of mass $1 \mathrm{~kg}$. It is struck by a bullet of mass $10^{-2} \mathrm{~kg}$ moving with a speed of $2 \times 10^2 \mathrm{~ms}^{-1}$. The bullet gets embedded into the bob. ...

A mass $m$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9 \mathrm{~m}$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2} \ma...

For particle P revolving round the centre O with radius of circular path \(\mathrm{r}\) and angular velocity \(\omega\), as shown in below figure, the projection of OP on the \(x\)-axis at time \(t\) is

A mass \(m\) is attached to two strings as shown in figure. The spring constants of two springs are \(\mathrm{K}_{1}\) and \(\mathrm{K}_{2}\). For the frictionless surface, the time period of oscillation of mass \(m\) is :

A simple pendulum with length \(100 \mathrm{~cm}\) and bob of mass \(250 \mathrm{~g}\) is executing S.H.M. of amplitude \(10 \mathrm{~cm}\). The maximum tension in the string is found to be \(\frac{x}{40} \mathrm{~N}\). The value of \(x\) i...

In the figure given below, a block of mass \(M=490 \mathrm{~g}\) placed on a frictionless table is connected with two springs having same spring constant \(\left(\mathrm{K}=2 \mathrm{~N} \mathrm{~m}^{-1}\right)\). If the block is horizontal...

The maximum potential energy of a block executing simple harmonic motion is \(25 \mathrm{~J}\). A is amplitude of oscillation. At \(\mathrm{A / 2}\), the kinetic energy of the block is

The general displacement of a simple harmonic oscillator is \(x = A\sin \omega t\). Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when \(t = {T \over \beta }\). The value of \(\beta\) is __...

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

For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is $1 \mathrm{~kg}$, the angular frequency is $\omega_{1}$. When the mass block is $2 \mathrm{~kg}$ the angular frequency is...

A particle of mass 250 g executes a simple harmonic motion under a periodic force \(\mathrm{F}=(-25~x)\mathrm{N}\). The particle attains a maximum speed of 4 m/s during its oscillation. The amplitude of the motion is ___________ cm.

T is the time period of simple pendulum on the earth's surface. Its time period becomes \(x\) T when taken to a height R (equal to earth's radius) above the earth's surface. Then, the value of \(x\) will be :