Oscillations — JEE Main PYQs

The equation of motion of a particle is given by $x = a\sin\left(50t + \frac{\pi}{3}\right)$ 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 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 $\frac{x}{7}$ 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)

[Figure: A uniform disc of mass M and radius R hanging from a horizontal axis A that is tangent to the disc at its topmost point; the disc oscillates about this axis.]

Match List - I with List - II.

List - I | List - II

A. $\sin^2 \omega t$ — I. Periodic with time period $T = \dfrac{\pi}{\omega}$ but not simple harmonic motion (SHM)

B. $\sin^3 (2\omega t)$ — II. Periodic with time period $T = \dfrac{2\pi}{\omega}$ but Not SHM

C. $\sin(\omega t) + \cos(\pi \omega t)$ — III. Periodic with time period $T = \dfrac{\pi}{\omega}$ and SHM

D. $\cos \omega t + \cos 2\omega t$ — IV. Non-periodic

Choose the correct answer from the options given below :

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 maximum energy in the spring are __________ Hz and ________ J, respectively. (Take $g = 10$ m/s$^2$)

The frequency of oscillation of a mass $m$ suspended by a spring is $\nu_1$. If the length of the spring is cut to half, the same mass oscillates with frequency $\nu_2$. The value of $\nu_2/\nu_1$ is ________.

A particle is subjected to two simple harmonic motions as :

$x_1=\sqrt{7}\sin 5t~\mathrm{cm}$

and $x_2=2\sqrt{7}\sin\left(5t+\frac{\pi}{3}\right)\mathrm{cm}$

where $x$ is displacement and $t$ is time in seconds. The maximum acceleration of the particle is $x\times10^{-2}~\mathrm{ms^{-2}}$. The value of $x$ is :

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

($\mu$ = coefficient of friction between the two blocks)

[Figure: block M on a frictionless table connected to a wall by a spring of constant k; smaller block m rests on top of M with friction coefficient $\mu$ between them]

A. The time period of small oscillation of the two blocks is $T = 2\pi\sqrt{\frac{(m+M)}{k}}$

B. The acceleration of the blocks is $a = -\frac{kx}{M+m}$ ($x$ = displacement of the blocks from the mean position)

C. The magnitude of the frictional force on the upper block is $\frac{m\mu|x|}{M+m}$

D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu(M+m)g}{k}$

E. Maximum frictional force can be $\mu(M + m)g$.

Choose the correct answer from the options given below:

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 period of the pendulum remains same on earth and the planet.

Reason (R) : The mass of the pendulum remains unchanged at Earth and the other planet.

In the light of the above statements, choose the correct answer from the options given below :

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

(Acceleration due to gravity, $g = 10\ \mathrm{m/s^2}$, density of water $= 10^3\ \mathrm{kg/m^3}$)

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 $\frac{D}{d}$ is

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

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Knowing initial position $x_0$ and initial momentum $p_0$ is enough to determine the position and momentum at any time t for a simple harmonic motion with a given angular frequency $\omega$.

Reason (R) : The amplitude and phase can be expressed in terms of $x_0$ and $p_0$.

In the light of the above statements, choose the correct answer from the options given below :

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:

[Figure: a spring of stiffness $k$ hanging from a ceiling with a block of mass $m$ (particle $P$) attached at its lower end]

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

A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_1 = 4$ s. $T_2$ would be it's time period if it is brought to a point which is at a height 2R from earth surface. Choose the correct relation [R = radius of earth] :

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is _________ cm/s.

An object of mass 0.2 kg executes simple harmonic motion along x axis with frequency of $\left(\frac{25}{\pi}\right)$ Hz. At the position $x = 0.04$ m the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is _________ cm.

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

A particle of mass 0.50 $kg$ 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 $\pi = \frac{22}{7}$)

If R is the radius of the earth and the acceleration due to gravity on the surface of earth is $g = \pi^2\ \mathrm{m/s^2}$, then the length of the second's pendulum at a height h = 2R from the surface of earth will be, :

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 m is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2}$ is ________.

A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is $\sqrt{\alpha}$ cm, where $\alpha =$ ________.

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 pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is $4m$, then the time period of small oscillations will be _______ $s$. [take $g = \pi^2\ ms^{-2}$]

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

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

[Figure: mass M suspended from the ceiling by a combination of springs — two springs of constant k in parallel, in series with another spring of constant k, all in parallel with a spring of constant k]