Core Syllabus Module

Oscillations

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JEE Advanced 1991

LEVELJEE Advanced

Two non-viscous, incompressible and immiscible liquids of densities $ρ$ and $1.5 ρ$ are poured into the two limbs of a circular tube of radius $R$ and small cross-section kept fixed in a vertical plane as shown in figure. Each liquid occupies one-fourth the circumference of the tube. (1991) (a) Find the angle $θ$ that the radius to the interface makes with the vertical in equilibrium position. (b) If the whole liquid column is given a small displacement from its equilibrium position, show that the resulting oscillations are simple harmonic. Find the time period of these oscillations.

JEE Advanced 2024

LEVELJEE Advanced

Comprehension Passage

Two particles, 1 and 2, each of mass

m

, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at

x_{0}

, are oscillating with amplitude

a

and angular frequency

ω

. Thus, their positions at time

t

are given by

x_{1} (t) = (x_{0} + d) + a sin ω t

and

x_{2} (t) = (x_{0} - d) - a sin ω t

, respectively, where

d > 2 a

. Particle 3 of mass

m

moves towards this system with speed

u_{0} = aω /2

, and undergoes instantaneous elastic collision with particle 2, at time

t_{0}

. Finally, particles 1 and 2 acquire a center of mass speed

v_{c m}

and oscillate with amplitude

b

and the same angular frequency

ω

Question 1:

If the collision occurs at time $t_{0} = 0$ , the value of $v_{c m} / (aω)$ will be

Question 2:

If the collision occurs at time $t_{0} = π / (2 ω)$ , then the value of $4 b^{2} / a^{2}$ will be

JEE Advanced 2019

LEVELJEE Advanced

A block of mass 2M is attached to a massless spring with spring-constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are $a_{1}$ , $a_{2}$ and $a_{3}$ as shown in figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_{0}$ . Which of the following option(s) is/are correct?[g is the acceleration due to gravity. Neglect friction]

* Multiple Correct Options

(A)

x_{0} = \frac{4 M g}{k}

(B)

When spring achieves an extension of

\frac{x _{0}}{2}

for the first time, the speed of the block connected to the spring is

3 g \frac{M}{5 k}

(C)

a_{2} - a_{1} = a_{1} - a_{3}

(D)

At an extension of

\frac{x _{0}}{4}

of the spring, the magnitude of acceleration of the block connected to the spring is

\frac{3 g}{10}

JEE Advanced 2025

LEVELJEE Advanced

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$ , inside a ring of radius $R > r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $T = \frac{2 π}{ω}$ . The correct expression for $ω$ is ( $g$ is the acceleration due to gravity):

(A)

\frac{2}{3} (\frac{g}{R - r} + \frac{k}{m})

(B)

\frac{2 g}{3 ( R - r )} + \frac{k}{m}

(C)

\frac{1}{6} (\frac{g}{R - r} + \frac{k}{m})

(D)

\frac{1}{4} (\frac{g}{R - r} + \frac{k}{m})

JEE Main 2019

LEVELJEE Advanced

Two masses $m$ and $m /2$ are connected at the two ends of a massless rigid rod of length $l$ . The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of torsional constant $k$ , the restoring torque is $τ = k θ$ for angular displacement $θ$ . If the rod is rotated by $θ_{0}$ and released, the tension in it when it passes through its mean position will be

(A)

\frac{2 k θ _{0}^{2}}{l}

(B)

\frac{k θ _{0}^{2}}{l}

(C)

\frac{3 k θ _{0}^{2}}{l}

(D)

\frac{k θ _{0}^{2}}{2 l}

JEE Main 2019

LEVELJEE Advanced

A simple pendulum of length $1 m$ is oscillating with an angular frequency $10 rad/s$ . The support of the pendulum starts oscillating up and down with a small angular frequency of $1 rad/s$ and an amplitude of $1 0^{- 2} m$ . The relative change in the angular frequency of the pendulum is best given by

(A)

1 rad/s

(B)

1 0^{- 5} rad/s

(C)

1 0^{- 3} rad/s

(D)

1 0^{- 1} rad/s

JEE Advanced 2011

LEVELJEE Advanced

Comprehension Passage

Phase space diagrams are useful tools in analysing all kinds of dynamical problems. They are especially useful in studying the motion where position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is

x (t)

p (t)

curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Question 1:

The phase space diagram for a ball thrown vertically up from ground is

The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_{1}$ and $E_{2}$ are the total mechanical energies respectively. Then,

(A)

E_{1} = 2 E_{2}

(B)

E_{1} = 2 E_{2}

(C)

E_{1} = 4 E_{2}

(D)

E_{1} = 16 E_{2}

Question 3:

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

A particle of mass $m$ is attached to one end of a massless spring of force constant $k$ , lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t = 0$ with an initial velocity $u_{0}$ . When the speed of the particle is $0.5 u_{0}$ , it collides elastically with a rigid wall. After this collision (2013 Adv.)

* Multiple Correct Options

(A)

the speed of the particle when it returns to its equilibrium position is

u_{0}

(B)

the time at which the particle passes through the equilibrium position for the first time is

t = π \frac{m}{k}

(C)

the time at which the maximum compression of the spring occurs is

t = \frac{4 π}{3} \frac{m}{k}

(D)

the time at which the particle passes through the equilibrium position for the second time is

t = \frac{5 π}{3} \frac{m}{k}

JEE Advanced 2018

LEVELJEE Advanced

A spring block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0 Nm^{- 1}$ and the mass of the block is $2.0 kg$ . Ignore the mass of the spring. Initially, the spring is in an unstretched condition. Another block of mass $1.0 kg$ moving with a speed of $2.0 ms^{- 1}$ collides elastically with the first block. The collision is such that the $2.0 kg$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ............ .

JEE Advanced 2011

LEVELJEE Advanced

A metal rod of length $L$ and mass $m$ is pivoted at one end. A thin disc of mass $M$ and radius $R$ ( $R < L$ ) is attached at its centre to the free end of the rod. Consider two ways the disc is attached. Case A—the disc is not free to rotate about its centre and Case B—the disc is free to rotate about its centre. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is/are true?

* Multiple Correct Options

(A)

Restoring torque in case A = Restoring torque in case B

(B)

Restoring torque in case A < Restoring torque in case B

(C)

Angular frequency for case A > Angular frequency for case B

(D)

Angular frequency for case A < Angular frequency for case B

JEE Advanced 2016

LEVELJEE Advanced

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_{0}$ . Consider two cases : (i) when the block is at $x_{0}$ and (ii) when the block is at $x = x_{0} + A$ . In both the cases, a particle with mass $m$ ( $m < M$ ) is softly placed on the block after which they stick to each other. Which of the following statement(s) is (are) true about the motion after the mass $m$ is placed on the mass $M$ ?

* Multiple Correct Options

(A)

The amplitude of oscillation in the first case changes by a factor of

\frac{M}{m + M}

, whereas in the second case it remains unchanged

(B)

The final time period of oscillation in both the cases is same

(C)

The total energy decreases in both the cases

(D)

The instantaneous speed at

x_{0}

of the combined masses decreases in both the cases

JEE Advanced 2007

LEVELJEE Advanced

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II.

List-I

(P)

The object moves on the

x

-axis under a conservative force in such a way that its speed and position satisfy

v = c_{1} c_{2} - x^{2}

, where

c_{1}

and

c_{2}

are positive constants.

(Q)

The object moves on the

x

-axis in such a way that its velocity and its displacement from the origin satisfy

v = - k x

, where

k

is a positive constant.

(R)

The object is attached to one end of a mass-less spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration

a

. The motion of the object is observed from the elevator during the period it maintains this acceleration.

(S)

The object is projected from the earth's surface vertically upwards with a speed

2 \frac{G M _{e}}{R _{e}}

, where

M_{e}

is the mass of the earth and

R_{e}

is the radius of the earth. Neglect forces from objects other than the earth.

List-II

(1)

The object executes a simple harmonic motion.

(2)

The object does not change its direction.

(3)

The kinetic energy of the object keeps on decreasing.

(4)

The object can change its direction only once.

JEE Advanced 1993

LEVELJEE Advanced

Comprehension Passage

Two identical balls

A

and

B

, each of mass

0.1 kg

, are attached to two identical massless springs. The spring-mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius

0.06 m

. Each spring has a natural length of

0.06 π m

and spring constant

0.1 N/m

. Initially, both the balls are displaced by an angle

θ = π /6 rad

with respect to the diameter

P Q

of the circle (as shown in figure) and released from rest.

Question 1:

Calculate the frequency of oscillation of ball B.

Question 2:

Find the speed of ball A when A and B are at the two ends of the diameter PQ.

Question 3:

What is the total energy of the system?

JEE Advanced 1996

LEVELJEE Advanced

A thin rod of length $L$ and uniform cross-section is pivoted at its lowest point $P$ inside a stationary homogeneous and non-viscous liquid. The rod is free to rotate in a vertical plane about a horizontal axis passing through $P$ . The density $d_{1}$ of the material of the rod is smaller than the density $d_{2}$ of the liquid. The rod is displaced by small angle $θ$ from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters.

JEE Advanced 2020

LEVELJEE Advanced

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0, 0)$ . A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $p$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $ℓ$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $Δ ℓ ≪ ℓ$ from its equilibrium position and released, it is found to oscillate at frequency $\frac{1}{δ} \frac{k}{m}$ . The value of $δ$ is _________.

JEE Advanced 2026

LEVELJEE Advanced

A tank contains two immiscible liquids of densities $6 ρ$ and $2 ρ$ . The higher density liquid is filled up to a height $L /2$ from the bottom. A thin rod of density $ρ$ and length $L$ is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is $\frac{2 π}{n} \frac{L}{g}$ , where $g$ is the acceleration due to gravity. The value of $n$ is:

JEE Main 2015

LEVELJEE Advanced

An LCR circuit is equivalent to a damped pendulum. In an LCR circuit, the capacitor is charged to $Q_{0}$ and then connected to the $L$ and $R$ as shown below. If a student plots graphs of the square of maximum charge ( $Q_{Max}^{2}$ ) on the capacitor with time ( $t$ ) for two different values $L_{1}$ and $L_{2}$ ( $L_{1} > L_{2}$ ) of $L$ , then which of the following represents this graph correctly? (plots are schematic and not drawn to scale)

In the reported figure, two bodies A and B of masses $200 g$ and $800 g$ are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be ......... $rad/s$ when $k = 20 N/m$ .

JEE Main 2019

LEVELJEE Advanced

A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations. If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to

Two particles are executing simple harmonic motion of the same amplitude $A$ and frequency $ω$ along the x-axis. Their mean position is separated by distance $X_{0}$ ( $X_{0} > A$ ). If the maximum separation between them is ( $X_{0} + A$ ), the phase difference between their motion is

(A)

\frac{π}{3}

(B)

\frac{π}{4}

(C)

\frac{π}{6}

(D)

\frac{π}{2}

Micro-Topic Strategy Matrix

Simple Harmonic Motion (SHM)

Angular SHM

Force and Energy Method in SHM

Free, Forced, and Damped Oscillations

Chapter Elite-20 Challenge Zone

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the collision occurs at time t0​=0 , the value of vcm​/(aω) will be

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the collision occurs at time t0​=π/(2ω) , then the value of 4b2/a2 will be

Comprehension Passage

The phase space diagram for a ball thrown vertically up from ground is

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II.

List-I

List-II

Comprehension Passage

Calculate the frequency of oscillation of ball B.

Find the speed of ball A when A and B are at the two ends of the diameter PQ.

What is the total energy of the system?

A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations. If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to

If the collision occurs at time $t_{0} = 0$ , the value of $v_{c m} / (aω)$ will be

If the collision occurs at time $t_{0} = π / (2 ω)$ , then the value of $4 b^{2} / a^{2}$ will be