Core Syllabus Module

Laws of Motion

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JEE Advanced 1978

LEVELJEE Advanced

A rigid insulated wire frame in the form of a right angled triangle $A B C$ , is set in a vertical plane as shown in figure. Two beads of equal masses $m$ each and carrying charges $q_{1}$ and $q_{2}$ are connected by a cord of length $l$ and can slide without friction on the wires. Considering the case when the beads are stationary determine (a) (i) The angle $α$ (ii) The tension in the cord (iii) The normal reaction on the beads (b) If the cord is now cut what are the value of the charges for which the beads continue to remain stationary?

JEE Advanced 2016

LEVELJEE Advanced

Comprehension Passage

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity

ω

is an example of a non-inertial frame of reference. The relationship between the force

F_{rot}

experienced by a particle of mass

m

moving on the rotating disc and the force

F_{in}

experienced by the particle in an inertial frame of reference is,

F_{rot} = F_{in} + 2 m (v_{rot} \times ω) + m (ω \times r) \times ω

, where,

v_{rot}

is the velocity of the particle in the rotating frame of reference and

r

is the position vector of the particle with respect to the centre of the disc. Now, consider a smooth slot along a diameter of a disc of radius

R

rotating counter-clockwise with a constant angular speed

ω

about its vertical axis through its centre. We assign a coordinate system with the origin at the centre of the disc, the

X

-axis along the slot, the

Y

-axis perpendicular to the slot and the

z

-axis along the rotation axis (

ω = ω \hat{k}

). A small block of mass

m

is gently placed in the slot at

r = (R /2) \hat{i}

t = 0

and is constrained to move only along the slot.

Question 1:

LEVELJEE Advanced

In the figure masses $m_{1}$ , $m_{2}$ and $M$ are $20 kg$ , $5 kg$ and $50 kg$ respectively. The coefficient of friction between $M$ and ground is zero. The coefficient of friction between $m_{1}$ and $M$ and that between $m_{2}$ and ground is $0.3$ . The pulleys and the strings are massless. The string is perfectly horizontal between $P_{1}$ and $m_{1}$ and also between $P_{2}$ and $m_{2}$ . The string is perfectly vertical between $P_{1}$ and $P_{2}$ . An external horizontal force $F$ is applied to the mass $M$ . (Take $g = 10 m/s^{2}$ .) (a) Draw a free body diagram of mass $M$ , clearly showing all the forces. (b) Let the magnitude of the force of friction between $m_{1}$ and $M$ be $f_{1}$ and that between $m_{2}$ and ground be $f_{2}$ . For a particular force $F$ it is found that $f_{1} = 2 f_{2}$ . Find $f_{1}$ and $f_{2}$ . Write equations of motion of all the masses. Find $F$ , tension in the string and accelerations of the masses.

JEE Advanced 2016

LEVELJEE Advanced

Comprehension Passage

ω

is an example of a non-inertial frame of reference. The relationship between the force

F_{rot}

experienced by a particle of mass

m

moving on the rotating disc and the force

F_{in}

experienced by the particle in an inertial frame of reference is

F_{rot} = F_{in} + 2 m (v_{rot} \times ω) + m (ω \times r) \times ω

, where

v_{rot}

is the velocity of the particle in the rotating frame of reference and

r

is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius

R

rotating counter-clockwise with a constant angular speed

ω

about its vertical axis through its center. We assign a coordinate system with the origin at the centre of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis (

ω = ω \hat{k}

). A small block of mass

m

is gently placed in the slot at

r = (R /2) \hat{i}

t = 0

and is constrained to move only along the slot.

Question 1:

The distance $r$ of the block at time $t$ is :

(A)

\frac{R}{4} (e^{2 ω t} + e^{- 2 ω t})

(B)

\frac{R}{2} cos 2 ω t

(C)

\frac{R}{2} cos ω t

(D)

\frac{R}{4} (e^{ω t} + e^{- ω t})

Question 2:

The net reaction of the disc on the block is :

(A)

- m ω^{2} R cos ω t \hat{j} - m g \hat{k}

(B)

m ω^{2} R sin ω t \hat{j} - m g \hat{k}

(C)

\frac{1}{2} m ω^{2} R (e^{ω t} - e^{- ω t}) \hat{j} + m g \hat{k}

(D)

\frac{1}{2} m ω^{2} R (e^{2 ω t} - e^{- 2 ω t}) \hat{j} + m g \hat{k}

JEE Main 2021

LEVELJEE Advanced

An inclined plane making an angle of $3 0^{\circ}$ with the horizontal is placed in a uniform horizontal electric field $200 N/C$ as shown in the figure. A body of mass $1 kg$ and charge $5 mC$ is allowed to slide down from rest at a height of $1 m$ . If the coefficient of friction is $0.2$ , find the time taken by the body to reach the bottom. [Take, $g = 9.8 m/s^{2}$ , $sin 3 0^{\circ} = \frac{1}{2}, cos 3 0^{\circ} = \frac{3}{2}$ ]

(A)

0.92 s

(B)

0.46 s

(C)

2.3 s

(D)

1.3 s

JEE Main 2021, 1 Sep Shift-II

LEVELJEE Advanced

A block of mass $m$ slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is [Given, $m = 8 kg, M = 16 kg$ ] Assume all the surfaces shown in the figure to be frictionless.

(A)

\frac{4}{3} g

(B)

\frac{6}{5} g

(C)

\frac{3}{5} g

(D)

\frac{2}{3} g

JEE Main 2021, 18 March Shift-II

LEVELJEE Advanced

A solid cylinder of mass $m$ is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is (The coefficient of static friction, $μ_{s}$ , is 0.4)

(A)

\frac{7}{2} m g

(B)

5 m g

(C)

\frac{m g}{5}

(D)

0

JEE Advanced 2009

LEVELJEE Advanced

Column II shows five systems in which two objects are labelled as $X$ and $Y$ . Also in each case a point $P$ is shown. Column I gives some statements about $X$ and/or $Y$ . Match these statements to the appropriate system(s) from Column II

List-I

(P)

The force exerted by

X

Y

has a magnitude

M g

(Q)

The gravitational potential energy of

X

is continuously increasing.

(R)

Mechanical energy of the system

X + Y

is continuously decreasing.

(S)

The torque of the weight of

Y

about point

P

is zero.

List-II

(1)

Block

Y

of mass

M

left on a fixed inclined plane

X

, slides on it with a constant velocity.

(2)

Two ring magnets

Y

and

Z

, each of mass

M

, are kept in frictionless vertical plastic stand so that they repel each other.

Y

rests on the base

X

and

Z

hangs in air in equilibrium.

P

is the topmost point of the stand on the common axis of the two rings. The whole system is in a lift that is going up with a constant velocity.

(3)

A pulley

Y

of mass

m_{0}

is fixed to a table through a clamp

X

. A block of mass

M

hangs from a string that goes over the pulley and is fixed at point

P

of the table. The whole system is kept in a lift that is going down with a constant velocity.

(4)

A sphere

Y

of mass

M

is put in a non-viscous liquid

X

kept in a container at rest. The sphere is released and it moves down in the liquid.

(5)

A sphere

Y

of mass

M

is falling with its terminal velocity in a viscous liquid

X

kept in a container.

LEVELJEE Advanced

A circular disc with a groove along its diameter is placed horizontally. A block of mass $1 kg$ is placed as shown. The coefficient of friction between the block and all surfaces of groove in contact is $μ = \frac{2}{5}$ . The disc has an acceleration of $25 m/s^{2}$ . Find the acceleration of the block with respect to disc. Given $cos θ = \frac{4}{5}$ .

LEVELJEE Advanced

A block of mass $m$ rests on a horizontal floor with which it has a coefficient of static friction $μ$ . It is desired to make the body move by applying the minimum possible force $F$ . Find the magnitude of $F$ and the direction in which it has to be applied.

LEVELJEE Advanced

A long horizontal rod has a bead which can slide along its length and is initially placed at a distance $L$ from one end $A$ of the rod. The rod is set in angular motion about $A$ with a constant angular acceleration $α$ . If the coefficient of friction between the rod and bead is $μ$ , and gravity is neglected, then the time after which the bead starts slipping is

(A)

\frac{μ}{α}

(B)

\frac{μ}{α}

(C)

\frac{1}{μα}

(D)

infinitesimal

LEVELJEE Advanced

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass $2 m$ and $m$ just after the string is cut will be (2006)

(A)

\frac{g}{2}

upwards,

g

downwards

(B)

g

upwards,

\frac{g}{2}

downwards

(C)

g

upwards,

2 g

downwards

(D)

2 g

upwards,

g

downwards

LEVELJEE Advanced

A smooth semicircular wire track of radius $R$ is fixed in a vertical plane (figure). One end of a massless spring of natural length $\frac{3 R}{4}$ is attached to the lowest point $O$ of the wire track. A small ring of mass $m$ which can slide on the track is attached to the other end of the spring. The ring is held stationary at point $P$ such that the spring makes an angle $6 0^{\circ}$ with the vertical. The spring constant $k = \frac{m g}{R}$ . Consider the instant when the ring is making an angle $6 0^{\circ}$ with the vertical. The spring is released, (a) Draw the free body diagram of the ring. (b) Determine the tangential acceleration of the ring and the normal reaction. (1996)

LEVELJEE Advanced

Block $A$ of mass $m$ and block $B$ of mass $2 m$ are placed on a fixed triangular wedge by means of a massless, inextensible string and a frictionless pulley as shown in figure. The wedge is inclined at $4 5^{\circ}$ to the horizontal on both sides. The coefficient of friction between block $A$ and the wedge is $2/3$ and that between block $B$ and the wedge is $1/3$ . If the blocks $A$ and $B$ are released from rest, find (1997) (a) the acceleration of $A$ , (b) tension in the string and (c) the magnitude and direction of the force of friction acting on $A$ .

LEVELJEE Advanced

A hemispherical bowl of radius $R = 0.1 m$ is rotating about its own axis (which is vertical) with an angular velocity $ω$ . A particle of mass $1 0^{- 2} kg$ on the frictionless inner surface of the bowl is also rotating with the same $ω$ . The particle is at a height $h$ from the bottom of the bowl. (1993) (a) Obtain the relation between $h$ and $ω$ . What is the minimum value of $ω$ needed, in order to have a non-zero value of $h$ ? (b) It is desired to measure $g$ (acceleration due to gravity) using the set-up by measuring $h$ accurately. Assuming that $R$ and $ω$ are known precisely and that the least count in the measurement of $h$ is $1 0^{- 4} m$ , what is minimum possible error $Δ g$ in the measured value of $g$ ?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

The load A is being pulled with the help of two inextensible strings that pass over two fixed pulleys as shown in the figure. At an instant velocities of the ends of the string being pulled are $u_{1}$ and $u_{2}$ and the angle between the strings connected to the load is $θ$ , what is speed of the load?

JEE Advanced 2025

LEVELJEE Advanced

A projectile of mass 200 g is launched in a viscous medium at an angle $6 0^{\circ}$ with the horizontal, with an initial velocity of 270 m/s. It experiences a viscous drag force $F = - c v$ where the drag coefficient $c = 0.1$ kg/s and $v$ is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s. Taking $e = 2.7$ , the horizontal distance of the wall from the point of projection (in m) is ____

JEE Main 2020

LEVELJEE Advanced

A bead of mass $m$ stays at point $P (a, b)$ on a wire bent in the shape of a parabola $y = 4 C x^{2}$ and rotating with angular speed $ω$ (see figure). The value of $ω$ is (neglect friction)

(A)

2 2 g C

(B)

\frac{2 g}{C}

(C)

2 g C

(D)

\frac{2 g C}{ab}

JEE Main 2021, 17 March Shift-II

LEVELJEE Advanced

A body of mass $1 kg$ rests on a horizontal floor with which it has a coefficient of static friction $1/ 3$ . It is desired to make the body move by applying the minimum possible force $F$ newton. The value of $F$ will be ............... . (Round off to the nearest integer) (Take, $g = 10 ms^{- 2}$ )

JEE Main 2021, 24 Feb Shift-I

LEVELJEE Advanced

An inclined plane is bent in such a way that the vertical cross-section is given by $y = \frac{x ^{2}}{4}$ where, $y$ is in vertical and $x$ in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $μ = 0.5$ , the maximum height in cm at which a stationary block will not slip downward is .......... cm.

Micro-Topic Strategy Matrix

Newton's Laws of Motion

Inertia, Momentum, and Impulse

Equilibrium of Concurrent Forces

Static and Kinetic Friction

Dynamics of Circular Motion

Pseudo Force

Chapter Elite-20 Challenge Zone

Comprehension Passage

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance r of the block at time t is :

The net reaction of the disc on the block is :

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A bead of mass m stays at point P(a,b) on a wire bent in the shape of a parabola y=4Cx2 and rotating with angular speed ω (see figure). The value of ω is (neglect friction)

The distance $r$ of the block at time $t$ is :

A bead of mass $m$ stays at point $P (a, b)$ on a wire bent in the shape of a parabola $y = 4 C x^{2}$ and rotating with angular speed $ω$ (see figure). The value of $ω$ is (neglect friction)