Torque and Angular Momentum

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A force of $- F \hat{k}$ acts on $O$ , the origin of the coordinate system. The torque about the point $(1, - 1)$ is

F (\hat{i} - \hat{j})

(B)

- F (\hat{i} + \hat{j})

(C)

F (\hat{i} + \hat{j})

(D)

- F (\hat{i} - \hat{j})

LEVELBoard

Let $F$ be the force acting on a particle having position vector $r$ and $τ$ be the torque of this force about the origin. Then,

(A)

r \cdot τ = 0

and

F \cdot τ \neq = 0

(B)

r \cdot τ \neq = 0

and

F \cdot τ = 0

(C)

r \cdot τ \neq = 0

and

F \cdot τ \neq = 0

(D)

r \cdot τ = 0

and

F \cdot τ = 0

JEE Main 2019, 11 Jan Shift-I

LEVELJEE Advanced

A slab is subjected to two forces $F_{1}$ and $F_{2}$ of same magnitude $F$ as shown in the figure. Force $F_{2}$ is in $x y$ -plane while force $F_{1}$ acts along $Z$ -axis at the point $(2 \hat{i} + 3 \hat{j})$ . The moment of these forces about point $O$ will be

(A)

(3 \hat{i} + 2 \hat{j} - 3 \hat{k}) F

(B)

(3 \hat{i} - 2 \hat{j} + 3 \hat{k}) F

(C)

(3 \hat{i} - 2 \hat{j} - 3 \hat{k}) F

(D)

(3 \hat{i} + 2 \hat{j} + 3 \hat{k}) F

JEE Main 2014

LEVELJEE Main

A bob of mass $m$ attached to an inextensible string of length $l$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $ω$ rad/s about the vertical support. About the point of suspension,

(A)

angular momentum is conserved

(B)

angular momentum changes in magnitude but not in direction

(C)

angular momentum changes in direction but not in magnitude

(D)

angular momentum changes both in direction and magnitude

JEE Main 2019, 10 April Shift-I

LEVELJEE Main

A particle of mass $m$ is moving along a trajectory given by $x = x_{0} + a cos ω_{1} t$ and $y = y_{0} + b sin ω_{2} t$ . The torque acting on the particle about the origin at $t = 0$ is

(A)

zero

(B)

m (- x_{0} b + y_{0} a) ω_{1}^{2} \hat{k}

(C)

- m (x_{0} b ω_{2}^{2} - y_{0} a ω_{1}^{2}) \hat{k}

(D)

+ m y_{0} a ω_{1}^{2} \hat{k}

JEE Main 2021, 17 March Shift-I

LEVELJEE Advanced

A triangular plate is shown below. A force $F = 4 \hat{i} - 3 \hat{j}$ is applied at point P. The torque at point P with respect to point O and Q are

(A)

- 15 - 203, 15 - 203

(B)

15 + 203, 15 - 203

(C)

15 - 203, 15 + 203

(D)

- 15 + 203, 15 + 203

JEE Main 2019, 10 April Shift-II

LEVELJEE Main

The time dependence of the position of a particle of mass $m = 2$ is given by $r (t) = 2 t \hat{i} - 3 t^{2} \hat{j}$ . Its angular momentum, with respect to the origin, at time $t = 2$ is

(A)

36 \hat{k}

(B)

- 34 (\hat{k} - \hat{i})

(C)

- 48 \hat{k}

(D)

48 (\hat{i} + \hat{j})

JEE Main 2021, 17 March Shift-I

LEVELJEE Advanced

A mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass $M$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $ω$ . The angular momentum of $M$ about point $A$ is $L_{A}$ which lies in the positive z-direction and the angular momentum of $M$ about $B$ is $L_{B}$ . The correct statement for this system is

(A)

L_{A}

and

L_{B}

are both constant in magnitude and direction

(B)

L_{B}

is constant in direction with varying magnitude

(C)

L_{B}

is constant, both in magnitude and direction

(D)

L_{A}

is constant, both in magnitude and direction

JEE Main 2021, 16 March Shift-I

LEVELJEE Main

Consider a $20 kg$ uniform circular disc of radius $0.2 m$ . It is pin supported at its centre and is at rest initially. The disc is acted upon by a constant force $F = 20 N$ through a massless string wrapped around its periphery as shown in the figure. Suppose the disc makes $n$ number of revolutions to attain an angular speed of $50 rad s^{- 1}$ . The value of $n$ to the nearest integer, is .......... . (Given, in one complete revolution, the disc rotates by $6.28 rad$ .)

LEVELJEE Main

Angular momentum of the particle rotating with a central force is constant due to

(A)

constant force

(B)

constant linear momentum

JEE Main 2019, 12 Jan Shift-II

LEVELJEE Main

A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B, its angular momentum about O will be (Take, $g = 10 m/s^{2}$ )

JEE Main 2021, 25 July Shift-1

LEVELJEE Main

A solid disc of radius $20 cm$ and mass $10 kg$ is rotating with an angular velocity of $600 rpm$ , about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in $10 s$ is $____π \times 1 0^{- 1} N-m$ .

JEE Main 2019

LEVELJEE Advanced

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s, is close to

(A)

4.0 \times 1 0^{- 6}

N-m

(B)

2.0 \times 1 0^{- 5}

N-m

(C)

1.6 \times 1 0^{- 5}

N-m

(D)

7.9 \times 1 0^{- 6}

N-m

JEE Main 2021 (16 March Shift-II)

LEVELJEE Main

A force $F = 4 \hat{i} + 3 \hat{j} + 4 \hat{k}$ is applied on an intersection point of $x = 2$ plane and X-axis. The magnitude of torque of this force about a point $(2, 3, 4)$ is …… . (Round off to the nearest integer)

JEE Main 2020, 2 Sep Shift-I

LEVELJEE Main

Shown in the above figure, a rigid and uniform $1 m$ long rod $A B$ held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass $m$ and has another weight of mass $2 m$ hung at a distance of $75 cm$ from $A$ . The tension in the string at $A$ is

(A)

1 mg

(B)

2 mg

(C)

0.75 mg

(D)

0.5 mg

JEE Main 2021, 31 Aug Shift-1

LEVELBoard

Angular momentum of a single particle moving with constant speed along circular path

(A)

changes in magnitude but remains same in the direction

(B)

remains same in magnitude and direction

(C)

remains same in magnitude but changes in the direction

(D)

is zero

JEE Main 2020, 05 Sep Shift-I

LEVELJEE Main

A force $F = (\hat{i} + 2 \hat{j} + 3 \hat{k})$ N acts at a point $(4 \hat{i} + 3 \hat{j} - \hat{k})$ m. Then, the magnitude of torque about the point $(\hat{i} + 2 \hat{j} + \hat{k})$ m will be $x$ N-m. The value of $x$ is …… .

JEE Main 2020, 02 Sep Shift-I

LEVELJEE Advanced

A uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height $a$ ( $a < R$ ) by applying a force $F$ at its centre $O$ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of $F$ required is

(A)

M g \frac{a}{R}

(B)

M g 1 - \frac{a ^{2}}{R ^{2}}

(C)

M g (\frac{R}{R - a})^{2} - 1

(D)

M g 1 - (\frac{R - a}{R})^{2}

LEVELJEE Main

A particle performing uniform circular motion has angular momentum $L$ . If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is

(A)

\frac{L}{4}

(B)

2 L

(C)

4 L

(D)

\frac{L}{2}

JEE Main 2019, 11 Jan Shift-II

LEVELJEE Main

The magnitude of torque on a particle of mass $1 kg$ is $2.5 N-m$ about the origin. If the force acting on it is $1 N$ and the distance of the particle from the origin is $5 m$ , then the angle between the force and the position vector is (in radian)

(A)

\frac{π}{8}

(B)

\frac{π}{4}

(C)

\frac{π}{3}

(D)

\frac{π}{6}

LEVELJEE Main

A particle of mass $m$ is projected with a velocity $v$ making an angle of $3 0^{\circ}$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is

(A)

\frac{3}{2} \frac{m v ^{2}}{g}

(B)

zero

(C)

\frac{m v ^{3}}{2 g}

(D)

\frac{3}{16} \frac{m v ^{3}}{g}

JEE Advanced 1990

LEVELJEE Main

A particle of mass $m$ is projected with a velocity $v$ making an angle of $4 5^{\circ}$ with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is

(A)

zero

(B)

\frac{m v ^{3}}{4 2 g}

(C)

\frac{m v ^{3}}{2 g}

(D)

m 2 g h^{3}

JEE Advanced 1997

LEVELJEE Main

A mass $m$ is moving with a constant velocity along a line parallel to the $X$ -axis, away from the origin. Its angular momentum with respect to the origin

The position vector $r$ of particle of mass $m$ is given by the following equation $r (t) = α t^{3} \hat{i} + β t^{2} \hat{j}$ where, $α = \frac{10}{3} ms^{- 3}$ , $β = 5 ms^{- 2}$ and $m = 0.1 kg$ . At $t = 1 s$ , which of the following statement(s) is (are) true about the particle?

* Multiple Correct Options

(A)

The velocity

v

is given by

v = (10 \hat{i} + 10 \hat{j}) ms^{- 1}

(B)

The angular momentum

L

with respect to the origin is given by

L = - (5/3) \hat{k} Nms

(C)

The force

F

is given by

F = (\hat{i} + 2 \hat{j}) N

(D)

The torque

τ

with respect to the origin is given by

τ = - \frac{20}{3} \hat{k} Nm

JEE Advanced 2012

LEVELJEE Main

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x$ - $y$ plane with centre at $O$ and constant angular speed $ω$ . If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $L_{O}$ and $L_{P}$ respectively, then

(A)

L_{O}

and

L_{P}

do not vary with time

(B)

L_{O}

varies with time while

L_{P}

remains constant

(C)

L_{O}

remains constant while

L_{P}

varies with time

(D)

L_{O}

and

L_{P}

both vary with time

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A force of −Fk^ acts on O, the origin of the coordinate system. The torque about the point (1,−1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let F be the force acting on a particle having position vector r and τ be the torque of this force about the origin. Then,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle of mass m is moving along a trajectory given by x=x0​+acosω1​t and y=y0​+bsinω2​t. The torque acting on the particle about the origin at t=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A triangular plate is shown below. A force F=4i^−3j^​ is applied at point P. The torque at point P with respect to point O and Q are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The time dependence of the position of a particle of mass m=2 is given by r(t)=2ti^−3t2j^​. Its angular momentum, with respect to the origin, at time t=2 is

Angular momentum of the particle rotating with a central force is constant due to

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s, is close to

Angular momentum of a single particle moving with constant speed along circular path

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A force F=(i^+2j^​+3k^) N acts at a point (4i^+3j^​−k^) m. Then, the magnitude of torque about the point (i^+2j^​+k^) m will be x​ N-m. The value of x is …… .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle performing uniform circular motion has angular momentum L. If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A mass m is moving with a constant velocity along a line parallel to the X-axis, away from the origin. Its angular momentum with respect to the origin

A force of $- F \hat{k}$ acts on $O$ , the origin of the coordinate system. The torque about the point $(1, - 1)$ is

Let $F$ be the force acting on a particle having position vector $r$ and $τ$ be the torque of this force about the origin. Then,

A particle of mass $m$ is moving along a trajectory given by $x = x_{0} + a cos ω_{1} t$ and $y = y_{0} + b sin ω_{2} t$ . The torque acting on the particle about the origin at $t = 0$ is

A triangular plate is shown below. A force $F = 4 \hat{i} - 3 \hat{j}$ is applied at point P. The torque at point P with respect to point O and Q are

The time dependence of the position of a particle of mass $m = 2$ is given by $r (t) = 2 t \hat{i} - 3 t^{2} \hat{j}$ . Its angular momentum, with respect to the origin, at time $t = 2$ is

A force $F = (\hat{i} + 2 \hat{j} + 3 \hat{k})$ N acts at a point $(4 \hat{i} + 3 \hat{j} - \hat{k})$ m. Then, the magnitude of torque about the point $(\hat{i} + 2 \hat{j} + \hat{k})$ m will be $x$ N-m. The value of $x$ is …… .

A particle performing uniform circular motion has angular momentum $L$ . If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is

A mass $m$ is moving with a constant velocity along a line parallel to the $X$ -axis, away from the origin. Its angular momentum with respect to the origin