Angular Momentum and Law of Conservation of Angular Momentum

Q 1 :

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours 30 minutes. [2024]

(13) $By angular momentum conservation, I_{1} ω_{1} = I_{2} ω_{2}$

$\frac{2}{5} m r^{2} \cdot ω = \frac{2}{5} m {(\frac{3}{4} r)}^{2} \cdot ω^{'} \Rightarrow ω^{'} = \frac{16}{9} ω$

$\frac{2 π}{T^{'}} = \frac{16}{9} \times \frac{2 π}{T} \Rightarrow T^{'} = \frac{9 T}{16} = \frac{9}{16} \times 24 hours = \frac{9 \times 3}{2} hours$

$\Rightarrow T^{'} = \frac{27}{2} hr = 13 h r 30 mins$

Q 2 :

A body of mass 5 kg moving with a uniform speed $3 \sqrt{2} m s^{- 1}$ in X - Y plane along the line $y = x + 4$ . The angular momentum of the particle about the origin will be _____ $k g m^{2}$ $s^{- 1}$ . [2024]

(60) $y - x - 4 = 0$

$d_{1}$ is perpendicular distance of given line from origin.

$d_{1} =$ $| \frac{- 4}{\sqrt{1^{2} + 1^{2}}} |$ $\Rightarrow 2 \sqrt{2} m$

So, $| \vec{L} |$ $= m v d_{1}$

$= 5 \times 3 \sqrt{2} \times 2 \sqrt{2} kg m^{2} / s = 60 kg m^{2} / s$

Q 3 :

Two discs of moment of inertia $I_{1} = 4$ $k g m^{2}$ and $I_{2} = 2$ $k g m^{2}$ about their central axes and normal to their planes, rotating with angular speeds 10 rad/s and 4 rad/s respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is ________ J. [2024]

(24) Here angular momentum will remain conserve

$L = I ω$

$L_{i} = L_{f}$

$4 \times 10 + 2 \times 4 = (6) ω$

$48 = 6 ω$

$ω = 8 rad / \sec$

$Δ K E = K E_{f} - K E_{i}$

$Δ K = \frac{1}{2} (6) {(8)}^{2} - [(\frac{1}{2} 4 \times 10^{2}) + (\frac{1}{2} (2) \times {(4)}^{2})]$

$Δ K = 192 - 216$

$Δ K = - 24 J$

Q 4 :

A uniform rod AB of mass 2 kg and length 30 cm at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{π}{x} s$ , where $x =$ _____________ . [2024]

(8)

Q 5 :

A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $ω$ . If another disc of the same dimensions but of mass M/2 is placed gently on the first disc co-axially, then the new angular velocity of the system is _____. [2024]

$\frac{4}{5} ω$
$\frac{2}{3} ω$
$\frac{5}{4} ω$
$\frac{3}{2} ω$

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(2)

Q 6 :

A particle of mass m projected with a velocity 'u' making an angle of 30° 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 _____. [2024]

$\frac{\sqrt{3}}{16} \frac{m u^{3}}{g}$
$\frac{\sqrt{3}}{2} \frac{m u^{2}}{g}$
$\frac{m u^{3}}{\sqrt{2} g}$
zero

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(1)

Velocity of particle at maximum height

$v = u \cos 30 ° = \frac{u \sqrt{3}}{2}$

Maximum height, $h = \frac{u^{2} \sin^{2} 30 °}{2 g} = \frac{u^{2}}{8 g}$

Magnitude of angular momentum

$| \vec{L} |$ $= h m v$

$= (\frac{u^{2}}{8 g}) m (\frac{u \sqrt{3}}{2}) = \frac{\sqrt{3} m u^{3}}{16 g}$

Q 7 :

Consider a disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is _____ J. [2024]

(250)

Net torque about axis of rotation is zero

$\Rightarrow L_{i} = L_{f} \Rightarrow I_{i} ω_{i} = I_{f} ω_{f}$

$\Rightarrow (\frac{5 \times 2^{2}}{2}) \times 10 = (\frac{10 \times 2^{2}}{2}) ω_{f}$

$5 = ω_{f}$

$Δ E = E_{i} - E_{f} = \frac{1}{2} I_{i} ω_{i}^{2} - \frac{1}{2} I_{f} ω_{f}^{2}$

$= \frac{1}{2} (\frac{5 \times 2^{2}}{2}) \times 10^{2} - \frac{1}{2} (\frac{10 \times 2^{2}}{2}) \times 5^{2}$

$= 500 - 250 = 250 J$

Q 8 :

A body of mass 'm' is projected with a speed 'u' making an angle of 45° with the ground. The angular momentum of the body about the point of projection, at the highest point, is expressed as $\frac{\sqrt{2} m u^{3}}{X g}$ . The value of ‘X’ is _____. [2024]

(8)

$L = m u \cos θ \frac{u^{2} \sin^{2} θ}{2 g} = m u^{3} \frac{1}{4 \sqrt{2} g} \Rightarrow x = 8$

Q 9 :

Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side $a$ in free space. At t = 0, they are given an initial velocity ${\vec{V}}_{A} = V_{0} \vec{A C}, {\vec{V}}_{B} = V_{0} \vec{B A}$ and ${\vec{V}}_{C} = V_{0} \vec{C B}$ . Here $\vec{A C}, \vec{C B}$ and $\vec{B A}$ ware unit vectors along the edges of the triangle. If the three mases interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is : [2025]

$\frac{1}{2} a m V_{0}$
$3 a m V_{0}$
$\frac{\sqrt{3}}{2} a m V_{0}$
$\frac{3}{2} a m V_{0}$

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(3)

$\tan 30 ° = \frac{2 r}{a} = \frac{1}{\sqrt{3}} \Rightarrow r = \frac{a}{2 \sqrt{3}}$

Net angular momentum about point O

$L = (m v r_{⊥}) \times 3 = m v_{0} \frac{a}{2 \sqrt{3}} \times 3 = \frac{\sqrt{3}}{2} m v_{0} a$

Q 10 :

An object of mass 'm' is projected from origin in a vertical xy-plane at an angle 45° with the x-axis with an initial velocity $v_{0}$ . The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [g is acceleration due to gravity] [2025]

$\frac{m v_{0}^{3}}{2 \sqrt{2} g}$ along negative z-axis
$\frac{m v_{0}^{3}}{2 \sqrt{2} g}$ along positive z-axis
$\frac{m v_{0}^{3}}{4 \sqrt{2} g}$ along positive z-axis
$\frac{m v_{0}^{3}}{4 \sqrt{2} g}$ along negative z-axis

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(4)

At highest point

The speed is $v_{x} = v_{0} \cos 45 ° \Rightarrow v_{x} = \frac{v_{0}}{\sqrt{2}}$

$H = \frac{{(\frac{v_{0}}{\sqrt{2}})}^{2}}{2 g} = \frac{v_{0}^{2}}{4 g}$

So, angular momentum about origin is

$L = m v_{x} \cdot H$

$\Rightarrow L = m \times \frac{v_{0}}{\sqrt{2}} \times \frac{v_{0}^{2}}{4 g}$

$\Rightarrow L = \frac{m v_{0}^{3}}{4 \sqrt{2} g}$ (along negative z-axis)

Q 11 :

Let $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector $\vec{r} = a (\hat{i} \cos ω t + \hat{j} \sin ω t)$ . The direction of force is [2025]

Opposite to the direction of $\vec{r}$
Opposite to the direction of $\vec{L}$
Opposite to the direction of $\vec{P}$
Opposite to the direction of $\vec{L} \times \vec{P}$

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(1)

$\vec{r} = a \cos ω t \hat{i} + a \sin ω t \hat{j}$

$\vec{v} = \frac{d \vec{r}}{d t} = - a ω \sin ω t \hat{i} + a ω \cos ω t \hat{j}$

$\vec{a} = \frac{d \vec{v}}{d t} = - a ω^{2} \cos ω t \hat{i} - a ω^{2} \sin ω t \hat{j}$

$\vec{a} = - ω^{2} (a \cos ω t \hat{i} + a \sin ω t \hat{j}) = - ω^{2} \vec{r}$

$\vec{a} is antiparallel to \vec{r}$

Hence, $\vec{F}$ is opposite to the direction of $\vec{r}$

Q 12 :

The position vectors of two 1 kg particles, (A) and (B), are given by $\vec{r_{A}} = (α_{1} t^{2} \hat{i} + α_{2} \hat{j} + α_{3} t \hat{k}) m$ and $\vec{r_{B}} = (β_{1} t \hat{i} + β_{2} t^{2} \hat{j} + β_{3} \hat{t} \hat{k}) m$ , respectively; $(α_{1} = 1 m / s^{2}, α_{2} = 3 n m / s, α_{3} = 2 m / s, β_{1} = 2 m / s)$ , $(β_{2} = - 1 m / s^{2}, β_{3} = 4 p m / s)$ , where t is time, n and p are constants, At t = 1 s, $| {\vec{V}}_{A} |$ $=$ $| {\vec{V}}_{B} |$ and velocities ${\vec{V}}_{A}$ and ${\vec{V}}_{B}$ of the particles are orthogonal to each other. At t = 1 s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L} {kgm}^{2} s^{- 1}$ . The value of L is _________. [2025]

(90)

${\vec{r}}_{A} = (α_{1} t^{2} \hat{i} + α_{2} \hat{j} + α_{3} t \hat{k}) m$

$\Rightarrow \frac{d {\vec{r}}_{A}}{d t} = {\vec{V}}_{A} = (2 t \hat{i} + 3 n \hat{j} + 2 \hat{k})$

${\vec{r}}_{B} = (β_{1} t \hat{i} + β_{2} t^{2} \hat{j} + β_{3} \hat{t} \hat{k})$

$\Rightarrow \frac{d {\vec{r}}_{B}}{d t} = {\vec{V}}_{B} = (2 \hat{i} - 2 \hat{j} + 4 p \hat{k})$

At t = 1 s, ${\vec{V}}_{A} \cdot {\vec{V}}_{B} = 0$

4 – 6n + 8p = 0

$\Rightarrow 3 n = 2 + 4 p$ ... (i)

At t = 1 s, $| {\vec{V}}_{A} |$ $=$ $| {\vec{V}}_{B} |$

$4 + 9 n^{2} + 4 = 4 + 4 + 16 p^{2}$

From (i) and (ii), $p = \frac{- 1}{4} \Rightarrow n = \frac{1}{3}$

The angular momentum of particle (A) with respect to the position of particle (B)

$| \vec{L} |$ $= m_{A} ({\vec{r}}_{A / B} \times {\vec{V}}_{A})$

${\vec{r}}_{A / B} = (α_{1} - β_{1}) \hat{i} + (α_{2} - β_{2}) \hat{j} + (α_{3} - β_{3}) \hat{k}$

$= (1 - 2) \hat{i} + (1 + 1) \hat{j} + 3 \hat{k}$

$\vec{L} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 1 & 2 & 3 \\ 2 & 1 & 2 \end{matrix} |$ $= \hat{i} + 8 \hat{j} - 5 \hat{k}$

$| \vec{L} |$ $= \sqrt{1 + 64 + 25} = \sqrt{90}$

Q 13 :

A solid sphere with uniform density and radius R is rotating initially with constant angular velocity ( $ω_{1}$ ) about its diameter. After sometime during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become R/2 is $x ω_{1}$ . The value of x is _________. [2025]

(32)

When the sphere is of radius R, its mass is M, when radius is reduced to $\frac{R}{2}$ , mass will reduced to $\frac{M}{8}$

Angular momentum will remain conserve.

$L_{1} = L_{2} \Rightarrow I_{1} ω_{1} = I_{2} ω_{2}$

$(\frac{2}{5} M R^{2}) ω_{1} = (\frac{2}{5} (\frac{M}{8}) {(\frac{R}{2})}^{2}) ω_{2}$

$ω_{2} = 32 ω_{1}$ value of x is 32.

Q 14 :

A particle of mass 100 g is projected at time $t = 0$ with a speed $20 {ms}^{- 1}$ at an angle $45 °$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $t = 2 s$ is found to be $\sqrt{K} {kgm}^{2} /s$ . The value of $K$ is _________ . (Take $g = 10 {ms}^{- 2}$ ) [2023]

(800)

$Use Δ L = \int_{0}^{t} τ d t$

$L_{0} = \int_{0}^{2} m g (v_{x} t) d t$

$= m g v_{x} \frac{t^{2}}{2} = (0.1) (10) (10 \sqrt{2}) \frac{2^{2}}{2} = 20 \sqrt{2}$

$= \sqrt{800} {kg m}^{2} / s$

Q 15 :

A solid sphere of mass 500 g and radius 5 cm is rotated about one of its diameter with angular speed of 10 rad $s^{- 1}$ . If the moment of inertia of the sphere about its tangent is $x \times 10^{- 2}$ times its angular momentum about the diameter, then the value of $x$ will be ________. [2023]

(35)

$m = 500 g = 0.5 kg$

$R = 5 cm$

$ω = 10 rad/s$

$Moment of inertia about tangent = I_{T}$

$I_{T} = x \times 10^{- 2} L$

$\frac{7}{5} m R^{2} = x \times 10^{- 2} \frac{2}{5} m R^{2} ω$

$\frac{7}{2 ω} = x \times 10^{- 2} = \frac{7}{2 \times 10}$

Q 16 :

A circular plate is rotating in a horizontal plane about an axis passing through its center and perpendicular to the plate, with an angular velocity $ω$ . A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If E be the initial kinetic energy of the system, then the final kinetic energy will be $\frac{E}{x}$ . The value of $x$ is _______. [2023]

(3)

$KE = \frac{L^{2}}{2 I} \Rightarrow \frac{{KE}_{final}}{{KE}_{initial}} = \frac{I_{initial}}{I_{final}} \Rightarrow \frac{{KE}_{final}}{E} = \frac{1}{3}$

$\Rightarrow {KE}_{final} = \frac{E}{3}$

Q 17 :

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is: [2026]

$\frac{2 L}{m d^{2}}$
$\frac{3}{2} \frac{L}{m d^{2}}$
$\frac{4}{3} \frac{L}{m d^{2}}$
$\frac{2 L}{5 m d^{2}}$

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(2)

$L = I ω and ω = \frac{L}{I}$

$ω = \frac{L}{m {(\frac{2 d}{3})}^{2} + 2 m {(\frac{d}{3})}^{2}}$ $= \frac{L}{\frac{4}{9} m d^{2} + \frac{2}{9} m d^{2}} = \frac{L}{\frac{6 m d^{2}}{9}}$

$ω = \frac{3 L}{2 m d^{2}}$

Q 18 :

Two cars A and B each of mass $10^{3} kg$ are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ______ J·s. [2026]

$3.6 \times 10^{5}$
$10^{5}$
$2 \times 10^{5}$
$3 \times 10^{5}$

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(2)

$L = m . V_{rel} r_{⊥}$

$= 1000 \times (36 \times \frac{5}{18}) \times 10$

$= 10^{5} {kg m}^{2} / s$

Topic Question Set

Q 1 :

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours 30 minutes. [2024]

Q 2 :

A body of mass 5 kg moving with a uniform speed $3 \sqrt{2} m s^{- 1}$ in X - Y plane along the line $y = x + 4$ . The angular momentum of the particle about the origin will be _____ $k g m^{2}$ $s^{- 1}$ . [2024]

Q 4 :

A uniform rod AB of mass 2 kg and length 30 cm at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{π}{x} s$ , where $x =$ _____________ . [2024]

(2)

Q 6 :

A particle of mass m projected with a velocity 'u' making an angle of 30° 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 _____. [2024]

Q 8 :

A body of mass 'm' is projected with a speed 'u' making an angle of 45° with the ground. The angular momentum of the body about the point of projection, at the highest point, is expressed as $\frac{\sqrt{2} m u^{3}}{X g}$ . The value of ‘X’ is _____. [2024]

Q 11 :

Let $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector $\vec{r} = a (\hat{i} \cos ω t + \hat{j} \sin ω t)$ . The direction of force is [2025]

Q 17 :

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is: [2026]

Q 18 :

Two cars A and B each of mass $10^{3} kg$ are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ______ J·s. [2026]

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Topic Question Set

Q 1 : If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours 30 minutes. [2024]

Q 2 : A body of mass 5 kg moving with a uniform speed 32ms-1 in X - Y plane along the line y=x+4. The angular momentum of the particle about the origin will be _____ kgm2 s-1. [2024]

Q 4 : A uniform rod AB of mass 2 kg and length 30 cm at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be πxs, where x= _____________ . [2024]

(2)

Q 6 : A particle of mass m projected with a velocity 'u' making an angle of 30° 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 _____. [2024]

Q 8 : A body of mass 'm' is projected with a speed 'u' making an angle of 45° with the ground. The angular momentum of the body about the point of projection, at the highest point, is expressed as 2mu3Xg. The value of ‘X’ is _____. [2024]

Q 11 : Let L→ and P→ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector r→=a(i^ cos ωt+j^ sin ωt). The direction of force is [2025]

Q 15 : A solid sphere of mass 500 g and radius 5 cm is rotated about one of its diameter with angular speed of 10 rad s-1. If the moment of inertia of the sphere about its tangent is x×10-2 times its angular momentum about the diameter, then the value of x will be ________. [2023]

Q 17 : Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is: [2026]

Q 18 : Two cars A and B each of mass 103 kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ______ J·s. [2026]

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Q 1 :

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours 30 minutes. [2024]

Q 2 :

A body of mass 5 kg moving with a uniform speed $3 \sqrt{2} m s^{- 1}$ in X - Y plane along the line $y = x + 4$ . The angular momentum of the particle about the origin will be _____ $k g m^{2}$ $s^{- 1}$ . [2024]

Q 4 :

A uniform rod AB of mass 2 kg and length 30 cm at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{π}{x} s$ , where $x =$ _____________ . [2024]

Q 6 :

A particle of mass m projected with a velocity 'u' making an angle of 30° 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 _____. [2024]

Q 8 :

A body of mass 'm' is projected with a speed 'u' making an angle of 45° with the ground. The angular momentum of the body about the point of projection, at the highest point, is expressed as $\frac{\sqrt{2} m u^{3}}{X g}$ . The value of ‘X’ is _____. [2024]

Q 11 :

Let $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector $\vec{r} = a (\hat{i} \cos ω t + \hat{j} \sin ω t)$ . The direction of force is [2025]

Q 17 :

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is: [2026]

Q 18 :

Two cars A and B each of mass $10^{3} kg$ are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ______ J·s. [2026]