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]

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(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]

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(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]

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(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]

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(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

3

4

(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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2

3

4

(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]

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(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]

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(8)

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

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]

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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]

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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]

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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]

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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]

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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]

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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]

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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]

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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]

0%

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]

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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]