Angular momentum

Angular momentum

The quantity of the rotational motion, which is identical to mass (m) in the linear motion, is the moment of inertia (I). The quantity of the rotational motion, which is identical to the velocity (v) in the linear motion, is the angular velocity (ω). Thus, the rotating object has angular momentum that can be calculated using the equation:

L = I ω

L = angular momentum (kg m²/s), I = moment of inertia (kg m²), ω = angular velocity (rad/s)

Sample problems of the angular momentum

Sample problem 1.

A particle with a mass of 0.5 grams moves in a circle with a constant angular velocity of 2 rad/s. Determine the angular momentum of the particle if the radius of the particle’s path is 10 cm.

Solution:

The moment of inertia of the particle:

I = m r² = (0.5 x 10^-3 kg)(1 x 10^-1 m)² = (0.5 x 10^-3 kg)(1 x 10^-2 m²) = 0.5 x 10^-5 kg m²

The angular speed of the particle:

ω = 2 rad/s

The angular momentum of the particle:

L = (0.5 x 10^-5 kg m²)(2 rad/s) = 1 x 10^-5 kg m²/s

The law of conservation of angular momentum

The law of conservation of angular momentum states that if the resultant moment of force on a rigid body when rotating is zero, then the angular momentum of the rigid body when rotating is always constant. The law of conservation of angular momentum can be derived mathematically by modifying the equation of Newton’s second law of angular momentum. Here is the equation of Newton‘s second law on the angular momentum:

If the resultant moment of force is zero, then the equation above changes to:

I_t = the final moment of inertia, I_o = the initial moment of inertia, ω_t = the final angular speed, ω_o = the initial angular speed, L_t = the final angular momentum, L_o = the initial angular momentum.

Sample problems of the law of conservation of angular momentum

Sample problem 1.

A homogeneous solid cylinder disk is initially rotating on its axis with a speed of 4 rad/s. The mass and radius of the disk are 1 kg and 0.5 m. If above the plate is placed a ring that has a mass and radius of 0.2 kg and 0.1 m and the center of the ring,

just above the center of the disk, then the disc and ring will rotate together with the angular velocity of…..

Solution:

Moment of inertia of solid cylinder : I = 1⁄2 m r² = 1⁄2 (1 kg)(0.5 m)² = (0.5)(0.25) = 0.125 kg m²

Moment of inertia of ring : I = m r² = (0.2 kg)(0.1 m)² = (0.2)(0.01) = 0.002 kg m²

Initial angular momentum (L₁) = Final angular momentum (L₂)

I₁ ω₁ = I₂ ω₂

(0.125 kg m²)(4 rad/s) = (0.125 kg m² + 0.002 kg m²)(ω₂)

(0.5) = (0.127)(ω₂)

ω₂ = 0.5 : 0.127

ω₂ = 4 rad/s

Conceptual questions and asnwer

1. What is angular momentum?
   Answer: Angular momentum is a measure of the amount of rotation an object has, considering its mass and shape. It is the rotational analog of linear momentum.
2. How is angular momentum defined mathematically?
   Answer: Angular momentum (L) is defined as $L=r\times p$, where $r$ is the position vector from the axis of rotation to the point of application, and $p$ is the linear momentum.
3. Is angular momentum a scalar or vector quantity?
   Answer: Angular momentum is a vector quantity.
4. What is the unit of angular momentum in the SI system?
   Answer: The unit of angular momentum in the SI system is kilogram-meter squared per second (kg·m²/s).
5. How is angular momentum conserved?
   Answer: In a closed system with no external torques, the total angular momentum remains constant.
6. What is the principle of conservation of angular momentum?
   Answer: The principle states that the total angular momentum of a closed system remains constant unless acted upon by an external torque.
7. How does the spin of a figure skater relate to angular momentum?
   Answer: When a figure skater pulls their arms and legs close to their body during a spin, they decrease their moment of inertia, causing them to spin faster. This demonstrates the conservation of angular momentum.
8. What is the relationship between torque and angular momentum?
   Answer: Torque ($\tau$) is the rate of change of angular momentum. Mathematically, τ=dL/dt​.
9. Is it possible for an object to have angular momentum without angular velocity?
   Answer: No, if an object has angular momentum, it has angular velocity. The magnitude and direction of angular momentum depend on both the moment of inertia and angular velocity.
10. How does the moment of inertia affect angular momentum?
    Answer: Angular momentum (L) is the product of the moment of inertia (I) and angular velocity (ω). As $L=I\omega$, an increase in the moment of inertia for a given angular velocity will increase angular momentum.
11. What is the difference between spin and orbital angular momentum in quantum mechanics?
    Answer: Spin angular momentum is an intrinsic property of particles, like electrons, and does not arise from motion in space. Orbital angular momentum, on the other hand, arises from the motion of a particle around a central point.
12. What role does angular momentum play in the formation of planetary systems?
    Answer: As a molecular cloud collapses to form stars and planets, the conservation of angular momentum causes the material to flatten into a disk around the newborn star, from which planets can form.
13. How does a gyroscope maintain its orientation in space?
    Answer: A spinning gyroscope has angular momentum. When an external torque tries to change its orientation, the gyroscope precesses, or changes its axis of rotation, in a direction perpendicular to the applied torque due to the conservation of angular momentum.
14. What causes precession in a spinning top?
    Answer: Precession in a spinning top is caused by the torque due to gravity acting on the top’s mass center, which is offset from its pivot point. This torque results in a change in the direction of the top’s angular momentum, causing it to precess.
15. Why do stars flatten at their poles?
    Answer: As stars rotate, they experience a centrifugal force pushing matter outward. This effect is stronger at the equator than at the poles, leading to an oblate, or flattened, shape.
16. How is angular momentum related to Kepler’s second law?
    Answer: Kepler’s second law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This is a consequence of the conservation of angular momentum.
17. Why does a cat always land on its feet when it falls?
    Answer: Cats use the conservation of angular momentum. By twisting their bodies and changing their moments of inertia mid-air, they can reorient themselves to land on their feet without violating the conservation laws.
18. How is the magnetic moment related to angular momentum in electrons?
    Answer: The magnetic moment of an electron arises from its spin and orbital angular momentum. The magnetic moment is proportional to the electron’s angular momentum.
19. What is angular momentum in terms of polar coordinates?
    Answer: In polar coordinates, angular momentum $L$ for a point mass can be expressed as $L=mrv$, where $m$ is the mass, $r$ is the radial distance, and $v$ is the tangential velocity.
20. What happens to the angular momentum of a closed system when two objects collide?
    Answer: In a closed system where two objects collide, the total angular momentum before the collision is equal to the total angular momentum after the collision, assuming no external torques act on the system.

Problems and Solutions

1. Problem: A point mass m = 2 kg moves with a velocity v = 3 m/s in a circle of radius r = 4 m. Calculate its angular momentum about the center of the circle.

Solution: $L=m\times r\times v$ =2×4×3=24 kg.m²/s

2. Problem: A solid sphere of mass 3 kg and radius 0.5 m is rotating about its diameter with an angular velocity ω = 2 rad/s. What is its angular momentum?

Solution: For a solid sphere, I=2/5 ​m r²

$L=I\times \omega$

L=2/5×3×(0.5)²×2=0.6 kg.m²/s

3. Problem: A point mass of 3 kg is moving with a speed of 4 m/s in a straight line. What is its angular momentum about a point 5 m away from its line of motion?

Solution:

$L=m\times v\times r$

L=3×4×5=60 kg.m²/s

4. Problem: A disc of mass 5 kg and radius 3 m is rotating with an angular speed of 2 rad/s. Calculate its angular momentum.

Solution: For a disc, I=1/2​m r²

$L=I\times \omega$

L=1/2×5×3²×2=45 kg.m²/s

5. Problem: A rod of length 4 m and mass 2 kg rotates about one end with an angular velocity of 3 rad/s. Find its angular momentum.

Solution: For a rod about one end, I=1/3​ m l²

$L=I\times \omega$

L=1/3×2×4²×3=96 kg.m²/s

6. Problem: What is the change in angular momentum if a rotating body doubles its angular velocity?

Solution: The relation between angular momentum (L) and angular velocity (ω) is $L=I\times \omega$. If ω is doubled, the angular momentum will also double.

7. Problem: If the radius of a circular path of a moving particle doubles while keeping velocity constant, by what factor does the angular momentum change?

Solution: Angular momentum, $L=m\times r\times v$. If r doubles, L will also double.

8. Problem: A child of mass 30 kg is standing at the edge of a merry-go-round which is at rest. The radius of the merry-go-round is 2 m. If the child starts running at a speed of 2m/s with respect to the ground along the edge, find the angular momentum of the child with respect to the center.

Solution: $L=m\times r\times v$ =30×2×2=120 kg.m²/s

9. Problem: A cylinder of mass 6 kg and radius 2 m is rotating about its central axis with an angular velocity of 5 rad/s. Calculate its angular momentum.

Solution: For a cylinder, I=1/2 ​m r²

$L=I\times \omega$ =1/2×6×2²×5=120 kg.m²/s

10. Problem: What will be the new angular momentum of a body if its moment of inertia is halved and its angular velocity is tripled?

Solution: Given $L=I\times \omega$, if $I$ is halved and $\omega$ is tripled, the new $L$ will be 1/2×3=1.5 × the original.

11. Problem: A hoop of mass 2kg and radius 1m is rotating with an angular speed of 4 rad/s. Calculate its angular momentum.

Solution: For a hoop, I=m r²

$L=I\times \omega$

L=2×1²×4=8 kg.m²/s

12. Problem: A fan has blades of length 0.5m and total mass 1kg. If it rotates with an angular speed of 10 rad/s, determine its angular momentum.

Solution: Assuming the fan blades are like rods rotating about one end, I=1/3 ​ml²

$L=I\times \omega$ =1/3×1×0.5²×10=0.833 kg.m²/s

13. Problem: Calculate the angular momentum of the earth about its own axis due to its rotation. Given: Mass of earth =5.972×10²⁴ kg, Radius $r=6371\text{km}$, ω=7.27×10⁻⁵rad/s.

Solution: Assuming earth as a solid sphere, I=2/5 ​mr²

$L=I\times \omega$ =2/5×5.972×10²⁴×(6371×10³)²×7.27×10⁻⁵

L≈7.07×10³³kg.m²/s

14. Problem: A child moves towards the center of a rotating platform. Does the angular momentum increase, decrease or remain the same?

Solution: Angular momentum is conserved. As the child moves towards the center, the moment of inertia decreases. To conserve angular momentum, the angular velocity increases. Thus, the angular momentum remains the same.

15. Problem: If the kinetic energy of a rotating body is doubled, by what factor does its angular momentum change?

Solution: Kinetic energy K= 1/2 ​Iω² and Angular momentum $L=I\omega$. If kinetic energy is doubled and $I$ remains constant, ω² is doubled. This implies that $\omega$ is increased by a factor of √2​. Thus, $L$ also increases by a factor of √2​.

16. Problem: A bicycle wheel of radius 0.3 m has a rim of mass 1.5 kg. If it rotates with an angular speed of 5 rad/s, find its angular momentum.

Solution: Assuming the rim as a hoop, I=mr²

$L=I\times \omega$ =1.5×0.32×5=6.75 kg.m²/s

L=1.5×0.32×5=6.75kg.m²/s

17. Problem: An ice skater pulls her arms inward while spinning. What happens to her angular momentum?

Solution: The skater’s angular momentum is conserved. By pulling her arms in, she decreases her moment of inertia, and thus she spins faster (increased angular velocity) to conserve her angular momentum.

18. Problem: A flywheel has a moment of inertia of 0.2 kg.m²kg.m² and is rotating with an angular velocity of 10 rad/s. Find its angular momentum.

Solution:

$L=I\times \omega$ =0.2×10=2 kg.m²/s

L=0.2×10=2 kg.m²/s

19. Problem: A system has a net external torque of zero acting on it. What can you say about its angular momentum?

Solution: If the net external torque is zero, then according to the conservation of angular momentum, the total angular momentum of the system remains constant.

20. Problem: A point mass m = 5 kg moves with a velocity v = 3 m/s perpendicular to the line joining it to a point P. If the distance from the point P is 4 m, find its angular momentum about point P.

Solution:

$L=m\times r\times v$ =5×4×3=60 kg.m²/s

L=5×4×3=60kg.m²/s