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Angular momentum is the rotary inertia of an object about an axis. It is the analogue of linear momentum. The earth exhibits angular momentum as a result of its revolution around the sun and rotation. Angular momentum is a vector quantity as it is dependent on both magnitude and direction. Angular momentum is a conserved quantity as there is constant angular momentum in a closed system. The formula for angular momentum is,
L = r x p
Where,
L is the angular velocity
r is the radius (the distance between the object and the fixed point it revolves around)
p is the linear momentum.
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Key Takeaways: Angular Momentum, Orbital Angular Momentum, Spin angular momentum, linear momentum, vector quantity, conserved quantity.
What is Angular Momentum?
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- Angular Momentum is defined as the property of any rotating object given by the product of the moment of inertia and angular velocity of the rotating object.
- It is a vector and has both magnitude and direction. The common examples of angular momentum are the rotation and revolution of Earth.
- The angular momentum is denoted by \(\overrightarrow{L}\)
- Its SI unit is Kgm2s-1.
- The dimensional formula of angular momentum is represented as [M][L]2[T]-1
There are two types of angular momentum:
- Spin angular momentum. (E.g. Rotation)
- Orbital angular momentum. (E.g. Revolution)
The Earth's annual revolution around the Sun reflects orbital angular momentum, whereas its daily rotation about its axis indicates spin angular momentum.
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| Read More about Angular Motions | ||
|---|---|---|
| Radial Acceleration | Angular acceleration | Moment of force |
| Rolling motion | Angular Displacement | Angular Speed |
Formula of Angular Momentum
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An object can encounter angular momentum in two ways. They are as follows:
Point Object
The object that accelerates around a fixed point is known as a point object. For example, the earth revolves around the sun.
The angular momentum of a point object is determined as:
L = r x p
Where,
- L is the angular velocity
- r is the radius (the distance between the object and the fixed point it revolves around)
- p is the linear momentum.
Extended Object
The object which is rotating around a fixed point is called an extended object. For example, the earth rotates around its axis. The angular momentum is determined as:
L = I x ω
Where,
- L is the angular momentum.
- I is rotational inertia.
- ω is the angular velocity.
Angular Momentum Quantum Number
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- Angular Momentum Quantum Number is a quantum number that specifies the angular momentum of an atomic orbital and characterizes its size and shape.
- The typical value of angular momentum quantum numbers is between 0 and 1.
- Azimuthal quantum numbers or secondary quantum numbers are alternatives for angular momentum quantum numbers.
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Right-Hand Thumb Rule
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Right-hand thumb rule indicates that if a person positions his or her hand in such a way that the fingers point in the direction of rotation, then the fingers on that hand curl towards the direction of rotation, and the thumb points in the direction of angular momentum (L), angular velocity, and torque.
In short,
- If you place your right hand in such a way that the fingers are pointing in the direction of r.
- Then the curl points in the direction of linear momentum (p).
- The direction of angular momentum is depicted by the outstretched thumb (L).
Right Hand Thumb Rule
Angular Momentum and Torque
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Consider a point mass attached to a string, which is linked to a point. If we add a torque to the string, the point mass will continue to rotate around the center. The particle of mass m will travel at a perpendicular velocity V, which is the velocity perpendicular to the radius of the circle. The radius r is the distance between the particle's center of rotation and the radius of the circle.
Relation between Angular Momentum and Torque
The magnitude of angular momentum ‘L’ is determined as:
L = rmv sin Φ
→ r p⊥
→ rmv⊥
→ r⊥p
→ r⊥mv
Where,
- Φ is the angle between r and p
- p⊥ and v⊥ are the components of p and v perpendicular to r.
- r⊥ is the perpendicular distance between the fixed point and the extension of p.
According to the equation, the angular momentum of the body only changes when a net torque is applied. When no torque is applied, the body's perpendicular velocity is determined by the radius of the circle. Thus,
- For a shorter radius, velocity will be high.
- For a higher radius, velocity will be low.
Examples of Angular Momentum
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Some of the real life applications of angular momentum include:
Ice-skater
When the ice skaters initiate a spin, they position their hands and legs widely apart from their body's center. When they require greater angular velocity to spin, they bring their hands and legs closer to their body. Due to this, the angular momentum is conserved and they spin faster.
Example of Angular Momentum
Gyroscope
The principle of angular momentum is used by a gyroscope to maintain its position. It makes use of a three-degree-of-freedom spinning wheel. It locks on to the orientation when turned at a rapid speed, and it will not deviate from it. This is useful in space applications where a spacecraft's attitude is an important criterion to regulate.
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Things to Remember
- Angular Momentum is defined as the property of any rotating object given by the product of the moment of inertia and angular velocity of the rotating object.
- Mathematically, the magnitude of angular momentum ‘L’ can be expressed as L = rmv sin Φ
- Angular Momentum is of two types- Spin angular momentum and orbital angular momentum.
- The revolution of Earth around the sun reflects orbital angular momentum, whereas the rotation about its axis indicates spin angular momentum.
- The angular momentum of a point object is expressed as L = r x p.
- The angular momentum of an extended object is expressed as L = I x ω.
- When no torque is applied, the perpendicular velocity of a body is determined by the radius of the circle.
- For a shorter radius, velocity will be high and for a higher radius, velocity will be low.
Sample Questions
Ques: What is the position of the centre of mass of a uniform triangular lamina? (1 Mark)
Ans: The centre of mass is at the centroid of the triangular lamina.
Ques: State the condition for translational equilibrium of a body? (1 Mark)
Ans: For translational equilibrium of a body the vector sum of all the forces acting on the body must be zero.
Ques: (i) Do the centre of mass of a body necessarily lie inside the body? Give any example.
(ii) A system is in stable equilibrium. What can we say about Its potential energy? (2 Marks)
Ans: (i) No, the centre of mass of a body need not necessarily lie inside the body. For example – A ring.
(ii) For a system in stable equilibrium, potential energy is minimum.
Ques: (i) What is the amount of torque on the planet due to the gravitational force of the sun?
(ii) Why do we prefer to use a wrench with a longer arm? (2 Marks)
Ans: (i) The torque on the planet due to the gravitational force of the sun is 0.
(ii) We prefer to use a wrench with a longer arm to increase torque.
Ques: A planet revolves around a massive star in a highly elliptical orbit. Is its angular momentum constant over the entire orbit? Give a reason? (2 Marks)
Ans: A planet revolves under the effect of gravitational force around the star. Since the force is radial and does not contribute to torque. So, in the absence of an external torque, angular momentum of a planet remains constant.
Ques: A wheel 0.5m in radius is moving with a speed of 12m/s. find its angular speed? (2 Marks)
Ans: Given that:
v=12 m/s, r= 0.5 m
We know that,
v=rω
ω=v/r= 12/0.5
ω = 24 rad/sec
Ques: Calculate the angular momentum of a pulley of 2 kg, radius 0.1 m, rotating at a constant angular velocity of 4 rad/sec. (2 Marks)
Ans: Given that: m = 2 kg and r = 0.1 m
We know that:
I = 1/2mr2
I= 0.01 kg.m2
Angular momentum is given by L=Iω, thus, substituting the values we get L=0.04 kg.m2.s-1.
Ques: Two particles travel in parallel directions, as shown below. What is the total angular momentum of the system with respect to O? (2 Marks)
Ans: The total angular momentum is zero at point O. When the two particles are traveling, one particle moves in the clockwise direction relative to O and one moves in the counter clockwise direction. Also, both particles are the same distance apart from the axis, and angle between the radius and the velocity of the particle. Thus, the two particles always have equal and opposite angular momenta at O, hence total angular momentum of the system is zero.
Ques: (i) What is the SI unit of angular momentum?
(ii) What is the dimensional formula of angular momentum?
(iii) Give the expression for Angular momentum. (3 Marks)
Ans: (i) Dimensional formula of angular momentum (L) is [M][L]2[T]-1
(ii) Angular momentum can be expressed as:
L = I×ω
or L = r×p
Here L is angular momentum, I is moment of Inertia, ω is angular velocity, r is the radius and p is linear momentum.
(iii) The SI unit of angular momentum is Kg.m2.s-1
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