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Angular Momentum Formula

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  • Last Updated : 01 Feb, 2022

The angular momentum formula will be discussed in detail in this article. The rotating equivalent of linear momentum is the angular momentum formula. Both notions are concerned with the speed at which something moves. It also addresses the difficulty of changing the speed. Linear momentum, on the other hand, has just two variables: mass and velocity. Starting with a fairly similar equation, angular momentum is calculated. As a result, it may appear to have similar complexity at first glance. As we’ll see, it’s a lot more difficult than linear momentum and incorporates a lot more variables.

Angular Momentum

Angular Momentum is also called as Moment of Linear Momentum. Angular momentum is defined as, the attribute of any rotating object that is determined by the product of moment of inertia and angular velocity.

Angular momentum or moment of linear momentum is a rotational mechanics variable that is comparable to linear momentum. A torque is similar to a force’s moment. If p is a particle’s instantaneous linear momentum in a circular motion, L = r × p represents its angular momentum at that time, where r is the position vector from the rotation axis.

Kg.m2.s-1 is the SI unit of measurement.

The Dimensional formula is, L2MT-1.

Moment of Inertia

Moment of Inertia is defined as the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation, which expresses a body’s potential to resist angular acceleration.

Although the term ‘moment of inertia’ may be perplexing, everyone is already familiar with the ideas involved in calculating it. Assume there are two wooden rods. They are the same weight and diameter, but one is one foot long and the other is ten feet long. If thrown it overhand towards a target, which one will spin faster? Does spinning a ball around the head with a 2-foot rope or a 5-foot rope require more energy? What if the ball’s mass is increased by a factor of two?

Didn’t one realize it was going to be easier to spin with the smaller rod and shorter rope? So if started calculating an object’s moment of inertia. While one may associate the term “moment” with time, it actually relates to torque or twisting in mathematics and physics. The moment of inertia describes how difficult it is to twist an object about a specific axis. The shape, mass, and rotational axis of every item define its moment of inertia.

Moment of Inertia of Point mass, I = mr2

Moment of Inertia of the rod, I = mL2 / 12 (Center)

Expression for Angular Momentum in Terms of the Moment of Inertia

A stiff object spinning with a constant angular speed ω around an axis perpendicular to the plane of paper is seen in the diagram. Let’s assume that the object is made up of N particles with masses m1, m2,….mN and perpendicular distances r1, r2,….rN from the rotation axis. As the object rotates, all of these particles conduct UCM with the same angular speed ω but different line speeds v1 = r1ω, v2 = r2ω, … vN = rNω.

Individual velocity directions, v1, v2, and so on, are along the tangents of their respective tracks. The initial particle’s linear momentum is p1 = m1v1 = m1r1ω. It travels in the same direction as v1

The magnitude of its angular momentum is consequently L1 = p1r1 = m1r12ω. Similarly, L2 = m2r22ω, L3 = m3r32ω, … LN = mNrN2ω.

All of these angular momenta are directed along the axis of rotation for a rigid body with a fixed axis of rotation, which may be determined using the right-hand thumb rule. Their magnitudes may be summed algebraically because they all have the same direction. As a result, the magnitude of the body’s angular momentum is given by,

L = m1r12ω + m2r22ω +….+ mNrN2ω

∴ L = (m1r12 + m2r22 +…+ mNrN2)ω = Iω

The moment of inertia of the body around the given axis of rotation is I = m1r12 + m2r22 +…+ mNrN2. If the moment of inertia I replace mass, the angular momentum statement L = Iω is comparable to the linear momentum expression p = mv, which is its physical significance.

Angular momentum Quantum Number

Azimuthal quantum number or secondary quantum number is interchangeable with angular momentum quantum number. It is a quantum number that determines the angular momentum of an atomic orbital as well as its size and form. The most common value is between 0 and 1.

Right-Hand Thumb Rule

  • If you position your right hand so that the fingers point in the direction of r.
  • The curl is thus oriented in the direction of linear momentum (p).
  • The outstretched thumb depicts the direction of angular momentum (L).

Examples of Angular Momentum

  • Gyroscope

To retain its orientation, a gyroscope uses the angular momentum principle. It works with a three-degree-of-freedom spinning wheel. It locks on to the orientation when turned at a rapid speed and will not stray from it. This is useful in space applications where controlling the attitude of a spacecraft is critical.

  • Ice-skater 

When ice skaters start a spin, their hands and legs are spread wide apart from their body’s center. They put their hands and legs closer to their bodies when spinning requires a higher angular velocity. They spin faster as a result of the conservation of angular momentum.

Torque

The force that may cause an object to rotate along an axis is measured by torque. In linear kinematics, force is what propels an object forward. Angular acceleration is also caused by torque. As a result, torque can be thought of as the linear force’s rotating equivalent. The axis of rotation is the point around which an object rotates. Torque is defined as a force’s proclivity to turn or twist in physics.

The formula of Torque,

τ = r × F

Conservation of Angular Momentum

Everyone has seen the conservation of linear momentum, which states that in the absence of an external unbalanced force, the linear momentum of an isolated system is conserved. In rotational dynamics, torque and angular momentum are similar to force and linear momentum, as previously stated. With the right modifications, this can be turned into angular momentum conservation.

As seen above, angular momentum or moment of linear momentum of the system is gives, L = r × p

Where, r = position vector from the axis of rotation, p = linear momentum.

Differentiating with respect to time, 

dL/dt = d/dt(r × p)

∴ dL/dt = r × (dp/dt) + (dr/dt) × p ⇢ (equation a)

Now, (dr/dt) = v and (dp/dt) = F put in equation a.

∴ dL/dt =  r × F + m(v × v)

Now, (v × v) = 0

∴ dL/dt = r × F

But, r × F is torque.

∴ τ = dL/dt

Thus, if τ = 0, dL/dt = 0 or L = constant.

As a result, in the absence of uneven external torque τ, angular momentum L is conserved. This is the principle of angular momentum conservation, which is similar to linear momentum conservation.

Conservation of angular momentum is demonstrated in ballet dancing performances, circus acrobats, and sports such as ice skating and swimming pool diving. L = Iω = I(2πη) is constant in all of these cases. As a result, increasing the moment of inertia I lower the angular speed and, as a result, the frequency of revolution η. In addition, decreasing the moment of inertia raises the frequency.

Things to Keep in Mind When It Comes to Angular Momentum

  • Angular Momentum is defined as the attribute of any rotating object that is determined by the product of moment of inertia and angular velocity.
  • The magnitude of angular momentum ‘L’ can be calculated using the following formula:  L = rmv sin Φ
  • Spin and orbital angular momentum are the two types of angular momentum.
  • L = r × p is the formula for calculating the angular momentum of a point object.
  • L = I × ω is the formula for calculating the angular momentum of a long object.
  • The radius of the circle determines a body’s perpendicular velocity when no torque is applied.

Sample Questions

Question 1: An asteroid with a mass of 1.7 × 105 kg and a relative velocity of 25 km/s collides with the Earth at the equator, tangentially and in the direction of Earth’s rotation, and becomes lodged there. Calculate the percent change in the Earth’s angular speed as a result of the collision using angular momentum.

Solution:

Ma =1.7 × 105 kg

va = 25 km/s ((1000m/s) / (1km/s)) = 2.5 × 104 m/s

Consider,

R = 1.50 × 1011 m

ME = 5.97 × 1024 kg

An asteroid’s moment of inertia is calculated as follows:

Ia = MaR2

R denotes the distance between the Sun and the planet Earth, which is 1.50 × 1011 m.

The Earth’s moment of inertia can be written as,

IE = 2/5 MER2

ME denotes the mass of the Earth, which is 5.97 × 1024 kg.

An asteroid’s angular velocity can be calculated as follows:

ωa = va/R

Apply the principle of momentum conservation. The equation can be written as follows:

Iaωa + IEωE = (Ia + IE)ω ⇢ (Equation 1)

Here, ωa represents an asteroid’s angular velocity, ωE represents an asteroid’s angular velocity, and ω is the net angular velocity.

The moment of inertia of the satellite in relation to Earth Ia can now be ignored. Then, using equation (1), the conservation of angular momentum can be written as,

Iaωa + IEωE = IEω

ω = (Iaωa +IEωE) / IE  ⇢  (Equation 2)

The Earth’s angular velocity can be computed as,

ωE =1 rev/yr (2π rad / 1rev)(1yr / 86400s)

≈ 7.27 × 10-5 rad/s

The following results are obtained by plugging the relevant terms into equation (2):

ω = (MaR2 × va/R + 2/5MER2 × ωE) / (2/5MER2)

When the values are plugged into the equation above, 

ω ≈ 7.27 × 10-5 rad/s

After the collision, the percentage change in the earth’s angular speed can be written as,

%C = (ω−ωE)/ωE × 100%

%C ≈ 0%

The angular speed changes by a small proportion in this case. As a result, the percentage change in the Earth’s angular speed as a result of the collision is around 0%.

Question 2: Define Angular momentum.

Answer:

Angular momentum is defined as the attribute of any rotating object that is determined by the product of moment of inertia and angular velocity.

Question 3: With an angular speed of 100rad/s, a solid cylinder of mass 20kg revolves around its axis. The cylinder’s radius is 0.25m in diameter. What is the K.E for cylinder rotation? How large is the cylinder’s angular momentum around its axis? 

Solution:

Moment of inertia of solid cylinder,

I = M × (R2 / 2)

∴ I = 20 × ((0.25)2 / 2)

∴ I = 0.625 kgm2

Kinetic Energy,

K = (1/2)Iω2

∴ K = 1/2 × 0.625 × 100 × 100

∴ K = 3125 joules

Angular momentum = Iω

∴ Angular momentum = 0.625 × 100

∴ Angular momentum = 6.25 kgm2/s

Question 4: Explain the Examples of Angular Momentum.

Answer :

  • Gyroscope:

To retain its orientation, a gyroscope uses the angular momentum principle. It works with a three-degree-of-freedom spinning wheel. It locks on to the orientation when turned at a rapid speed and will not stray from it. This is useful in space applications where controlling the attitude of a spacecraft is critical.

  • Ice-skater:

When ice skaters start a spin, their hands and legs are spread wide apart from their body’s center. They put their hands and legs closer to their bodies when spinning requires a higher angular velocity. They spin faster as a result of the conservation of angular momentum.

Question 5: Explain Angular momentum Quantum Number.

Answer:

Azimuthal quantum number or secondary quantum number is interchangeable with angular momentum quantum number. It is a quantum number that determines the angular momentum of an atomic orbital as well as its size and form. The most common value is between 0 and 1.

Question 6: Explain Torque.

Answer:

The force that may cause an object to rotate along an axis is measured by torque. In linear kinematics, force is what propels an object forward. An angular acceleration is also caused by torque. As a result, torque can be thought of as the linear force’s rotating equivalent. The axis of rotation is the point around which an object rotates. Torque is defined as a force’s proclivity to turn or twist in physics.

Formula of Torque,

τ = r × F

Question 7: Explain the Right-hand thumb Rule in brief.

Answer:

Right hand thumb Rule:

  • If one position their right hand so that the fingers point in the direction of r.
  • The curl is thus oriented in the direction of linear momentum (p).
  • The outstretched thumb depicts the direction of angular momentum (L).

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