Angular Momentum Formula

What is Angular Momentum Formula?

Angular momentum is a fundamental concept in physics that describes the rotational motion of objects. When it comes to understanding how objects behave when they rotate, the angular momentum formula is crucial. This formula provides a quantitative measure of the amount of rotation an object has, considering its mass, shape, and speed.

The angular momentum formula is expressed as

L = Iω

• L represents the angular momentum
• I is the moment of inertia (a measure of an object’s resistance to changes in its rotation rate).
• ω (omega) is the angular velocity, or how fast the object spins.

There is another way to express angular momentum for point particles, which is

L = r × p

• r is the position vector of the particle relative to the point about which the momentum is measured
• p is the linear momentum of the particle.

The concept of angular momentum was first introduced by Sir Isaac Newton in his seminal work Principia Mathematica. He described it as the “quantity of motion” of rotating bodies, which today we define more precisely with the formulas above. Newton’s development of the laws of motion and universal gravitation laid the groundwork for later physicists to explore and formalize the principles of angular momentum in various physical contexts.

Application of Angular Momentum Formula

1. Astronomy: Astronomers use the formula to study the orbits of planets and stars. It helps explain why planets continue to orbit the sun without drifting away or spiraling into it.
2. Mechanical Engineering: In designing gears and rotors, engineers apply the angular momentum formula to ensure stability and efficiency in their rotation.
3. Sports: The formula is used to improve the performance of athletes in sports involving rotation, such as figure skating or diving, by analyzing and enhancing their rotational movements.
4. Physics Education: Educators use angular momentum to teach students about conservation laws and rotational motion, fundamental concepts in physics.
5. Robotics: Robots that need to perform precise movements, especially those involving turning or pivoting, rely on calculations of angular momentum for accurate control.

Example Problems on Angular Momentum Formula

Problem 1: Spinning Wheel

Question: A wheel with a moment of inertia of 0.75 kg·m² spins at a rate of 150 revolutions per minute (rpm). Calculate its angular momentum.

Solution:

First, convert rpm to radians per second: 𝜔 =150 × (2𝜋 / 60) =15.7 rad/s

Use the angular momentum formula 𝐿=𝐼 x 𝜔:

𝐿 = 0.75 × 15.7 = 11.775 kg\cdotpm²/s

Answer: The angular momentum of the wheel is 11.775 kg·m²/s.

Problem 2: Particle

Question: A particle of mass 2 kg is moving at a velocity of 4 m/s in a straight line. It passes through a point which is 0.5 m perpendicular distance from the origin. Find the angular momentum of the particle about the origin.

Solution:

The linear momentum p of the particle is given by 𝑝 = 𝑚𝑣 = 2 × 4 = 8 kg\cdotpm/s.

Using 𝐿 = 𝑟 × 𝑝, and since the perpendicular distance r and p are at right angles, 𝐿 = 𝑟 x 𝑝:

𝐿 = 0.5 × 8 = 4 kg\cdotpm²/s

Answer: The angular momentum of the particle about the origin is 4 kg·m²/s.

Problem 3: Rotating Disk

Question: A solid disk with a radius of 0.2 m has a mass of 3 kg. If it rotates at 200 rpm, calculate its angular momentum. Assume the disk is a solid cylinder with 𝐼=(1 / 2) 𝑀𝑅².

Solution:

Convert rpm to radians per second:

𝜔 = 200 × (2𝜋 / 60) = 20.94 rad/s

Calculate the moment of inertia for the disk:

𝐼 = (1 / 2) × 3 × (0.2)² = 0.06 kg\cdotpm²

Calculate angular momentum using 𝐿=𝐼𝜔:

𝐿 = 0.06 × 20.94 = 1.2564 kg\cdotpm²/s

Answer: The angular momentum of the disk is approximately 1.2564 kg·m²/s.

FAQs

What is the Formula for Angular Momentum About Any Point?

The formula for angular momentum about any point is L = r × p, where r is the position vector and p is the linear momentum.

How Do You Calculate Angular Momentum?

Calculate angular momentum using L = Iω for rigid bodies, where I is the moment of inertia and ω is the angular velocity.

Does Angular Momentum Change with Radius?

Yes, angular momentum changes with radius; it is directly proportional to the radius (r) in the formula L = r × p.