What is moment of momentum?

Short Answer:

The moment of momentum is the rotational equivalent of linear momentum. It is defined as the product of the linear momentum of a particle and its perpendicular distance from a fixed axis or point.

In simple terms, moment of momentum represents the rotational motion possessed by a body. It indicates how difficult it is to stop a rotating object. Mathematically, it is expressed as

where  is the perpendicular distance from the axis,  is mass, and  is linear velocity. The moment of momentum is also called angular momentum.

Detailed Explanation :

Moment of Momentum

The moment of momentum (also known as angular momentum) is a fundamental concept in rotational dynamics. It describes the quantity of rotational motion that a body possesses about a fixed point or axis. Just as linear momentum is associated with motion in a straight line, moment of momentum is associated with rotational motion.

It depends on both the linear momentum (mv) of the body and its distance (r) from the axis of rotation. Therefore, it combines both the mass distribution and the rotational speed of the body. The larger the mass or the farther it is from the axis, the greater its moment of momentum.

Definition

The moment of momentum can be defined as:

“The moment of the linear momentum of a particle about any point or axis is called the moment of momentum about that point or axis.”

Mathematically,

where,
= Moment of momentum (also written as angular momentum)
= Position vector of the particle (distance from the axis)
= Linear momentum of the particle

Hence,

The direction of the moment of momentum is perpendicular to the plane containing  and , and it is determined using the right-hand rule — if the fingers of the right hand curl from  to , the thumb points in the direction of .

Units and Dimensions

  • SI Unit: kilogram meter² per second (kg·m²/s)
  • CGS Unit: gram centimeter² per second (g·cm²/s)
  • Dimensional Formula: [M¹L²T⁻¹]

These units indicate that moment of momentum is proportional to both the rotational inertia and the rate of rotation.

Moment of Momentum for a Particle

Consider a particle of mass  moving with velocity  at a perpendicular distance  from a fixed point .
Then,

or, if the velocity is perpendicular to ,

This means the moment of momentum of a particle is proportional to its mass, velocity, and distance from the axis.

Moment of Momentum for a Rigid Body

For a rigid body rotating about a fixed axis with angular velocity :

Let the body be made up of small particles, each having mass  and located at a distance  from the axis. The moment of momentum of each particle is:

The total moment of momentum for the whole body is:

Since  (moment of inertia),

Thus, for a rigid body,

This relation shows that the moment of momentum increases with greater mass distribution (moment of inertia) and faster rotation (angular velocity).

Law of Conservation of Moment of Momentum

The law of conservation of moment of momentum states that:

“If no external torque acts on a system, the total moment of momentum of the system remains constant.”

Mathematically,

This means that when the external torque acting on a rotating system is zero, its angular momentum (moment of momentum) remains unchanged.

Examples:

  • A spinning skater pulls in her arms to rotate faster.
  • A rotating planet continues to spin at a constant rate if no external torque acts on it.

Relation Between Torque and Moment of Momentum

Torque () is directly related to the rate of change of the moment of momentum.
According to Newton’s Second Law for Rotational Motion:

This means that torque is the time rate of change of moment of momentum.
If torque is zero,  remains constant — leading to the law of conservation of angular momentum.

Hence, torque and moment of momentum are interconnected — torque changes the rotational motion, while moment of momentum measures the rotation’s quantity.

Physical Significance

The moment of momentum represents the rotational inertia and motion strength of a rotating body.

  • A larger  means the body resists changes in its rotational state more strongly.
  • It plays the same role in rotational motion as linear momentum does in straight-line motion.
  • It is a conserved quantity in the absence of external torque, which means rotational systems maintain their angular momentum naturally.

Examples of Moment of Momentum

  1. Spinning Flywheel:
    A flywheel in an engine stores angular momentum, which helps maintain uniform speed during power strokes.
  2. Rotating Earth:
    The Earth’s rotation about its axis has a large moment of momentum, which remains constant due to negligible external torque.
  3. Skater or Diver:
    When a skater pulls in their arms, their moment of inertia decreases, and their angular speed increases to conserve moment of momentum.
  4. Gyroscope:
    The stability of a gyroscope is due to its large moment of momentum, which resists changes in its rotational axis.
  5. Planetary Motion:
    Each planet maintains constant angular momentum about the Sun, explaining Kepler’s laws of planetary motion.

Applications in Engineering

  • Used in flywheels and rotors to store rotational energy.
  • Important in vehicle stability and balance systems using gyroscopes.
  • Helps in designing turbines and rotating machinery for smooth operation.
  • Applied in aerospace engineering for satellite motion and attitude control.
  • Used in mechanical systems to analyze torque, angular speed, and energy conservation.
Conclusion

The moment of momentum is the product of the linear momentum of a body and its perpendicular distance from a fixed axis. It is also known as angular momentum and represents the rotational motion possessed by a body. The moment of momentum plays a vital role in rotational mechanics and is given by  for a rigid body. Its conservation explains many natural and engineering phenomena involving rotation, such as gyroscope stability, planetary motion, and machine operation under torque.