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Angular Momentum: Calculation

Momentum 

Momentum is a physical quantity associated with the object (system of particles). It is defined as time required to bring any object in motion to rest by applying constant force to it. 

How to calculate Momentum of a system of particles?

Mathematically, momentum (P) of any particle is the product of its mass and velocity. Mass(m) is a scalar quantity and velocity (v) is a vector quantity and the overall momentum of the body is a vector quantity.

P=mv

This mathematical expression leads to the SI unit of momentum, which is Kg. m/s

For example:

The momentum of an elephant weighing 2300 Kg, running with the velocity of 7 m/s will be

Momentum of elephant (P) = Mass of elephant (M) . Velocity of the elephant (v)

P = 2300kg7m/s = 16100 kg m/s

Following can be inferred from the mathematical expression of the momentum:

  • Velocity of the object and momentum are directly proportional to each other.
  • Mass and Momentum of the object are directly proportional to each other.
  • Mass and velocity of the object are inversely proportional to each other.

For any object, the total momentum is equivalent to the vector sum of individual momenta of each of the particles constituting the body.

Linear Momentum and Angular Momentum

  • When an object is in linear motion, it possesses linear momentum, whereas if an object is in curvilinear motion, it possesses angular momentum. 
  • The underlying difference between the two types of momenta is, linear momentum is a property of the system of particles in motion, with respect to a stationary or moving reference point, whereas angular momentum is the property of the system of particles which not only change their position w.r.t. a reference point but also undergoes change in direction and position.
  • The Linear momentum of the system of particles remains conserved, provided there is no application of external force on it. Angular momentum on the other hand remains conserved, provided there is no application of torque on the object.
  • The application of resultant force on the system of particles results in rate of change of linear momentum, whereas the application of resultant torque on the system of particles results in the rate of change of angular momentum.

Mathematical expression for Angular Momentum

Any particle of mass (m) moving with a velocity (v) on a curvilinear path possess angular momentum (L) w.r.t. a reference point. Reference points can be stationary or in motion.

Angular momentum (L) can be calculated as cross product of linear momentum (P) and position vector (r) of the particle from the reference point.

L = r mv = r P

Thus, the SI unit of angular momentum is Kgm2S-1. Considering angular momentum is the cross product of two vectors, hence the direction of angular momentum vector will be perpendicular to the direction of both the vectors, position vector and velocity vector.

Any rigid body is basically a system of particles. The vector sum of angular momentum of all the particles of the rigid body rotating about an axis gives the total angular momentum of the rigid body. The axis of rotation is perpendicular to the plane on which position and velocity vectors lie.

L= jrj mjvj

Using the relationship between linear velocity (v) and angular velocity () we get,

L= jr2j mjj

Considering all the particles are rotating about the same axis, the angular velocity of all the particles will be the same.

L= j(r2j mj)ωj

The quantity j(r2j mj)  is the object’s moment of inertia (I). 

  • L= I

For constant external torque, the angular momentum of the system of particles remain conserved, hence the rate of change of angular momentum dLdx signifies the resultant of torque acting on the system of particles.

= dLdx

Angular Momentum: In real life context

  • The magical spin of ice skaters: Ice skaters while initiating a spin keeps hands and legs far apart from the centre of mass of their body. Further to increase the angular velocity of their spin, they get their hands and legs closer to the centre of mass of their body. This modification in the body posture helps them to conserve their angular momentum and spin as fast as possible. The same phenomenon can be observed when anyone rotates while sitting on an office chair with wheels.
  • The bullets have grooves on their surface. The grooves help the bullet spin with maximum possible angular velocity while coming out of the rifled barrel. The increased spin in the bullet results in easy piercing.
  • Gyroscope is an instrument used in space applications related to the altitude of the spacecraft. This instrument works on the principle of conservation of angular momentum to maintain specific orientation.