We have already seen that linear momentum plays an important role in translational motion of bodies. Similarly, another quantity known as angular momentum has important role in the study of rotational motion.

A particle is said to possess an angular momentum about a reference axis if it so moves that it’s angular position changes relative to that reference axis.

The angular momentum L of a particle of mass m moving with velocity v and momentum p (Fig. 5.10) relative to the origin O is defined as

angular-momentum

Fig: 5.10

L = r x p                 ………………….       (5.18)

 Where r is the position vector of the particle at that instant relative to the origin O. angular momentum is a vector quantity. Its magnitude is

L = rp sin ϴ = m rv sinӨ

Where Ө is the angle between r and p. the direction of L is perpendicular to the plane formed by r and p and its sense is given by the right hand rule of vector product. SI unit of angular momentum is kg angular-moment-equation-01or J s.

For your information

The-sphere in-rotating (a)right-hand-rule-in (b)

The sphere in (a) is rotating in the sense given by the gold arrow. Its angular velocity and angular momentum are taken to be upward along the rotational axis as shown by the right-hand rule in (b).

 If the particle is moving in a circle of radius r with uniform angular velocity ω, then angle between r and tangential velocity is 90ᵒ. Hence

L = mrv sin 90ᵒ = mrv

But                         v = rω

Hence                   L = m r2 ω

Now consider a symmetric rigid body rotating about a fixed axis through the centre of mass as shown in Fig 5.11. Each particle of the rigid body rotates about the same axis in a circle with an angular velocity ω. The magnitude of the angular momentum of the particle of mass mi is mi vi ri about the origin O. the direction of Li is the same as that of ω. Since vi = ri2 ω. Summing this over all particles gives the total angular momentum of the rigid body.

angular-moment-equation-02

rigid-body-rotating

Fig: 5.11

Where I is the moment of inertia of the rigid body about the axis of rotation.

Physicists usually make a distinction between spin angular momentum (Ls) and orbital angular momentum (L˳). The spin angular momentum is the angular momentum is s spinning body, while orbital angular momentum is associated with the motion of a body along a circular path.

The difference is illustrated in Fig. 5.12. In the usual circumstances concerning orbital angular momentum, the orbital radius is large as compared to the size of the body, hence, the body may be considered to be a point object.

orbital-angular-momentum

Fig: 5.12

Example 5.4: the mass of Earth is 6.00 x angular-moment-equation-03 kg. The distance r from Earth to the Sun is 1.50 x angular-moment-equation-04 m. as seen from the dirction of the North Star, the Earth revolves counter-clockwise around the Sun. determine the orbital angular momentum of the Earth about the Sun, assuming that it traverses a circular orbit about the Sun once a year (3.16 x angular-moment-equation-05 ).

Solution: To find the Earth’s orbital angular momentum we must first know its orbital speed from the given data. When the Earth moves around a circle of radius r, it travels a distance of 2πr in one year, its orbital speed v₀ is thus

angular-moment-equation-06

The sign is positive because the revolution is counter clockwise.