# Conservation of angular momentum

If **τ**_{ext}* = ***o, **Eq. (28b) reduces to

d**L/**d*t = *0

or **L = **constant ………………………………………………………………………………(29a)

Thus, if the total external torque on a system of particles is zero, then the total angular momentum of the system is conserved. i.e., remains constant. Eq. (29a) is equivalent to three scalar equations.

*L _{x} =K_{1} , L_{y} =K_{2}, L_{z} =K_{3}* …………………………………………………………………….(29b)

Here *K _{1} , K_{2}, *and

*K*are constants :

_{3}*L*and

_{x}, L_{y }*L*are the components of the total angular momentum vector

_{z }**L**along the

*x, y*and

*z*axes respectively. The statement that the total angular momentum is conserved means that each of these three components is conserved.

Eq. (29a) is the rotational analogue of Eq. (18a) i.e., the conservation law of the total linear momentum for a system of particles. Like Eq. (18a), it has applications in many practical situations. We shall look at a few of the interesting applications later on in this chapter.

*Example 5*

Find the torque of a force (7**i **+ 3**j **– 5**k**) from the origin. The force acts on a particle whose position vector is **i **–** j + k**.

##### Answer

Here **r = i – j + k**

and **F = 7i + 3j – 5k**

*Example 6*

Show that the angular momentum about any point of a single particle moving with constant velocity remain constant throughout the motion.

##### Answer

Let the particle with velocity **v **be at point P at some instant *t. *We want to calculate the angular momentum of the particle about an arbitrary point O.

The angular momentum is **l = r × ***m***v. **Its magnitude is *mvr *sin*θ, *where *θ *is the angle between **r **and **v **as shown in Fig. 19. Although the particle changes position with time, the line of direction of **v **remains the same and hence OM = *r *sin*θ, *is a constant.

Further, the direction of **l **is perpendicular to the plane of **r **and **v. **It is in to the page of the figure. This direction does not change with time.

Thus, **l **remains the same in magnitude and direction and is therefore conserved. Is there any external torque on the particle?