# Feynman Lectures Vol1: 18-2 Rotation of a rigid body - conservation of angular momentum

http://www.feynmanlectures.caltech.edu/I_18.html

I am not able to understand how angular momentum is conserved. Without an external torque, the angular velocity will be constant. So, if I spin some object on some frictionless surface, it will keep spinning forever? But then, in section 18-2 Rotation of a rigid body, Feynman goes on to calculate the work done by the rotating object.

$$\Delta W=F_x\Delta x+F_y\Delta y$$(18.10)

When the object rotates, there is a change in the direction of velocity. So, without a constant external force, how can the body maintain its angular momentum? If without external torque an object will maintain its angular velocity, then how can we calculate "work done" by moving a certain degrees? Ie, if at 45 degrees it has a certain kinetic energy, it must be having the same energy after turning 50 degrees if we say angular momentum has not changed. How is it that here change in direction of velocity not considered as acceleration? What am I missing here?

## 2 Answers

If you consider a small part of a rotating body, then you are correct, it is accelerating as it rotates, because it is travelling around a circle.

The forces which cause the acceleration are the internal stresses in the body.

However if you consider the whole rotating body, you can ignore the work done by the internal stresses. If you imagine that you cut the body at any point, the internal forces on either side of the cut are equal and opposite, and therefore they do opposite amounts of work since the motion of both sides of the cut is the same. The total amount of work done by the internal forces is therefore $0$.

I don't remember if Feynman discusses that anywhere, but usually it is covered in a classical mechanics when you consider a finite sized rigid body to be a collection of point particles, apply Newton's laws to each particle, and then sum (or integrate) over the whole body.

Also, don't forget if a point particle is moving uniformly in a circle, even an external force which is accelerating it towards the center of the circle does no work, because the direction of the displacement is always perpendicular to the direction of the force.

Mathematically, as $\frac{dP}{dt} = F_{net}$ and we assume $F_{net}$ to be 0, P must be const.

Velocity must be constant - which is Newton's first law {P is momentum, F is force}

Similarly, as $\frac{dL}{dt} = T_{net}$ ; if $T_{net} = 0$; L must be constant. {L is angular momentum; T is torque}

Conservation of angular momentum is more general as conservation of momentum is a case of conservation of angular momentum where the radius is Infinity.