# Why is net torque zero in this case?

As an application of conservation of angular momentum around a fixed axis the following example was given in our textbook. If a girl sitting on a swivel chair starts rotating with her arms streched as shown then her angular speed reduces. Similarly as she brings her arms closer angular velocity automatically increases.

My book explains the following reason, Due to constant angular momentum we have that $$Iw$$ is constant. As we stretch the arms, the radius increases and as a consequence $$w$$ decreases and vice versa.

My question:-

Why at all is the angular momentum conserved in this case? In other words why is net torque zero?

The net torque is approximately zero if we assume the friction between the chair and its vertical pivot is negligible or is small enough to not change the angular momentum during the process of stretching arms. Beside this friction, non of the other external forces applied to the girl+chair system (gravity or the normal force from the ground) have a torque with a component in the vertical axis.

• But the girl is moving streching her arms. Isn't the $r×F$ along the vertical axis? Commented Apr 10, 2022 at 11:04
• @Rahul That force is in the radial direction, so its torque is zero. More importantly, that's an internal force, and for total angular momentum, we only need to consider external forces, that's the most important point about this problem. Commented Apr 10, 2022 at 12:19
• I forgot to mention that I assume the condition of symmetric about the axis of rotation. So assuming that , Is it true that a torque that's generated by a force is cancelled by torque generated by force which is opposite to it. If it's so, then is the angular momentum also zero? As all velocity vectors also can be cancelled Commented Apr 10, 2022 at 13:04
• Yes, the forces are cancelled by each other because of Newton's third law. But the angular momentum is not zero. Just like an object moving with constant velocity, even when the total applied force is zero, it can be moving. Commented Apr 10, 2022 at 13:06