# Non-conservation of angular momentum example but is an external torque really required?

Say a man is twisting his body using internal body muscle contractions and torque interactions between his feet and ground (no slipping).

From the answers of a previous question I raised, these ground reaction forces/torques will not do any work on the man (and also no work done to the ground by him) but yet he is increasing his angular momentum and rotational kinetic energy. Can I therefore assume this means he is just transforming other forms of internal body energy to increase his rotational kinetic energy (especially if no work is done on him via the ground)?

So is the real reason why his angular momentum is not conserved due to a change from one form of internal energy to rotational kinetic energy? If yes, would it be incorrect to state that the reason the man’s angular momentum is not conserved is because of an external torque applied to him (ie. the system) from the ground?

Can I therefore assume this means he is just transforming other forms of internal body energy to increase his rotational kinetic energy (especially if no work is done on him via the ground)?

Yes

So is the real reason why his angular momentum is not conserved due to a change from one form of internal energy to rotational kinetic energy?

No, the angular momentum of the man is not conserved due to the external torque from the ground. Angular momentum and energy are completely separate quantities and each are independently conserved. You cannot convert energy into angular momentum. It makes no sense to attribute the change in angular momentum to a conversion of energy.

The man’s angular momentum changed. That change is due to the external torque from the ground.

Separately the man’s KE increased. No work was done on the man, so that increase in KE must have come from a corresponding decrease in some other form of internal energy.

The two are separate conservation laws that are individually and separately obeyed.

• Unsure what you mean when you say you cannot convert energy into angular momentum.The ground is doing no work on the man, therefore there is no increase in his own internal energy. Yet he is increasing his rotational kinetic energy using muscle contractions without any net change in his total internal energy (if we ignore internal friction in his joints and muscles).Therefore can't we assume that there is some transformation of his internal energy (potential energy?) into extra rotational kinetic energy?I think there is a mathematical relationship between the rotational KE and angular momentum Jan 11 at 0:02
• @Dubious angular momentum is not a type of energy and energy is not a type of angular momentum. They cannot be converted into each other. They are each separately conserved. Potential energy is a type of energy and can be converted into kinetic energy, not angular momentum.
– Dale
Jan 11 at 0:13
• Rotational KE is proportional to MOI and angular velocity (squared) , while angular momentum is proportional to MOI and angular velocity. Doesn't this prove there is a mathematical relationship between both? Jan 11 at 0:26
• The external torque exerted by the ground is to stop the man creating equal and opposite torques within his own body (ie. to prevent conservation of angular momentum). But I think his increase in rotational KE 'element' also causes an increase in his angular momentum. Jan 11 at 0:41
• @Dubious that is not how it works. Only external torque changes a system’s angular momentum. Only external work changes a system’s energy. One force may produce both torque and work, but they are still separate concepts and separately conserved quantities. An increase in rotational KE does not cause an increase in angular momentum. E.g. an ice skater spinning faster by pulling in their arms, rotational KE increases but angular momentum is constant.
– Dale
Jan 11 at 0:55

The man's angular momentum after rotationally pushing his feet against the floor will be exactly countered by an opposite momentum induced in the planet earth (to which the floor was attached).

• Yes I understand that bit if the system is Earth/Man - conservation of angular momentum will hold. My question was different where I was asking whether an external torque is really required for 'Non-Conservation of Angular Momentum' for the the man only. Jan 10 at 18:11