An ice skater spins with his arms outstretched. When he pulls his arms in and raises them above his head he spins much faster than before. Did a torque act on the ice skater? If not how could his angular velocity have increased?

No torque was exerted on the skater to increase the spin rate. The angular momentum I*w remained the same. The angular velocity w increased while the moment of intertia (I) decreased.

Thank you.

Yes, when the ice skater pulls his arms in and raises them above his head, a torque acts on him, causing an increase in his angular velocity. To understand why this happens, we need to consider the principle of conservation of angular momentum.

Angular momentum is the rotational equivalent of linear momentum. It is a property of rotating objects and is defined as the product of moment of inertia (I) and angular velocity (Ļ‰). According to the law of conservation of angular momentum, the total angular momentum of a system remains constant unless acted upon by an external torque.

When the ice skater is spinning with his arms outstretched, his moment of inertia is higher because his arms are extended, creating a larger "rotating mass" around his body. This means that he needs to exert more effort to rotate his body due to the increased resistance caused by the larger moment of inertia.

When the ice skater pulls his arms in and raises them above his head, he reduces his moment of inertia by bringing the mass of his arms closer to his body. Since angular momentum is conserved, reducing the moment of inertia results in an increase in angular velocity. As the skater reduces his rotational mass, he is able to spin faster, demonstrating the conservation of angular momentum.

So, in conclusion, the ice skater's angular velocity increases when he pulls his arms in and raises them above his head due to the conservation of angular momentum. This change in angular velocity is made possible by the torque exerted on the skater as a result of the change in his moment of inertia.