Final answer:
The angular velocity of an object rotating in a circle increases as the radius decreases due to the conservation of angular momentum, assuming no external torque is applied.
Step-by-step explanation:
When a small mass m on a string is rotating in a circle and the string is shortened, the angular velocity of the object increases. This can be understood through the conservation of angular momentum which states that if no external torque is applied to a system, the angular momentum of the system remains constant. Angular momentum is a product of the rotational inertia (moment of inertia) and the angular velocity. As the string is shortened, the moment of inertia decreases because the mass m is moving closer to the axis of rotation. To keep the angular momentum constant, the angular velocity must increase proportionally.
This concept is parallel to what occurs when a figure skater pulls in their arms during a spin, resulting in a faster spin, and is also analogous to a child moving inward on a merry-go-round, causing it to rotate faster. Both of these examples reflect the principles of conservation of angular momentum in the absence of external friction or torque.