Final answer:
The angular velocity of a rotating object like a sphere or a merry-go-round can be determined using the conservation of angular momentum. However, without an initial rotational speed or external forces given for the sphere in the example question, its angular velocity cannot be calculated. For the merry-go-round, adding a child will change its angular velocity due to the change in total moment of inertia.
Step-by-step explanation:
Understanding Angular Velocity in Rotational Dynamics
When dealing with a rotating sphere or any object in rotational motion, one of the key concepts to understand is angular velocity. Angular velocity is a measure of the rate of rotation, specifying how fast an object is rotating about an axis. If the initial angular velocity of an object is known, and we apply the law of conservation of angular momentum, we can solve for the final angular velocity when a mass is added or removed from the object. However, in the example question provided about the sphere, there is insufficient information to calculate its angular velocity as no initial rotational speed or external forces are given.
Applying this concept to a playground merry-go-round, when a child jumps onto the merry-go-round, the system's total angular momentum is conserved. Since the child is initially at rest, their addition to the merry-go-round will affect its angular velocity. The conservation of angular momentum can be represented by the equation I1 * w1 = I2 * w2, where I denotes the moment of inertia and w denotes the angular velocity. To solve for the new angular velocity, we can rearrange this equation to w2 = (I1 * w1) / I2, accounting for the change in moment of inertia due to the added mass of the child.