Final answer:
The problem involves calculating the change in angular velocity of a merry-go-round when four people step onto it, and when they jump off, by using the conservation of angular momentum.
Step-by-step explanation:
To address the question about the change in angular velocity when four people each with a mass of 66 kg step onto the edge of a 3.5-m-diameter merry-go-round, we must apply the principle of conservation of angular momentum. This principle states that if no external torques act on a system, the total angular momentum of that system is conserved. The initial angular momentum of the merry-go-round is given by the product of its moment of inertia and angular velocity before the people step onto it.
For part A of the question, this initial angular momentum is calculated as:
Linitial = Imerry-go-round × ωinitial.
When the four people step onto the merry-go-round, they add to its moment of inertia. The new total moment of inertia Itotal is the sum of the merry-go-round's initial moment of inertia and the moment of inertia of the four people treated as point masses at the radius (diameter/2). We can write the new moment of inertia as:
Itotal = Imerry-go-round + 4 × mperson × (radius)^2.
To find the new angular velocity, we set the initial angular momentum equal to the final angular momentum: Linitial = Itotal × ωnew and solve for ωnew.
For part B, if the people initially on the merry-go-round jump off, we'd apply the same principle in reverse. Since they are leaving the system and thus taking their moment of inertia with them, the merry-go-round's moment of inertia would decrease to its original value, and its angular velocity would increase according to the conservation of angular momentum.