Final answer:
To determine the mass of the merry-go-round, one must calculate the angular acceleration based on the torque produced by the child's force and the time taken to reach the given angular speed, and then use the moment of inertia formula for a disk to solve for the mass.
Step-by-step explanation:
The question requires solving a problem of rotational kinematics and dynamics to find the mass of a merry-go-round based on a child exerting a known force and the resulting angular speed. To start, we know that the force applied causes the merry-go-round to acquire a certain angular speed over a given time, implying an angular acceleration. The torque τ produced by the child's force is equal to the force multiplied by the radius (r) of the merry-go-round (τ = rF). The angular acceleration (α) can be determined using the formula α = τ/I, where I is the moment of inertia of the merry-go-round.
Since the merry-go-round starts from rest and reaches an angular speed (ω) in a time (t), we can use the kinematic equation ω = αt to find α. Once we have α, we can rearrange the formula to solve for the moment of inertia (I = τ/α). The moment of inertia for a disk-shaped object is given by I = (1/2)mR², allowing us to solve for the mass (m) of the merry-go-round.
To convert angular speed from rev/s to rad/s, we can use the conversion 1 rev/s = 2π rad/s. The final step involves substituting the known values and calculations into the equations to derive the mass of the merry-go-round, making sure to use consistent units throughout.