Question

A child exerts a tangential 44.8N force on the rim of a disk-shaped merry-go-round with a radius of 2.20m .

If the merry-go-round starts at rest and acquires an angular speed of 0.6250 rev/s in 4.50 s , what is its mass?

Answer 1

To find the mass of the merry-go-round, one must apply the relationship between force, torque, and angular acceleration, and use the moment of inertia for a disk to solve for the mass.

Step-by-step explanation:

To determine the mass of the merry-go-round, we must use the concept of angular momentum and apply the principle of torque and angular acceleration. The child applies a tangential force which generates a torque (τ) given by τ = r × F, where r is the radius of the merry-go-round and F is the force applied.

This torque leads to angular acceleration (α) according to Newton's second law for rotation: τ = Iα, where I is the moment of inertia of the merry-go-round. The angular acceleration is calculated from the change in angular velocity (ω) over time (t): α = Δω / Δt. We know the final angular velocity in terms of revolutions per second, but we'll need to convert it to radians per second to match the standard units for angular quantities. The moment of inertia for a solid disk is I = (1/2) × m × r², where m is the mass. We can solve for mass (m) by combining these equations and using the known quantities.

Now, let's calculate the moment of inertia and, subsequently, the mass of the merry-go-round:

1. First, convert the final angular velocity from rev/s to rad/s: ω = 0.6250 rev/s × (2π rad/rev) = 3.93 rad/s.
2. Next, we calculate angular acceleration: α = ω / t = 3.93 rad/s / 4.50 s = 0.873 rad/s².
3. Knowing that torque τ = r × F and α = τ / I, we find I = (r × F) / α.
4. Lastly, we use I = (1/2) × m × r² to solve for the mass m, rearranging to m = 2I / r².

After substituting the given values (r = 2.20 m, F = 44.8 N, and α = 0.873 rad/s²), we can determine the mass of the merry-go-round.