Question

a 2.7-m-diameter merry-go-round with rotational inertia 130 kg⋅m2kg⋅m2 is spinning freely at 0.50 rev/srev/s . four 25-kg children sit suddenly on the edge of the merry-go-round.Part A Find the new angular speed. Express your answer using two significant figures. O AU A O O ? V = rev/s

Answer 1

To find the new angular speed when four 25-kg children sit suddenly on the edge of the merry-go-round, we can use the principle of conservation of angular momentum. The final angular speed is approximately 0.178 rev/s.

Step-by-step explanation:

To find the new angular speed when four 25-kg children sit suddenly on the edge of the merry-go-round, we can use the principle of conservation of angular momentum. The initial angular momentum is given by Li = Iiωi, where Ii is the initial moment of inertia and ωi is the initial angular speed.

Given:

• Diameter of the merry-go-round (D) = 2.7 m
• Radius (r) = D / 2 = 2.7 m / 2 = 1.35 m
• Initial moment of inertia (Ii) = 130 kg·m2
• Initial angular speed (ωi) = 0.50 rev/s
• Number of children (n) = 4
• Mass of each child (m) = 25 kg

The total mass added to the merry-go-round is M = nm = 4 * 25 kg = 100 kg. The final moment of inertia can then be calculated as If = Ii + MR2, where R is the radius of the extended merry-go-round including the children's seating area. With four children sitting at the edge, R = r + 1 m (assuming each child occupies a space of 0.25 m).

Substituting the given values into the formulas and calculating, we find that the final moment of inertia is If = 130 kg·m2 + 100 kg * (1.35 m + 1 m)2 = 130 kg·m2 + 100 kg * 2.35 m2 = 130 kg·m2 + 235 kg·m2 = 365 kg·m2.

The final angular speed can be calculated as ωf = Li / If = Iiωi / If, where Li = Iiωi.

Substituting the given values, we get ωf = 130 kg·m2 * 0.50 rev/s / 365 kg·m2 ≈ 0.178 rev/s.

Answer 2

In summary, the new angular speed of the merry-go-round after the four children sit on the edge can be calculated using conservation of angular momentum. The initial angular momentum is equal to the final angular momentum, allowing for the calculation of the new angular velocity.

Step-by-step explanation:

The question involves the concept of conservation of angular momentum in Physics, specifically in a rotational dynamics context. When four 25-kg children sit suddenly on the edge of a 2.7-m-diameter merry-go-round with rotational inertia of 130 kg·m2 that's spinning freely at 0.50 rev/s, the angular velocity changes due to the fact that the moment of inertia of the system increases.

To find the new angular speed, we use the formula:

L = Iinitial × ωinitial = Ifinal × ωfinal

Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. Assuming no external torque and that angular momentum is conserved, we calculate Ifinal by adding the moment of inertia of the children sitting at the rim of the merry-go-round to the original moment of inertia of the system. With the new moment of inertia, we can solve for ωfinal.