Question

A student of mass \(52 \mathrm{~kg}\) wants to measure the mass of a playground merry-go-round, which consists of a solid metal disk of radius \(R=1.5 \mathrm{~m}\) that is mounted in a horizontal position on a low-friction axle. She tries an experiment: She runs with speed \(v=6.8 \mathrm{~m} / \mathrm{s}\) toward the outer rim of the merry-go-round and jumps on to the outer rim, as shown in the figure. The merry-go-round is initially at rest before the student jumps on and rotates at \(1.3 \mathrm{rad} / \mathrm{s}\) immediately after she jumps on. You may assume that the student's mass is concentrated at a point. a) What is the mass of the merry-go-round? b) If it takes 35 s for the merry-go-round to come to a stop after the student has jumped on, what is the average torque due to friction in the axle? c) How many times does the merry-go-round rotate before it stops, assuming that the torque due to friction is constant?

Short answer

Answer: The mass of the merry-go-round is approximately 254.62 kg, the average torque due to friction in the axle is approximately 30.40 Nm, and the merry-go-round rotates approximately 29.4 times before coming to a stop.

Step-by-step solution

a) Finding the mass of the merry-go-round

Using the conservation of angular momentum, we can write the equation as: Initial angular momentum = Final angular momentum m_s * v * R = (m_s + M) * I * ω where m_s is the mass of the student, v is the student's speed, R is the merry-go-round radius, I is the moment of inertia of the merry-go-round, M is the mass of the merry-go-round, and ω is the angular speed. It is given that student's mass is concentrated at a point so we can ignore her moment of inertia. The moment of inertia of a solid disk is: \( I = \frac{1}{2} M R^2 \) Solving the equation for the mass of the merry-go-round (M), we have: \( M = \frac{2 m_s v R - m_s R^2 \omega}{R^2 \omega} \) Now, substitute the values: \( M = \frac{2 * 52 * 6.8 * 1.5 - 52 * 1.5^2 * 1.3}{1.5^2 * 1.3} \) Calculating the above expression, we get: M ≈ 254.62 kg So, the mass of the merry-go-round is approximately 254.62 kg.

b) Finding the average torque due to friction in the axle

The change in angular momentum of the merry-go-round will be equal to the torque multiplied by the time it takes to come to a stop, which can be written as: ΔL = τ * t The initial angular momentum is equal to (m_s + M) * I * ω, and the final angular momentum is 0 because it comes to a stop. Therefore, ΔL = (m_s + M) * I * ω Calculating the torque τ, we have: τ = ΔL / t Substitute the values and compute the torque: τ = [(52 + 254.62) * (1/2) * 254.62 * 1.5^2 * 1.3] / 35 Calculating the above expression, we get: τ ≈ 30.40 Nm So, the average torque due to friction in the axle is approximately 30.40 Nm.

c) Finding the number of rotations before the merry-go-round stops

The work done by frictional torque to stop the merry-go-round equals the initial rotational kinetic energy. So, we can write the equation as: τ * θ = (1/2) * (m_s + M) * I * ω^2 Solving the equation for θ (the number of radians turned before stopping), we have: θ = \(\frac{(m_s + M) * I * \omega^2}{2 * \tau}\) Now, substitute the values: θ = \(\frac{(52 + 254.62) * (1/2) * 254.62 * 1.5^2 * 1.3^2}{2 * 30.40} \) Calculating the above expression, we get: θ ≈ 184.67 rad To find the number of rotations (n), divide the number of radians by 2π: n = θ / (2 * π) Calculating the above expression, we get: n ≈ 29.4 So, the merry-go-round rotates approximately 29.4 times before coming to a stop.

Key concepts

Moment of Inertia

The moment of inertia is a crucial concept when studying rotational dynamics. It's akin to mass in linear motion, as it quantifies how much torque is required for an object to achieve a certain angular acceleration.
The moment of inertia depends on the mass distribution relative to the axis of rotation. For simple shapes, formulas exist for calculating it:

• Solid Disk: The formula for a solid disk is given by \( I = \frac{1}{2} M R^2 \), where \( M \) is the mass and \( R \) is the radius.

The merry-go-round in our problem is a solid disk. Therefore, we use this formula to calculate how resistant it is to rotational acceleration.
Remember, just as more mass implies more inertia, a greater spread of that mass from the axis increases the moment of inertia, making it harder to begin rotation.

Torque

Torque is another fundamental concept in rotational mechanics. It is the force that causes an object to rotate around an axis. You can think of it as the rotational equivalent of linear force.
Mathematically, torque \( \tau \) is calculated as the force applied multiplied by the distance from the pivot point (lever arm). It is given by:

• \( \tau = F \times r \)

In our exercise, the average torque due to friction is responsible for bringing the merry-go-round to a stop over time. The relationship is that the change in angular momentum of the system is equal to this torque multiplied by the time period during which it acts.
The equation \( \Delta L = \tau \times t \) shows this directly. Here, \( \Delta L \) is the change in angular momentum, \( \tau \) is the torque, and \( t \) is the time it acts.

Conservation of Angular Momentum

The principle of conservation of angular momentum is a fascinating aspect of physics. It states that when no external torque acts on a system, the total angular momentum remains constant.
In the given exercise, the conservation of angular momentum is used to find the merry-go-round's mass. Initially, the student has a linear momentum since she is moving towards the merry-go-round. Once she jumps on, her momentum converts to angular momentum.
The equation for conservation is written as:

• Initial angular momentum = Final angular momentum

When the student jumps onto the merry-go-round, the system starts rotating. By setting the initial angular momentum (married with the running student's linear momentum) equal to the final angular momentum (of both the student and the merry-go-round rotating together), we can solve for variables like the mass of the merry-go-round.
It's a powerful tool in mechanics and highlights how movements fit together in rotational systems where external impacts are minimal or absent.