Merry-Go-Round - Conservation of Angular Momentum Problem

Summary

In this engaging physics lesson, students explore the dynamics of a merry-go-round with a 25 kg child moving from its edge to the center. By applying the principle of conservation of angular momentum, they calculate the increase in the system's final angular velocity to 2.4 radians per second. The discussion highlights the relationship between rotational inertia and angular velocity, with an emphasis on how the child’s movement reduces the system’s inertia, leading to increased rotational speed. The lesson also covers the change in kinetic energy, explaining how the child’s work in moving inward leads to a net increase in the system’s energy.

Takeaways

• 😀 The conservation of angular momentum is the key principle in this problem. Angular momentum remains constant in the absence of external torque, such as friction from the axle.
• 😀 The system consists of a 25 kg child and a 255 kg merry-go-round, where the child moves from the edge to the center, affecting the system's angular velocity.
• 😀 The rotational inertia of the merry-go-round remains constant throughout the problem, while the rotational inertia of the child changes as they move inward.
• 😀 Angular velocity of the merry-go-round increases as the child moves to the center, due to conservation of angular momentum. The system's total angular momentum must remain constant.
• 😀 The equation for the rotational inertia of a system of particles is the sum of the mass of each particle multiplied by the square of the distance from the axis of rotation.
• 😀 When treating the child as a point particle, their rotational inertia is calculated as the mass of the child times the square of the distance from the center of the merry-go-round.
• 😀 The change in kinetic energy of the system is positive because the child does work to move to the center, which increases the system’s kinetic energy.
• 😀 The final angular velocity of the system (merry-go-round + child) is calculated by considering both the initial and final moments of inertia and angular velocities.
• 😀 Work done by the child in moving inward results in an increase in the system's kinetic energy, as shown by the positive change in kinetic energy.
• 😀 The simplified model assumes the child has zero rotational inertia when at the center of the merry-go-round, though in real life, the child would have some rotational inertia even at the center.

Q & A

What is the role of angular momentum in this problem?

Angular momentum is conserved because there is no external torque acting on the system. The system consists of the child and the merry-go-round, and as the child moves from the edge to the center, the angular momentum of the system remains constant.

Why is the child treated as a point particle?

The child is treated as a point particle to simplify the calculation of rotational inertia. This assumption allows us to focus only on the distance from the axis of rotation without accounting for the child's actual size.

How does the change in the child's position affect the rotational inertia?

As the child moves closer to the center, their distance from the axis of rotation decreases, which reduces their rotational inertia. This decrease in rotational inertia causes an increase in the angular velocity of the system to conserve angular momentum.

Why does the final angular velocity of the system increase?

The final angular velocity increases because, as the child moves inward, the rotational inertia of the system decreases. Since angular momentum is conserved, a decrease in rotational inertia must be compensated by an increase in angular velocity.

What does the equation for the rotational inertia of the system of particles tell us?

The equation for the rotational inertia of a system of particles tells us that the rotational inertia is the sum of the masses of the particles multiplied by the square of their distances from the axis of rotation.

What assumption is made about the merry-go-round's axle?

It is assumed that the merry-go-round's axle has negligible friction, meaning it does not exert external torque on the system. This ensures that angular momentum is conserved throughout the process.

Why is the change in kinetic energy positive?

The change in kinetic energy is positive because the child must do work to move inward. As the child's rotational inertia decreases, the system's angular velocity increases, which results in a positive change in the system's kinetic energy.

What is the relationship between the radius of the merry-go-round and the change in kinetic energy?

The radius of the merry-go-round affects the change in kinetic energy because a larger radius increases the amount of work needed to move the child inward, leading to a larger change in kinetic energy.

How is the final angular velocity of the system calculated?

The final angular velocity is calculated using the conservation of angular momentum. The equation takes into account the initial and final rotational inertias of the child and the merry-go-round, and it results in a final angular velocity of 2.4 radians per second.

What is the significance of the child doing work on the system?

The child doing work on the system is significant because it increases the system's kinetic energy. The work done by the child to move towards the center of the merry-go-round results in an increase in the rotational kinetic energy of the system.