Question

An ice skater rotating on frictionless ice brings her hands into her body so that she rotates faster. Which, if any, of the conservation laws hold? a) conservation of mechanical energy and conservation of angular momentum b) conservation of mechanical energy only c) conservation of angular momentum only d) neither conservation of mechanical energy nor conservation of angular momentum

Short answer

a) conservation of mechanical energy only b) conservation of mechanical energy and conservation of angular momentum c) conservation of angular momentum only d) neither conservation of mechanical energy nor conservation of angular momentum Answer: c) conservation of angular momentum only

Step-by-step solution

Conservation of Mechanical Energy

Mechanical energy is conserved when the sum of kinetic energy and potential energy remains constant throughout the motion. In the case of the ice skater, there is no change in her height while she rotates, so her gravitational potential energy remains constant. Her initial and final total mechanical energy can be given by: Initial: \(E_{initial}=K_{initial}+U_{initial}\) Final: \(E_{final}=K_{final}+U_{final}\) We need to compare these two expressions to determine if the mechanical energy is conserved.

Conservation of Angular Momentum

Angular momentum is conserved when there is no external torque acting on the system. The ice skater is said to be on frictionless ice, meaning that there will be no external torque acting on her. Her initial and final total angular momentum can be given by: Initial: \(L_{initial}=I_{initial}ω_{initial}\) Final: \(L_{final}=I_{final}ω_{final}\) We need to compare these two expressions to determine if the angular momentum is conserved.

Comparing Initial and Final Mechanical Energy

As we have already mentioned, the ice skater's potential energy remains constant during her motion. Therefore, we can compare her initial and final kinetic energy. Kinetic energy is given by the expression: \(K = \frac{1}{2}Iω^2\) Initially, the ice skater has her hands stretched out, and her moment of inertia is larger. When she brings her hands closer to her body, her moment of inertia decreases. However, her angular speed \(\omega\) increases so that her angular momentum is conserved. There is a change in kinetic energy, which implies that her mechanical energy is not conserved.

Comparing Initial and Final Angular Momentum

As we noted earlier, there is no external torque acting on the system, and thus her angular momentum should be conserved. Since her moment of inertia decreases and angular speed increases, it appears that the product of the two remains constant for the initial and final state. This means her angular momentum is conserved.

Choose the Correct Answer

Based on our observations in Steps 3 and 4, we found that the mechanical energy is not conserved, while the angular momentum is conserved. Therefore, the correct answer is: c) conservation of angular momentum only

Key concepts

Conservation of Mechanical Energy

In physics, one of the most fundamental principles is the conservation of mechanical energy. This law asserts that the total mechanical energy (the sum of kinetic and potential energies) in an isolated system remains constant over time, in the absence of non-conservative forces, such as friction or air resistance.

Take for instance our ice skater: As she pirouettes, we may observe changes in her speed but not in her height above the ice, implying a constant potential energy and a varying kinetic energy. If she were in a completely isolated system, we'd expect her mechanical energy to be conserved. However, it's crucial to remember that specific actions, such as an ice skater pulling her arms in, can convert potential energy into kinetic energy and vice versa, without adding or losing energy to the system as a whole.

Conservation of Angular Momentum

Closely related to the previous concept is the conservation of angular momentum, which is particularly salient in situations involving rotation or revolutions. According to this principle, if no external torques act upon a system, the total angular momentum remains constant. An external torque refers to a force that causes an object to rotate.

In the case of our twirling skater, the frictionless ice means that no external forces cause her rotation to speed up or slow down. Her initial state, with arms outstretched, and final state, with arms pulled in, may look different, but the invisible quantity, angular momentum, sticks to its initial value, regardless of her body position.

Kinetic Energy

Delving deeper into the dynamics of the skater's performance, kinetic energy plays a starring role. It is the energy an object possesses due to its motion. Calculated using the formula
\(K = \frac{1}{2}mv^2\)
where \(m\) is the mass and \(v\) is the velocity, it showcases how speed influences an object's energy. In rotational motion, like an ice skater spinning, the kinetic energy is instead related to angular velocity \(\omega\) and the moment of inertia \(I\) and is given by:
\(K = \frac{1}{2}I\omega^2\).

As our skater pulls her arms in and her moment of inertia decreases, her angular velocity must increase to conserve angular momentum, which in turn causes a spike in her kinetic energy.

Moment of Inertia

The concept of moment of inertia is crucial to understanding the rotation of bodies. It is essentially the rotational equivalent of mass for linear motion – a measure of an object's resistance to changes in its rotation rate. The moment of inertia depends on the object's mass distribution relative to the rotation axis. For a point mass, it's calculated by
\(I = mr^2\)
where \(m\) is mass and \(r\) is the distance from the rotation axis.

For our skater, the initial, arms-outspread position means a high moment of inertia. When she draws her arms in, she’s not only changing her shape but also reducing the radius through which her mass is distributed, therefore reducing her moment of inertia – leading to the accelerated rotation observed by the audience.