An ice skater spinning with outstretched arms has an angular...

An ice skater spinning with outstretched arms has an angular speed of $4.0 \mathrm{rad} / \mathrm{s}$. She tucks in her arms, decreasing her moment of inertia by $7.5 \% .$ (a) What is the resulting angular speed? (b) By what factor does the skater's kinetic energy change? (Neglect any frictional effects.) (c) Where does the extra kinetic energy come from?

An ice skater spinning with outstretched arms has an angular speed of $4.0 \mathrm{rad} / \mathrm{s}$. She tucks in her arms, decreasing her moment of inertia by $7.5 \% .$
(a) What is the resulting angular speed? (b) By what factor does the skater's kinetic energy change? (Neglect any frictional effects.)
(c) Where does the extra kinetic energy come from?

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is given by the formula (1/2)I?², where I is the moment of inertia and ? is the angular velocity. Changes in moment of inertia and angular speed affect the kinetic energy of the system. When the moment of inertia decreases and angular speed increases, the kinetic energy typically increases, leading to questions about the source of the additional energy.

Work Done by Internal Forces

The extra kinetic energy observed when the moment of inertia decreases is due to work done by internal forces within the system. In systems like a spinning skater, the muscular forces exerted while changing the posture do work, thereby converting chemical energy into additional rotational kinetic energy, even though the overall angular momentum remains constant.

Conservation of Angular Momentum

This principle states that if no external torques act on a system, its total angular momentum remains constant. When the distribution of mass changes, as in a change in moment of inertia, the angular velocity adjusts inversely to conserve the overall angular momentum. This fundamental concept is crucial in understanding rotational dynamics in isolated systems.

Moment of Inertia

The moment of inertia is a measure of how mass is distributed relative to the axis of rotation. It quantifies an object’s resistance to changes in its rotational motion. A decrease in the moment of inertia, such as when mass is moved closer to the axis, leads to an increase in angular speed if angular momentum is conserved.

Transcript

00:01 In this problem, we'll see how a skater manages to spin their self up into a faster spin by tucking their arms closer to their body.

00:12 The concept we use here is conservation of angular momentum, which is something that you can apply if there's no net external torque, no external torque acting on a body, or if that external torque is negligent.

00:31 Which we assume is true because this person is on ice and there's nothing creating a force far away from the axis of the spin.

00:47 Reminder, angular momentum is l and it is calculated as the product of the moment of inertia times the angular velocity, the rotational analogs of mass times linear velocity.

01:06 But here we know the initial omega angular velocity of the spin before the tuck is four rads per second.

01:18 And what we know is that the moment of inertia drops by 7 .5%.

01:27 So that would bring it down to a factor of, 92 .5 % of the initial moment of inertia.

01:42 Okay, there we go.

01:45 Okay, but we'll want to set the initial and final angular momenta equal to each other, and it is the omega -2 that we are trying to determine.

02:07 So we can solve for that.

02:13 Thankfully, we don't have to know.

02:16 The absolute value of the moment of inertia, just how much it dropped by, and the initial angular velocity, we do know absolutely.

02:39 So we can work out that, yes, their final angular velocity is a little bit bigger, goes up to 4 .3 rads per second.

02:54 And it's an interesting question.

02:56 This is not an elastic type of situation where mechanical energy is conserved necessarily...

Please give Ace some feedback