since angular momentum is conserved, an ice-skater who throws her arms out will a. rotate more slowly. b. rotate more quickly. c. rotate at the same rate. d. stop rotating entirely.

Moment of inertia, often symbolized as \( I \), is a measure of how much resistance an object has to changes in its rotational motion. Think of it as the rotational equivalent of mass in linear motion. The greater the moment of inertia, the harder it is to spin the object or change its spin rate.
For the ice skater, when she extends her arms out, she increases her moment of inertia because she is spreading her mass farther from her rotational axis. The moment of inertia for an extended object is higher compared to when the mass is closer to the axis of rotation. In simpler terms, by throwing her arms out, the skater makes it more difficult for herself to rotate.
Moment of inertia depends on two key factors:

• The mass of the object.
• How that mass is distributed in relation to the axis of rotation.

In the case of the ice skater, extending the arms away from her body distributes the mass over a larger distance, thereby increasing her moment of inertia.

Angular velocity, usually denoted by \( \omega\ \), describes how fast an object rotates or spins. It's the rate at which the angle changes, measured in radians per second (rad/s).
If we think about the ice skater, her angular velocity tells us how many rotations she completes in a given amount of time. When she pulls her arms in, she spins faster; when she extends her arms out, she spins more slowly. This change in rotational speed is directly related to the conservation of angular momentum.
Consider this equation for angular momentum: \[ L = I \times \omega \]Where:

• \( L \): Angular momentum
• \( I \): Moment of inertia
• \( \omega \): Angular velocity

Since the angular momentum \( L \) is conserved, if the moment of inertia \( I \) increases (arms extended), the angular velocity \( \omega \) must decrease to keep \( L \) constant. This relationship ensures that the skater will rotate more slowly when she extends her arms.

The primary physics principles at work in this scenario involve conservation laws, particularly the conservation of angular momentum. In a system free from external torques, the total angular momentum remains constant. For our ice skater, this means that her initial angular momentum before extending her arms is the same as her angular momentum after.
This principle is observed in various physical scenarios and everyday activities, such as:

• A spinning student sitting on a swivel chair. Arms pulled in results in faster spinning; arms out means slower spinning.
• A figure skater's spins during a performance.
• A diver tucking and untucking their body to control somersault speed.

Moreover, the relationship among moment of inertia, angular velocity, and the conservation principle is fundamental in understanding rotational motion in physics.
By grasping these core concepts, students can better understand how rotational dynamics work and predict the outcomes in similar situations involving angular momentum conservation.