An ice skater is spinning about a vertical axis with arms fully extended. If the arms are pulled in closer to the body, in which of the following ways are the angular momentum and kinetic energy of the skater affected?Answer ChoicesAngular MomentumKinetic EnergyAIncreasesIncreasesBIncreasesRemains ConstantCRemains ConstantIncreasesDRemains ConstantRemains Constant

To answer the question, we need to consider the principles of angular momentum and kinetic energy in rotational motion.

1. Angular Momentum: Angular momentum ($L$) is given by the product of the moment of inertia ($I$) and the angular velocity ($\omega$): $L = I \omega$. According to the conservation of angular momentum, if no external torque acts on the system, the angular momentum remains constant. When the skater pulls their arms in, the moment of inertia decreases. To conserve angular momentum, the angular velocity must increase.

2. Kinetic Energy: Rotational kinetic energy ($K$) is given by $K = \frac{1}{2} I \omega^2$. When the skater pulls their arms in, the moment of inertia decreases, but the angular velocity increases. Since the increase in angular velocity is squared in the kinetic energy formula, the overall kinetic energy increases.

Given these principles, the correct answer is:

Angular Momentum: Remains Constant Kinetic Energy: Increases

Therefore, the correct choice is:

C Remains Constant Increases

To answer this question, we need to consider the principles of angular momentum and kinetic energy in rotational motion.

1. Angular Momentum (L):

   • Angular momentum is given by the product of the moment of inertia (I) and the angular velocity (ω): $L = I \cdot \omega$.
   • According to the conservation of angular momentum, if no external torque acts on the system, the angular momentum remains constant.

2. Moment of Inertia (I):

   • The moment of inertia depends on the mass distribution relative to the axis of rotation. When the skater pulls their arms in, the mass is closer to the axis, reducing the moment of inertia.

3. Angular Velocity (ω):

   • To conserve angular momentum, if the moment of inertia decreases, the angular velocity must increase.

4. Kinetic Energy (K):

   • Rotational kinetic energy is given by $K = \frac{1}{2} I \cdot \omega^2$.
   • When the skater pulls their arms in, the decrease in moment of inertia and the increase in angular velocity result in an increase in kinetic energy.

Given these principles, let's analyze the effects:

• Angular Momentum: Remains constant (due to conservation of angular momentum).
• Kinetic Energy: Increases (due to the increase in angular velocity).

Therefore, the correct answer is:

Angular Momentum: Remains Constant Kinetic Energy: Increases

The correct choice is:

C Remains Constant Increases