an ice skater spins about a vertical axis at an angular speed of 15 rad/s when her arms are outstreched . she then quickly pulls her arms into her sides in a very small time interval so that the frictional force due to ice are negligible.her initial moment of inertia about the axis of rotation is 1.72kgm^2 and her final moment of inertia is 0.061kgm^2. what is the change in her angular speed? what is the change in her kinetic energy?explain this change in kinetic energy.

Because angular momentum is conserved,

I*w = (Moment of intertia)*(angular speed) remains constant
Therefore
1.72*(15 rad/s)= 0.061*(final angular speed)
Final angular speed = 15*1.72/0.061
= 423 rad/s

Kinetic energy = (1/2)*I*w^2
increases by the same (w2/w1) factor, sinc I*w is constant.

The added kinetic energy comes from work done by the skater pulling in her arms and free leg.

To calculate the change in angular speed and kinetic energy of the ice skater, we can use the conservation of angular momentum and the conservation of mechanical energy.

1. Change in Angular Speed:

The conservation of angular momentum states that the initial angular momentum is equal to the final angular momentum. Mathematically, it can be expressed as:

Initial Angular Momentum = Final Angular Momentum

Angular Momentum = Moment of Inertia × Angular Speed

Initial Moment of Inertia × Initial Angular Speed = Final Moment of Inertia × Final Angular Speed

Substituting the given values:

1.72 kgm² × 15 rad/s = 0.061 kgm² × Final Angular Speed

Solving for the final angular speed:

Final Angular Speed = (1.72 kgm² × 15 rad/s) / 0.061 kgm²

Therefore, the change in her angular speed is the difference between the final and initial angular speeds:

Change in Angular Speed = Final Angular Speed - Initial Angular Speed

2. Change in Kinetic Energy:

The change in kinetic energy can be calculated by subtracting the initial kinetic energy from the final kinetic energy. The formula for kinetic energy is:

Kinetic Energy = 1/2 × Moment of Inertia × Angular Speed²

Initial Kinetic Energy = 1/2 × 1.72 kgm² × (15 rad/s)²
Final Kinetic Energy = 1/2 × 0.061 kgm² × Final Angular Speed²

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy

Now, to understand this change in kinetic energy, we need to consider the work done by the skater in pulling her arms inward.

When the skater pulls her arms into her sides, she decreases her moment of inertia. This results in a decrease in the kinetic energy of the system since the final moment of inertia is smaller than the initial moment of inertia, as given in the question.

The work-energy principle states that the work done on an object is equal to its change in kinetic energy. In this case, the skater does work against her own body by decreasing her moment of inertia.

As a result, the skater's kinetic energy decreases when she pulls her arms in. The energy she had while being spread out is transformed into potential energy in her body.