A skater spinning with angular speed of 1.5 rad/s draws in her outstretched arms thereby reducing her moment of inertia by a factor a 3.

a) Find her angular speed.
b)What is the ratio of her final angular momentum to her initial angular momentum?
c)Did her mechanical energy change? (Yes but why?)

I forgot to add one more part.

d) Determine the ratio of her final kinetic energy to her initial kinetic energy.

I*w stays the same due to the requirement for angular momentum conservation.

If I is cut in half, w must double.

KE is (1/2)I*w^2 = (1/2)(I*w)^2/I

Since I drops by half while I*w is constant, the kinetic energy must also double.

The extra KE comes from work done by the skater pulling in her arms. There is a lot of centrifugal resistance.

a) To find her angular speed after reducing her moment of inertia by a factor of 3, we can use the principle of conservation of angular momentum. According to this principle, the product of the moment of inertia and the angular speed should remain constant. Therefore, if the moment of inertia decreases by a factor of 3, the angular speed should increase by a factor of 3.

Initial angular speed = 1.5 rad/s
Final angular speed = (Initial angular speed) * (Factor of decrease in moment of inertia)
Final angular speed = 1.5 rad/s * 3 = 4.5 rad/s

b) The ratio of her final angular momentum to her initial angular momentum can be determined by using the formula for angular momentum:

Angular momentum = Moment of inertia * Angular speed

The ratio of the final angular momentum to the initial angular momentum is equal to:

(final moment of inertia * final angular speed) / (initial moment of inertia * initial angular speed)

Since the final moment of inertia is 1/3rd of the initial moment of inertia and the final angular speed is 3 times the initial angular speed, the ratio simplifies to:

(final moment of inertia * final angular speed) / (initial moment of inertia * initial angular speed)
= (1/3 * 4.5) / (1 * 1.5)
= 1/3

Therefore, the ratio of her final angular momentum to her initial angular momentum is 1/3.

c) Yes, her mechanical energy changed. Mechanical energy is the sum of kinetic energy and potential energy in a system. In this case, the skater reduced her moment of inertia by drawing in her arms, which means she decreased her potential energy. As a result, her mechanical energy changes because the potential energy is converted into kinetic energy due to the increase in her angular speed.

To solve this problem, we need to use the conservation of angular momentum and the principle of conservation of mechanical energy.

a) To find the skater's final angular speed, we can use the conservation of angular momentum, which states that the product of moment of inertia and angular speed should remain constant. Let's denote the initial moment of inertia as Ii, the initial angular speed as ωi, the final moment of inertia as If, and the final angular speed as ωf.

According to the problem, the skater reduces her moment of inertia by a factor of 3. This means that If = (1/3) * Ii. Since angular momentum is conserved, we have:

Ii * ωi = If * ωf

Substituting the values, we get:

Ii * ωi = (1/3) * Ii * ωf

Simplifying the equation:

ωf = 3 * ωi

Therefore, the skater's final angular speed is 3 times her initial angular speed.

b) The ratio of the final angular momentum to the initial angular momentum can be determined by dividing the final angular momentum (Lf) by the initial angular momentum (Li). Angular momentum is given by the product of moment of inertia and angular speed.

Li = Ii * ωi
Lf = If * ωf

Substituting the values:

Li / Lf = (Ii * ωi) / (If * ωf)

Since If = (1/3) * Ii and ωf = 3 * ωi, we have:

Li / Lf = (Ii * ωi) / ((1/3) * Ii * 3 * ωi)

Simplifying the equation:

Li / Lf = 1 / (1/3)

Li / Lf = 3

Therefore, the ratio of the final angular momentum to the initial angular momentum is 3.

c) The mechanical energy of a system is conserved only if there are no external forces doing work on it. In this case, when the skater draws in her outstretched arms, the work is done by internal forces (muscles). This work reduces the skater's potential energy but increases her kinetic energy. Therefore, her mechanical energy changes as the skater's potential energy decreases and her kinetic energy increases. Hence, the mechanical energy of the skater changes in this situation.