You are standing on a merry-go-round of mass 500 kg and radius 2 m. Initially you are at 1.2 m from the center and you are rotating with an angular speed of 45 rad/s. (a) If you move in to a point 0.2 m from the center, what speed will you be rotating at? Assume your mass is 60 kg. (b) Does the kinetic energy of you and the disk increase, decrease or stay the same?

To solve this problem, we need to apply the principle of conservation of angular momentum, which states that the total angular momentum of a system remains constant if there are no external torques acting on it.

(a) Let's find the initial angular momentum of the merry-go-round. The formula for angular momentum is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

The moment of inertia of a solid disk rotating about an axis through its center is given by I = (1/2)MR^2, where M is the mass of the disk and R is the radius.

Given:
Mass of the merry-go-round (disk), M = 500 kg
Radius of the merry-go-round (disk), R = 2 m
Initial distance from the center, r1 = 1.2 m
Initial angular velocity, ω1 = 45 rad/s
Mass of the person on the merry-go-round, m = 60 kg

First, let's find the moment of inertia of the merry-go-round:
I = (1/2)MR^2
I = (1/2)(500 kg)(2 m)^2
I = 1000 kg·m^2

Next, let's find the initial angular momentum of the merry-go-round:
L1 = Iω1
L1 = 1000 kg·m^2 * 45 rad/s
L1 = 45000 kg·m^2/s

When the person moves inward to a distance of 0.2 m from the center, the new distance becomes r2 = 0.2 m.

To find the new angular velocity ω2, we use the conservation of angular momentum: L1 = L2

Let's calculate the new angular velocity ω2:
L1 = L2
I1ω1 = I2ω2

Substituting the expressions for moment of inertia (I) and rearranging the equation, we get:
(1/2)MR1^2 * ω1 = (1/2)MR2^2 * ω2

Simplifying the equation further:
R1^2 * ω1 = R2^2 * ω2
(1.2 m)^2 * 45 rad/s = (0.2 m)^2 * ω2

Now we can solve for ω2:
ω2 = (1.2 m)^2 * 45 rad/s / (0.2 m)^2

Calculating ω2:
ω2 = 1296 rad/s

Therefore, when you move to a point 0.2 m from the center, you will be rotating with an angular speed of 1296 rad/s.

(b) The kinetic energy (KE) of an object is given by KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the linear velocity.

To determine if the kinetic energy of you and the disk increases, decreases, or stays the same, we need to compare the initial and final kinetic energies.

Initial linear velocity:
v1 = ω1 * r1

Substituting the given values:
v1 = 45 rad/s * 1.2 m
v1 = 54 m/s

Final linear velocity:
v2 = ω2 * r2

Substituting the calculated values:
v2 = 1296 rad/s * 0.2 m
v2 = 259.2 m/s

Now, let's calculate the initial and final kinetic energies:

Initial kinetic energy (KE1) = (1/2)m1v1^2
KE1 = (1/2)(60 kg)(54 m/s)^2

Final kinetic energy (KE2) = (1/2)m2v2^2
KE2 = (1/2)(60 kg)(259.2 m/s)^2

Calculating KE1 and KE2:
KE1 = 87480 J
KE2 = 4835712 J

Comparing the initial and final kinetic energy:

KE1 < KE2

The kinetic energy of you and the disk increases when you move closer to the center.