A child of mass 40 kg stands beside a circular platform of mass 100 kg and radius 1.8 m spinning at 2.6 rad/s. Treat the platform as a disk. The child steps on the rim.

a) What is the new angular speed;
rad/s

b) She then walks to the center and stay there. What is the angular velocity of the platform then?
rad/s

c) What is the change in kinetic energy when she walks from the rim to the center of the platform?

To solve this problem, we need to apply the principles of conservation of angular momentum and conservation of energy.

a) What is the new angular speed?

To find the new angular speed, we can use the principle of conservation of angular momentum. The initial angular momentum of the system is given by:

L_initial = I_initial * ω_initial

where L_initial is the initial angular momentum, I_initial is the initial moment of inertia, and ω_initial is the initial angular speed.

The moment of inertia of the system can be calculated using the formula for the moment of inertia of a disk:

I = 0.5 * m * r^2

where m is the mass of the platform and r is its radius.

Given:
Mass of the platform (m) = 100 kg
Radius of the platform (r) = 1.8 m
Initial angular speed (ω_initial) = 2.6 rad/s

Using the formula for the moment of inertia, we can calculate the initial moment of inertia (I_initial):

I_initial = 0.5 * m * r^2
= 0.5 * 100 kg * (1.8 m)^2

Substituting the values, we find:

I_initial = 0.5 * 100 kg * (1.8 m)^2 = 162 kg * m^2

Now, we can calculate the initial angular momentum (L_initial):

L_initial = I_initial * ω_initial
= 162 kg * m^2 * 2.6 rad/s

Substituting the values, we find:

L_initial = 162 kg * m^2 * 2.6 rad/s

b) What is the angular velocity of the platform when the child walks to the center?

When the child walks to the center, the moment of inertia of the system changes. The new moment of inertia (I_final) is given by:

I_final = I_platform + I_child

where I_platform is the moment of inertia of the platform and I_child is the moment of inertia of the child.

The moment of inertia of the platform (I_platform) can be calculated using the formula for a disk as mentioned earlier. Since the child walks to the center, the moment of inertia of the child (I_child) can be considered negligible.

Given:
Mass of the child (m_child) = 40 kg
Radius of the platform (r) = 1.8 m

Using the formula for the moment of inertia, we can calculate the moment of inertia of the platform (I_platform):

I_platform = 0.5 * m_platform * r^2
= 0.5 * 100 kg * (1.8 m)^2

Substituting the values, we find:

I_platform = 0.5 * 100 kg * (1.8 m)^2 = 162 kg * m^2

Now, the new moment of inertia (I_final) is:

I_final = I_platform + I_child (negligible)
= 162 kg * m^2 + 0 kg * m^2

Thus, I_final = 162 kg * m^2

The angular momentum is conserved, so we can use the principle of conservation of angular momentum to find the final angular velocity (ω_final) of the platform:

L_initial = L_final

I_initial * ω_initial = I_final * ω_final

Substituting the values, we have:

162 kg * m^2 * 2.6 rad/s = 162 kg * m^2 * ω_final

Solving for ω_final, we find:

ω_final = (162 kg * m^2 * 2.6 rad/s) / (162 kg * m^2)

Simplifying, we get:

ω_final = 2.6 rad/s

c) What is the change in kinetic energy?

To calculate the change in kinetic energy, we need to compare the initial and final kinetic energies of the system.

The initial kinetic energy (KE_initial) is given by:

KE_initial = 0.5 * I_initial * ω_initial^2

Substituting the values, we find:

KE_initial = 0.5 * (162 kg * m^2) * (2.6 rad/s)^2

The final kinetic energy (KE_final) is:

KE_final = 0.5 * I_final * ω_final^2

Substituting the values, we find:

KE_final = 0.5 * (162 kg * m^2) * (2.6 rad/s)^2

Then, the change in kinetic energy (ΔKE) is given by:

ΔKE = KE_final - KE_initial

Substituting the values, we have:

ΔKE = (0.5 * (162 kg * m^2) * (2.6 rad/s)^2) - (0.5 * (162 kg * m^2) * (2.6 rad/s)^2)

Simplifying, we find:

ΔKE = 0

Therefore, the change in kinetic energy is zero.