Question

A person with mass m_p = 70 kg stands on a spinning platform disk with a radius of R = 2.1 m and mass m_d= 187 kg. The disk is initially spinning at ω = 1.4 rad/s. The person then walks 2/3 of the way toward the center of the disk (ending 0.7 m from the center).  

1. What is the total moment of inertia of the system about the center of the disk when the person stands on the rim of the disk? I got 721
2. What is the total moment of inertia of the system about the center of the disk when the person stands at the final location 2/3 of the way toward the center of the disk?
3. What is the final angular velocity of the disk?
4. What is the change in the total kinetic energy of the person and disk? (A positive value means the energy increased.)
5. What is the centripetal acceleration of the person when she is at R/3?
6. If the person now walks back to the rim of the disk, what is the final angular speed of the disk?

Answer 1

What is the total moment of inertia of the system about the center of the disk when the person stands on the rim of the disk?

The moment of inertia of the uniform disk is
I = ½mr²

The moment of inertia of the person acts like a point mass so is

I = mr²

The total moment of inertia

I₀ = ½mdr² + mpr²
I₀ = r²(½md + mp)
I₀ = 2.1²(187/2 + 70)
I₀ = 721.035 kg•m²


What is the total moment of inertia of the system about the center of the disk when the person stands at the final location 2/3 of the way toward the center of the disk?

I₁ = ½mdr² + mpr₁²
I₁ = ½mdr² + mp(r/3)²
I₁ = r² (md/2 + mp/9)

I₁ = 2.1²(187/2 + 70/9)
I₁ = 446.635 kg•m²


What is the final angular velocity of the disk?

angular momentum will be conserved

I₀ω₀ = I₁ω₁
ω₁ = I₀ω₀ / I₁

ω₁ = 721.035(1.4) / 446.635
ω₁ = 2.26012...
ω₁ = 2.26 rad/s

What is the change in the total kinetic energy of the person and disk?

KE = ½Iω²

KE₀ = ½Iω₀²
KE₀ = ½(721.035)1.4²
KE₀ = 706.6143 J

KE₁ = ½(446.635)2.26²
KE₁ = 1140.6162 J

ΔKE = KE₁ - KE₀
ΔKE = 1140.6162 - 706.6143
ΔKE = 434.003 J

What is the centripetal acceleration of the person when she is at R/3?

ac = ω²r
ac = 2.26²(2.1/3)
ac = 3.575 m/s²


If the person now walks back to the rim of the disk, what is the final angular speed of the disk?

conservation of momentum will still occur so original velocity will be restored.

ω = 1.4 rad/s