To solve this problem, we apply the principle of conservation of angular momentum, which states:

$L_{\text{initial}} = L_{\text{final}}$

Given Data:

• Mass of cockroach ($m_{\text{c}}$) = $5.0 \, \text{kg}$
• Mass of disk ($M_{\text{d}}$) = $40.0 \, \text{kg}$
• Radius of disk ($R_{\text{d}}$) = $1.5 \, \text{m}$
• Initial angular speed ($\omega_{\text{initial}}$) = $2.0 \, \text{rad/s}$
• Initial position of the cockroach = $r_{\text{initial}} = 1.5 \, \text{m}$
• Final position of the cockroach = $r_{\text{final}} = 0.5 \, \text{m}$

Step 1: Moment of Inertia Calculation

Initial Moment of Inertia:

The moment of inertia for the system consists of two parts:

1. Moment of inertia of the disk: $\frac{1}{2} M_{\text{d}} R_{\text{d}}^2$
2. Moment of inertia of the cockroach: $m_{\text{c}} r_{\text{initial}}^2$

$I_{\text{initial}} = \frac{1}{2} M_{\text{d}} R_{\text{d}}^2 + m_{\text{c}} r_{\text{initial}}^2$

Substitute the values: $I_{\text{initial}} = \frac{1}{2} (40.0) (1.5)^2 + (5.0) (1.5)^2$

$I_{\text{initial}} = 45.0 + 11.25 = 56.25 \, \text{kg·m}^2$

Final Moment of Inertia:

The cockroach moves closer to the center. The final moment of inertia becomes: $I_{\text{final}} = \frac{1}{2} M_{\text{d}} R_{\text{d}}^2 + m_{\text{c}} r_{\text{final}}^2$

Substitute the values: $I_{\text{final}} = \frac{1}{2} (40.0) (1.5)^2 + (5.0) (0.5)^2$

$I_{\text{final}} = 45.0 + 1.25 = 46.25 \, \text{kg·m}^2$

Step 2: Conservation of Angular Momentum

The angular momentum before and after the cockroach moves must remain constant: $I_{\text{initial}} \omega_{\text{initial}} = I_{\text{final}} \omega_{\text{final}}$

Substitute the known values: $(56.25)(2.0) = (46.25) \omega_{\text{final}}$

$112.5 = 46.25 \omega_{\text{final}}$

Solve for $\omega_{\text{final}}$: $\omega_{\text{final}} = \frac{112.5}{46.25} \approx 2.43 \, \text{rad/s}$

Final Answer:

The new angular speed of the disk is approximately:

$\boxed{2.43 \, \text{rad/s}}$

Would you like a breakdown of the concepts or more questions? Here are 5 related problems:

1. What happens to the angular velocity if the cockroach moves even closer to the center?
2. How does the angular momentum of the system remain constant despite a change in rotational inertia?
3. What would be the new angular velocity if the cockroach stops at $0.75 \, \text{m}$ instead of $0.50 \, \text{m}$?
4. What is the physical significance of conservation of angular momentum in everyday systems?
5. How would the result change if the disk were not uniform?

Tip: Always check whether forces like friction or external torques are acting in angular momentum problems—they can affect