To find the torque that the axle must apply to prevent the disk from rotating, we need to calculate the net torque due to the two hanging masses and set it equal in magnitude and opposite in direction to the torque applied by the axle.

Given data:

• Mass of the left hanging block: $m_1 = 15$ kg
• Mass of the right hanging block: $m_2 = 10$ kg
• Mass of the disk: $M = 5.0$ kg (not directly relevant to torque calculation)
• Diameter of the disk: $d = 50$ cm $= 0.50$ m
• Radius of the disk: $r = \frac{d}{2} = 0.25$ m
• Axle is located at half the radius: $r_{\text{axle}} = \frac{r}{2} = 0.125$ m
• Acceleration due to gravity: $g = 9.81$ m/s²

Step 1: Calculate the torques due to the hanging masses

Torque is given by: $\tau = r \times F$ where $r$ is the radius and $F = mg$ is the force due to gravity.

Torque from the 15 kg mass:

$\tau_1 = (0.25)(15 \times 9.81)$ $\tau_1 = 0.25 \times 147.15 = 36.79 \text{ Nm (counterclockwise)}$

Torque from the 10 kg mass:

$\tau_2 = (0.25)(10 \times 9.81)$ $\tau_2 = 0.25 \times 98.1 = 24.53 \text{ Nm (clockwise)}$

Step 2: Find the net torque

$\tau_{\text{net}} = \tau_1 - \tau_2$ $\tau_{\text{net}} = 36.79 - 24.53 = 12.26 \text{ Nm (counterclockwise)}$

Step 3: Torque applied by the axle

Since the axle must counteract this net torque, and the axle is located at $r_{\text{axle}} = 0.125$ m:

$\tau_{\text{axle}} = -\tau_{\text{net}}$ $\tau_{\text{axle}} = -12.26 \text{ Nm}$

Thus, the torque that the axle must apply to prevent the disk from rotating is $-12.26$ Nm (clockwise).

Would you like a breakdown of any part of this calculation?

Here are five related questions to explore further:

1. How would the torque change if the axle were at the center instead of halfway?
2. What happens if the mass of the disk is doubled? Does it affect the torque required at the axle?
3. How do we calculate the angular acceleration of the disk if it were allowed to rotate?
4. What is the tension in each rope when the system is in equilibrium?
5. How would the torque requirement change if the disk were not a uniform solid disk?

Tip:

Torque is always calculated with respect to a chosen pivot point. Changing the pivot location can drastically alter the torque contributions!