20 Ν 40 N A disk is initially rotating counterclockwise around a fixed axis with angular speed wo- At time t = 0, the two forces shown in the figure above are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?
20 Ν 40 N A disk is initially rotating counterclockwise around a fixed axis with angular speed wo- At time t = 0, the two forces shown in the figure above are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?
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Transcribed Image Text:### Angular Velocity of a Rotating Disk Under External Forces
#### Problem Statement:
A disk is initially rotating counterclockwise around a fixed axis with angular speed \( \omega_0 \).
**Description of the Diagram:**
In the provided diagram, we can observe the following:
- The disk is subjected to two vertical forces.
- A 20 N force applied vertically downward on the left side.
- A 40 N force applied vertically downward on the right side.
- The center of the disk is marked, with \( \omega_0 \) indicating the initial angular velocity in the counterclockwise direction.
#### Question:
At time \( t = 0 \), the two forces shown in the figure above are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?
### Explanation:
To understand how the angular velocity of the disk changes over time due to the applied forces, we need to use the principles of torque and angular acceleration:
1. **Torque Calculation:**
Torque (\( \tau \)) is calculated by the formula \( \tau = r \cdot F \).
- Torque due to the 20 N force:
\( \tau_1 = r \cdot 20 \, \text{N} \)
(The radius r of the disk can be assumed if not given, but will be constant for all forces applied)
- Torque due to the 40 N force:
\( \tau_2 = r \cdot 40 \, \text{N} \)
2. **Direction of Forces:**
- The 20 N force creates a clockwise torque.
- The 40 N force also creates a clockwise torque.
3. **Net Torque:**
Since both torques are in the same direction (clockwise), we add them:
\( \tau_{\text{net}} = r \cdot 20 + r \cdot 40 \)
\( \tau_{\text{net}} = r \cdot 60 \)
4. **Effect on Angular Velocity:**
The net torque will cause an angular acceleration in the clockwise direction, opposing the initial counterclockwise rotation. This means the angular velocity \( \omega \) of the disk will decrease over time, eventually reaching zero and then becoming negative (clockwise).
### Conclusion:
If we
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