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![Title: Understanding Torque: Maximizing the Torque About a Pivot
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**Understanding Torque: Maximizing the Torque About a Pivot**
**Question:**
As shown in the figure, a given force is applied to a rod in several different ways. In which case is the torque about the pivot \( P \) due to this force the greatest?
**Figure Explanation:**
The figure consists of five different scenarios where a force \( \vec{F} \) is applied to a rod at different angles and positions relative to the pivot point \( P \).
1. Scenario 1: The force \( \vec{F} \) is applied perpendicular to the rod at its end.
2. Scenario 2: The force \( \vec{F} \) is applied parallel to the rod at its end.
3. Scenario 3: The force \( \vec{F} \) is applied perpendicular to the rod closer to the pivot point.
4. Scenario 4: The force \( \vec{F} \) is applied at an angle \( \theta \) to the rod at its end.
5. Scenario 5: The force \( \vec{F} \) is applied in line with the rod at its end.
**Answer Choices:**
- A. 1
- B. 2
- C. 3
- D. 4
- E. 5
**Detailed Explanation:**
Torque (\( \tau \)) is the rotational equivalent of force and is given by the equation:
\[ \tau = r \cdot F \cdot \sin(\theta) \]
where:
- \( r \) is the distance from the pivot point to the point where the force is applied.
- \( F \) is the magnitude of the force applied.
- \( \theta \) is the angle between the force and the lever arm (rod).
To maximize the torque, \( \sin(\theta) \) should be maximized (which is 1 when \( \theta = 90^\circ \)). Additionally, \( r \) must be maximized. Therefore, the configuration where the force is applied perpendicular to the rod (giving \( \theta = 90^\circ \)) and at the maximum distance from \( P \) will result in the greatest torque.
**Answer:**
The greatest torque is in scenario **A. 1**, where the force \( \vec{F} \) is applied](https://content.bartleby.com/qna-images/question/0f7775f3-edad-4f27-ba1b-f7886a7505bd/4f01106a-acbc-4fac-9b6e-1826e82d86ff/x5oum7r_processed.jpeg)
Transcribed Image Text:Title: Understanding Torque: Maximizing the Torque About a Pivot
---
**Understanding Torque: Maximizing the Torque About a Pivot**
**Question:**
As shown in the figure, a given force is applied to a rod in several different ways. In which case is the torque about the pivot \( P \) due to this force the greatest?
**Figure Explanation:**
The figure consists of five different scenarios where a force \( \vec{F} \) is applied to a rod at different angles and positions relative to the pivot point \( P \).
1. Scenario 1: The force \( \vec{F} \) is applied perpendicular to the rod at its end.
2. Scenario 2: The force \( \vec{F} \) is applied parallel to the rod at its end.
3. Scenario 3: The force \( \vec{F} \) is applied perpendicular to the rod closer to the pivot point.
4. Scenario 4: The force \( \vec{F} \) is applied at an angle \( \theta \) to the rod at its end.
5. Scenario 5: The force \( \vec{F} \) is applied in line with the rod at its end.
**Answer Choices:**
- A. 1
- B. 2
- C. 3
- D. 4
- E. 5
**Detailed Explanation:**
Torque (\( \tau \)) is the rotational equivalent of force and is given by the equation:
\[ \tau = r \cdot F \cdot \sin(\theta) \]
where:
- \( r \) is the distance from the pivot point to the point where the force is applied.
- \( F \) is the magnitude of the force applied.
- \( \theta \) is the angle between the force and the lever arm (rod).
To maximize the torque, \( \sin(\theta) \) should be maximized (which is 1 when \( \theta = 90^\circ \)). Additionally, \( r \) must be maximized. Therefore, the configuration where the force is applied perpendicular to the rod (giving \( \theta = 90^\circ \)) and at the maximum distance from \( P \) will result in the greatest torque.
**Answer:**
The greatest torque is in scenario **A. 1**, where the force \( \vec{F} \) is applied
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