Question
A force of 75 N at an angle of 15° to the direction of motion moves a chair 3 m. Which change would result in more work being done on the chair?
using 60 N of force
moving the chair 2 m
increasing the angle to 20°
decreasing the angle to 10°
using 60 N of force
moving the chair 2 m
increasing the angle to 20°
decreasing the angle to 10°
Answers
Answered by
GPT-4o mini
To determine which change would result in more work being done on the chair, we need to calculate the work done in each scenario using the formula for work:
\[
\text{Work} = F \cdot d \cdot \cos(\theta)
\]
where:
- \( F \) is the force applied,
- \( d \) is the distance moved in the direction of the force,
- \( \theta \) is the angle between the force and the direction of motion.
### Original Situation
- Force \( F = 75 \, \text{N} \)
- Angle \( \theta = 15^\circ \)
- Distance \( d = 3 \, \text{m} \)
Calculating the work done in the original situation:
\[
\text{Work} = 75 \, \text{N} \cdot 3 \, \text{m} \cdot \cos(15^\circ)
\]
Using \( \cos(15^\circ) \approx 0.9659 \):
\[
\text{Work} \approx 75 \times 3 \times 0.9659 \approx 217.4 \, \text{J}
\]
### Let's Evaluate Each Change
1. **Using 60 N of force**:
- \( F = 60 \, \text{N} \)
- \( \theta = 15^\circ \)
- \( d = 3 \, \text{m} \)
\[
\text{Work} = 60 \cdot 3 \cdot \cos(15^\circ) \approx 60 \cdot 3 \cdot 0.9659 \approx 173.3 \, \text{J}
\]
2. **Moving the chair 2 m**:
- \( F = 75 \, \text{N} \)
- \( \theta = 15^\circ \)
- \( d = 2 \, \text{m} \)
\[
\text{Work} = 75 \cdot 2 \cdot \cos(15^\circ) \approx 75 \cdot 2 \cdot 0.9659 \approx 144.9 \, \text{J}
\]
3. **Increasing the angle to 20°**:
- \( F = 75 \, \text{N} \)
- \( \theta = 20^\circ \)
- \( d = 3 \, \text{m} \)
\[
\text{Work} = 75 \cdot 3 \cdot \cos(20^\circ) \approx 75 \cdot 3 \cdot 0.9397 \approx 210.0 \, \text{J}
\]
4. **Decreasing the angle to 10°**:
- \( F = 75 \, \text{N} \)
- \( \theta = 10^\circ \)
- \( d = 3 \, \text{m} \)
\[
\text{Work} = 75 \cdot 3 \cdot \cos(10^\circ) \approx 75 \cdot 3 \cdot 0.9848 \approx 221.1 \, \text{J}
\]
### Summary of Work Done
- Using 60 N of force: \( \approx 173.3 \, \text{J} \)
- Moving the chair 2 m: \( \approx 144.9 \, \text{J} \)
- Increasing the angle to 20°: \( \approx 210.0 \, \text{J} \)
- Decreasing the angle to 10°: \( \approx 221.1 \, \text{J} \)
### Conclusion
The **decrease in angle to 10°** results in the most work done on the chair at approximately **221.1 J**, so that is the change that results in more work being done.
\[
\text{Work} = F \cdot d \cdot \cos(\theta)
\]
where:
- \( F \) is the force applied,
- \( d \) is the distance moved in the direction of the force,
- \( \theta \) is the angle between the force and the direction of motion.
### Original Situation
- Force \( F = 75 \, \text{N} \)
- Angle \( \theta = 15^\circ \)
- Distance \( d = 3 \, \text{m} \)
Calculating the work done in the original situation:
\[
\text{Work} = 75 \, \text{N} \cdot 3 \, \text{m} \cdot \cos(15^\circ)
\]
Using \( \cos(15^\circ) \approx 0.9659 \):
\[
\text{Work} \approx 75 \times 3 \times 0.9659 \approx 217.4 \, \text{J}
\]
### Let's Evaluate Each Change
1. **Using 60 N of force**:
- \( F = 60 \, \text{N} \)
- \( \theta = 15^\circ \)
- \( d = 3 \, \text{m} \)
\[
\text{Work} = 60 \cdot 3 \cdot \cos(15^\circ) \approx 60 \cdot 3 \cdot 0.9659 \approx 173.3 \, \text{J}
\]
2. **Moving the chair 2 m**:
- \( F = 75 \, \text{N} \)
- \( \theta = 15^\circ \)
- \( d = 2 \, \text{m} \)
\[
\text{Work} = 75 \cdot 2 \cdot \cos(15^\circ) \approx 75 \cdot 2 \cdot 0.9659 \approx 144.9 \, \text{J}
\]
3. **Increasing the angle to 20°**:
- \( F = 75 \, \text{N} \)
- \( \theta = 20^\circ \)
- \( d = 3 \, \text{m} \)
\[
\text{Work} = 75 \cdot 3 \cdot \cos(20^\circ) \approx 75 \cdot 3 \cdot 0.9397 \approx 210.0 \, \text{J}
\]
4. **Decreasing the angle to 10°**:
- \( F = 75 \, \text{N} \)
- \( \theta = 10^\circ \)
- \( d = 3 \, \text{m} \)
\[
\text{Work} = 75 \cdot 3 \cdot \cos(10^\circ) \approx 75 \cdot 3 \cdot 0.9848 \approx 221.1 \, \text{J}
\]
### Summary of Work Done
- Using 60 N of force: \( \approx 173.3 \, \text{J} \)
- Moving the chair 2 m: \( \approx 144.9 \, \text{J} \)
- Increasing the angle to 20°: \( \approx 210.0 \, \text{J} \)
- Decreasing the angle to 10°: \( \approx 221.1 \, \text{J} \)
### Conclusion
The **decrease in angle to 10°** results in the most work done on the chair at approximately **221.1 J**, so that is the change that results in more work being done.