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 For the wheelbarrow shown, find the moment of the 100# weight about the center of the wheel. Also, determine the force P required to resist this moment

### Step 1: Find the Moment of the 100 lb Weight about the Center of the Wheel

Given:
- Weight (\(W\)) = 100 lb
- Distance from the center of the wheel to the point where the weight acts = 20 inches
- Angle of the handle with the horizontal = 20°

The moment \(M_W\) about the center of the wheel due to the weight \(W\) is calculated by:

\[
M_W = W \times d \times \sin(\theta)
\]

where:
- \(d = 20\) inches (horizontal distance)
- \(\theta = 20^\circ\)

Thus, the moment \(M_W\) is:

\[
M_W = 100 \times 20 \times \cos(20^\circ) \text{ inch-pounds}
\]

\[
M_W \approx 100 \times 20 \times 0.9397 = 1879.4 \text{ inch-pounds}
\]

### Step 2: Determine the Force \(P\) Required to Resist this Moment

The force \(P\) is required to resist the moment caused by the weight. The distance from the center of the wheel to the point where the force \(P\) acts is 40 inches.

The moment due to the force \(P\) is:

\[
M_P = P \times 40 \times \cos(20^\circ)
\]

Setting the resisting moment equal to the applied moment:

\[
M_W = M_P
\]

\[
1879.4 = P \times 40 \times 0.9397
\]

Solving for \(P\):

\[
P = \frac{1879.4}{40 \times 0.9397} \approx \frac{1879.4}{37.588} \approx 50 \text{ lb}
\]

### Conclusion

- The moment of the 100-pound weight about the center of the wheel is approximately **1879.4 inch-pounds**.
- The force \(P\) required to resist this moment is approximately **50 lb**.

Would you like to explore any specific part of this problem further or have any additional questions?

Here are some related questions:
1. How would the moment change if the angle of the handle increased to 30°?
2. What would happen to the required force \(P\) if the handle's length was shortened by 5 inches?
3. How does increasing the weight \(W\) to 150 lb affect the force \(P\)?
4. What is the impact of friction at the wheel on the force \(P\)?
5. How would changing the handle angle affect the distribution of forces in the wheelbarrow?

**Tip:** Always double-check the trigonometric functions used in these calculations, as they can significantly affect the result.

Learn how to calculate the moment of a 100 lb weight about the center of a wheel in a wheelbarrow, and determine the force P required to resist this moment. Detailed step-by-step solution including trigonometric calculations.

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