The figure shows 2 bolts A and B bolted to a sloping surface. A force of magnitude 26.4 pounds and a 220-inch pound couple are acting in the negative z-direction at point A. The force is 10 inches below the z-axis. Another force of magnitude 17 pounds and a 238-inch pound couple are acting on point B at the origin of a 3-dimensional system. the top edge of the surface measures 20 inches and is 16 inches above the z-axis. Determine the pitch of the wrench. (Round the final answer to two decimal places.)

The pitch *p* of the wrench is ___ in.

To solve this problem, we need to determine the **pitch** of the wrench, which is the ratio of the moment of the couple to the resultant force in the direction of the force system. The pitch \( p \) is calculated as:

\[
p = \frac{M_{\text{res}}}{F_{\text{res}}}
\]

where:
- \( M_{\text{res}} \) is the resultant moment (sum of the moments acting on the system).
- \( F_{\text{res}} \) is the magnitude of the resultant force (sum of the forces acting on the system).

### Step 1: Identify the forces and moments acting on points A and B
#### At point A:
- Force acting downward (\( F_A \)) is 26.4 pounds.
- A couple (\( M_A \)) of 220 inch-pounds.
- The force is 10 inches below the z-axis.

#### At point B (origin):
- Force acting downward (\( F_B \)) is 17 pounds.
- A couple (\( M_B \)) of 238 inch-pounds.

### Step 2: Calculate the resultant force (\( F_{\text{res}} \))
The forces are both acting downward (negative z-direction), so the resultant force is the sum of the magnitudes of these forces:

\[
F_{\text{res}} = F_A + F_B = 26.4 + 17 = 43.4 \, \text{pounds}
\]

### Step 3: Calculate the resultant moment (\( M_{\text{res}} \))
The resultant moment is the sum of the moments, including both the couples and the moment generated by the forces due to their positions.

1. The moment generated by the force at point A about the z-axis is:
\[
M_{F_A} = 26.4 \, \text{pounds} \times 10 \, \text{inches} = 264 \, \text{inch-pounds}
\]
2. The total moment at point A is:
\[
M_A + M_{F_A} = 220 + 264 = 484 \, \text{inch-pounds}
\]

3. The total moment at point B is simply the couple at B:
\[
M_B = 238 \, \text{inch-pounds}
\]

The resultant moment is the sum of these moments:

\[
M_{\text{res}} = 484 + 238 = 722 \, \text{inch-pounds}
\]

### Step 4: Calculate the pitch of the wrench
Now, the pitch of the wrench is given by:

\[
p = \frac{M_{\text{res}}}{F_{\text{res}}} = \frac{722}{43.4} \approx 16.63 \, \text{inches}
\]

### Final Answer:
The pitch \( p \) of the wrench is **16.63 inches**.

---

Do you want further details or have any questions? 

Here are 5 related questions you could explore:
1. What is the effect of changing the couple at point A on the pitch of the wrench?
2. How would the problem change if the force at B was not acting in the negative z-direction?
3. Can you generalize the pitch calculation for forces not acting in the same direction?
4. How does the distance of forces from the z-axis affect the moment calculation?
5. What would happen to the resultant moment if the force at A were moved to a different point?

**Tip:** Always check the direction of forces and moments carefully—they directly affect the calculation of the resultant system.

The figure shows 2 bolts A and B bolted to a sloping surface. A force of magnitude 26.4 pounds and a 220-inch pound couple are acting in the negative z-direction at point A. The force is 10 inches below the z-axis. Another force of magnitude 17 pounds and a 238-inch pound couple are acting on point B at the origin of a 3-dimensional system. The top edge of the surface measures 20 inches and is 16 inches above the z-axis. Determine the pitch of the wrench. (Round the final answer to two decimal places.)

Learn how to calculate the pitch of a wrench by determining the resultant moment and force in a system involving forces and couples acting on two bolts A and B on a sloping surface. This problem demonstrates key statics principles, including moment calculation and force summation, suitable for college-level engineering students.

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