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The problem shows a light rod supported horizontally by two vertical strings at points \( A \) and \( B \), with forces \( P \) and \( Q \) acting downwards at those points respectively. There are also two upward forces of 21 N and 25 N acting at distances 4 m and 6 m from the ends, with the rod being 10 meters long in total.

To solve for the forces \( P \) and \( Q \), we need to apply two principles of equilibrium:
1. **Translational equilibrium:** The sum of the vertical forces must equal zero.
2. **Rotational equilibrium:** The sum of the moments (torques) about any point must be zero.

### Step 1: Translational Equilibrium
The sum of the vertical forces must be zero, as the rod is in equilibrium:
\[
P + Q = 21 \, \text{N} + 25 \, \text{N} = 46 \, \text{N}.
\]

### Step 2: Rotational Equilibrium
We can choose any point to calculate the moments. Let's take moments about point \( A \) to eliminate \( P \) from the equation.

- Moment arm of \( P \) at point \( A \): 0 m (since it's at point A)
- Moment arm of \( Q \): 10 m (distance from A to B)
- Moment arm of the 21 N force: 4 m
- Moment arm of the 25 N force: \( 4 + 6 = 10 \) m

Using the fact that clockwise moments equal counterclockwise moments, we write the equation for moments around \( A \):
\[
Q \times 10 = 21 \times 4 + 25 \times 10.
\]

Substitute the values:
\[
Q \times 10 = 84 + 250 = 334.
\]
\[
Q = \frac{334}{10} = 33.4 \, \text{N}.
\]

### Step 3: Solve for \( P \)
Now that we know \( Q = 33.4 \, \text{N} \), we can substitute this into the equation for translational equilibrium:
\[
P + 33.4 = 46.
\]
\[
P = 46 - 33.4 = 12.6 \, \text{N}.
\]

### Final Answer:
- \( P = 12.6 \, \text{N} \)
- \( Q = 33.4 \, \text{N} \)

Would you like more details on any part of the solution or have any further questions?

Here are some related questions you could explore:
1. How would the solution change if one of the forces was applied at a different point?
2. How do you calculate moments when forces are applied at angles other than vertical?
3. What would happen if the rod were not "light" and had its own weight?
4. How do we calculate the tension in the strings if they are not vertical?
5. What changes if the rod is no longer horizontal?

**Tip:** When dealing with equilibrium problems, always start by balancing forces and then use moments to simplify calculations.

The diagram shows a light rod held horizontally in equilibrium by two vertical strings. Find the magnitudes of the forces P and Q.

Learn how to solve an equilibrium problem involving a light rod supported by two vertical strings, with forces P and Q acting at the ends. This solution covers principles of statics, including balancing forces and moments, and uses key formulas for translational and rotational equilibrium.

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