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We have a light rod that is held horizontally by two vertical strings at points \( A \) and \( B \), with forces \( P \) and \( Q \) acting at these points, respectively. There are two additional vertical forces acting on the rod: 21 N at a point \( C \) (4 m from \( A \)) and 25 N at a point \( D \) (10 m from \( C \) or 6 m from \( B \)).

### To find the magnitudes of forces \( P \) and \( Q \), we use the following principles:

1. **Equilibrium Condition:**
   - The rod is in static equilibrium, so the **sum of the vertical forces** must be zero.
   - The **sum of the moments** (torques) about any point must also be zero.

### Step 1: Vertical Force Equilibrium
The sum of the upward forces (\( P \) and \( Q \)) must equal the sum of the downward forces (21 N and 25 N):

\[
P + Q = 21 \, \text{N} + 25 \, \text{N} = 46 \, \text{N}
\]

### Step 2: Moment Equilibrium (taking moments about point \( A \))
The moment (torque) due to a force is given by \( \text{Force} \times \text{Distance from the point of rotation} \).

To eliminate \( P \), let's take moments about point \( A \):

\[
\text{Moment of } P = 0 \quad (\text{since it's applied at point } A)
\]
\[
\text{Moment of } Q = Q \times 20 \, \text{m} \quad (\text{distance from } A \text{ to } B \text{ is } 20 \, \text{m})
\]
\[
\text{Moment of 21 N} = 21 \, \text{N} \times 4 \, \text{m}
\]
\[
\text{Moment of 25 N} = 25 \, \text{N} \times 16 \, \text{m} \quad (\text{distance from } A \text{ to } D \text{ is } 16 \, \text{m})
\]

Since the system is in equilibrium, the sum of the moments must be zero:

\[
Q \times 20 = 21 \times 4 + 25 \times 16
\]
\[
Q \times 20 = 84 + 400 = 484
\]
\[
Q = \frac{484}{20} = 24.2 \, \text{N}
\]

### Step 3: Solving for \( P \)
Now that we know \( Q = 24.2 \, \text{N} \), we can substitute this value into the vertical force equilibrium equation:

\[
P + 24.2 = 46
\]
\[
P = 46 - 24.2 = 21.8 \, \text{N}
\]

### Final Answer:
- \( P = 21.8 \,

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

This problem involves finding the forces P and Q that hold a light rod in horizontal equilibrium. The solution covers static equilibrium conditions, including force balance and moment calculations. Suitable for high school physics students.

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