a uniform beam AB of mass 16kg and length 3m rests in equilibrium in a horizontal position. the beam is supported by a single pivot at C and a particle of mass 7kg is attached at B

calculate the distance AC to 1 d.p

To solve this problem, we need to determine the distance \( AC \) at which the pivot must be placed for the beam to be in equilibrium.

### Given Data:
- Mass of the beam, \( m_{\text{beam}} = 16 \) kg
- Length of the beam, \( L = 3 \) m
- Mass of the particle at B, \( m_{\text{particle}} = 7 \) kg

### Steps to find the distance \( AC \):

1. **Calculate the weight forces:**
   - Weight of the beam: \( W_{\text{beam}} = m_{\text{beam}} \times g = 16 \times 9.8 = 156.8 \) N
   - Weight of the particle at B: \( W_{\text{particle}} = m_{\text{particle}} \times g = 7 \times 9.8 = 68.6 \) N

2. **Identify the positions of the forces:**
   - The weight of the beam acts at its center of gravity, which is at the midpoint of the beam (1.5 m from either end).
   - The weight of the particle acts at point B (the end of the beam).

3. **Set up the torque equilibrium equation:**
   For the beam to be in equilibrium, the sum of the torques about any point must be zero. We choose point C (the pivot) to sum the torques.

   Let \( AC = x \).

   The distances from point C to the weights are:
   - Distance from C to the beam's center of gravity: \( (x - 1.5) \) m
   - Distance from C to the particle at B: \( (3 - x) \) m

   The torque equilibrium equation about point C is:
   \[
   W_{\text{beam}} \cdot (x - 1.5) = W_{\text{particle}} \cdot (3 - x)
   \]

4. **Substitute the known values and solve for \( x \):**
   \[
   156.8 \cdot (x - 1.5) = 68.6 \cdot (3 - x)
   \]
   \[
   156.8x - 235.2 = 205.8 - 68.6x
   \]
   \[
   156.8x + 68.6x = 205.8 + 235.2
   \]
   \[
   225.4x = 441
   \]
   \[
   x = \frac{441}{225.4} \approx 1.96 \text{ m}
   \]

Thus, the distance \( AC \) is approximately \( 2.0 \) meters to one decimal place.

### Final Answer:
The distance \( AC \) is \( 2.0 \) meters.

Would you like more details or have any questions?

Here are some additional questions you might want to explore:

1. How would the equilibrium be affected if the particle's mass at B were doubled?
2. What is the reaction force at the pivot point C?
3. How would the solution change if the beam were not uniform?
4. What is the total force acting on the pivot point C?
5. How would you calculate the torque if the pivot point were not in the horizontal plane?

**Tip:** Always ensure that the sum of torques in a static equilibrium problem equals zero for the system to be in balance.

Learn how to calculate the distance AC for a uniform beam in equilibrium with a particle attached at B. This problem involves statics and torque concepts, suitable for advanced high school students. Includes step-by-step solution and formula application.

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