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To solve this ballistic pendulum problem, let's break down the steps:

### Given:
- Mass of the bullet (\( m \)) = 11 g = 0.011 kg
- Mass of the pendulum (\( M \)) = 68 kg
- Deviation angle (\( \theta \)) = 27°
- Vertical distance (L) = 92 cm = 0.92 m

### Objective:
Find the speed \( v \) with which the bullet hits the pendulum.

### Solution Outline:

1. **Energy Conservation**: When the pendulum reaches its maximum height (deviated by 27°), the kinetic energy from the bullet's impact is entirely converted into gravitational potential energy. We can use this to find the height \( h \) that the pendulum rises.

   \[
   h = L \cdot (1 - \cos \theta)
   \]
   Substituting \( L = 0.92 \) m and \( \theta = 27^\circ \):

   \[
   h = 0.92 \cdot (1 - \cos 27^\circ)
   \]

2. **Calculate Potential Energy**: The combined mass of the bullet and pendulum system rises to height \( h \). The potential energy at this height is:

   \[
   PE = (M + m) \cdot g \cdot h
   \]

3. **Momentum Conservation**: Just after the bullet hits the pendulum, the combined system moves with speed \( v' \). By conserving momentum, we get:

   \[
   m \cdot v = (M + m) \cdot v'
   \]

4. **Relating Kinetic Energy to Potential Energy**: Use the kinetic energy at the moment of impact to find \( v \).

Would you like the full calculations for each step with exact values? 

### Related Questions:
1. How would the solution change if the mass of the bullet was different?
2. What if the deviation angle were larger or smaller than 27°?
3. How would we solve this if we didn’t know the mass of the pendulum?
4. How does this problem demonstrate the principle of conservation of momentum?
5. Could this method be applied to different projectile types?

### Tip:
Always convert units (grams to kilograms, cm to meters) to maintain consistency in calculations.

An 11 g bullet is fired with speed v into a 68 kg ballistic pendulum. The wires holding the pendulum deviate from the vertical by 27°. The vertical distance from the point of impact to the suspension points for the wires is 92 cm. Find the speed with which the bullet hits the pendulum.

Learn how to solve a ballistic pendulum problem involving a bullet of mass 11 g fired into a 68 kg pendulum, with a deviation angle of 27° and a vertical distance of 92 cm. This problem covers conservation of momentum, energy, and trigonometry concepts, suitable for Grades 11-12.

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