This is a physics problem involving a ballistic pendulum, where a bullet is fired into a pendulum and gets embedded in it, causing the pendulum to swing to a certain angle. We are given:

• Mass of the bullet, $m = 11 \, \text{g} = 0.011 \, \text{kg}$
• Mass of the pendulum, $M = 68 \, \text{kg}$
• The pendulum deviates by an angle of $\theta = 27^\circ$ after impact.
• The vertical distance of the pendulum's pivot point to the impact point, $L = 92 \, \text{cm} = 0.92 \, \text{m}$.

We need to find the speed $v_0$ with which the bullet hits the pendulum.

Step-by-Step Solution:

1. Convert the height reached by the pendulum from angle deviation:

   • When the pendulum reaches an angle $\theta$, it gains potential energy due to the height it reaches.
   • The height $h$ can be found using trigonometry: $h = L(1 - \cos \theta)$
   • Plugging in the values: $h = 0.92 \, \text{m} \times (1 - \cos 27^\circ)$

2. Calculate the potential energy at the highest point:

   • When the pendulum reaches the height $h$, the potential energy (PE) of the pendulum-bullet system is: $\text{PE} = (M + m) \cdot g \cdot h$
   • Use $g = 9.81 \, \text{m/s}^2$.

3. Determine the initial speed of the pendulum-bullet system after the collision:

   • At the point of collision, the system has only kinetic energy (KE), which later converts to potential energy at the highest point.
   • The initial kinetic energy (just after collision) is: $\text{KE} = \frac{1}{2} (M + m) v^2$
   • Since KE converts to PE at the maximum height: $\frac{1}{2} (M + m) v^2 = (M + m) \cdot g \cdot h$
   • Solving for $v$: $v = \sqrt{2 \cdot g \cdot h}$

4. Use conservation of momentum to find the bullet's initial speed $v_0$:

   • The momentum just before impact is equal to the momentum just after impact.
   • So, $m \cdot v_0 = (M + m) \cdot v$.
   • Solving for $v_0$: $v_0 = \frac{(M + m) \cdot v}{m}$

Substitute the values at each step to find the numerical result for $v_0$. Let me know if you'd like detailed calculations at each stage or further clarifications!

Additional Questions for Exploration:

1. How would the result change if the angle deviation was increased?
2. What happens to the potential energy if the length of the wire $L$ changes?
3. How does conservation of energy play a role in ballistic pendulum problems?
4. How would air resistance affect the bullet’s speed calculation?
5. What would be the effect on $v_0$ if the bullet mass was doubled?

Tip:

Always check unit conversions, especially with mass (e.g., grams to kilograms) and distance (e.g., centimeters to meters).