To solve the problem step by step, we will follow these steps:
### Step 1: Understand the Problem
We have two masses: a stationary mass \( m_1 = 3.9 \, \text{kg} \) suspended from a string of length \( L = 0.5 \, \text{m} \), and a moving mass \( m_2 = 0.1 \, \text{kg} \) with a velocity \( v_2 = 200 \, \text{m/s} \). When \( m_2 \) strikes \( m_1 \), they stick together. We need to find the maximum angle \( \theta \) through which the system swings just before the tension in the string becomes zero.
### Step 2: Apply Conservation of Momentum
When the moving mass strikes the stationary mass, we can apply the law of conservation of momentum:
\[
m_2 v_2 = (m_1 + m_2) v
\]
Substituting the values:
\[
0.1 \times 200 = (3.9 + 0.1) v
\]
Calculating the left side:
\[
20 = 4.0 v
\]
Solving for \( v \):
\[
v = \frac{20}{4.0} = 5 \, \text{m/s}
\]
### Step 3: Analyze the Motion After Collision
After the collision, the combined mass \( (m_1 + m_2) \) will swing in a circular motion. At the highest point of the swing, the tension \( T \) in the string will be zero. The forces acting on the mass at the highest point are gravity and the centripetal force required to keep the mass moving in a circular path.
### Step 4: Set Up the Equation for Tension
At the highest point, the tension in the string is given by:
\[
T = \frac{(m_1 + m_2) v^2}{L} - (m_1 + m_2) g \cos(\theta)
\]
Setting \( T = 0 \):
\[
0 = \frac{(m_1 + m_2) v^2}{L} - (m_1 + m_2) g \cos(\theta)
\]
### Step 5: Simplify the Equation
This simplifies to:
\[
\frac{(m_1 + m_2) v^2}{L} = (m_1 + m_2) g \cos(\theta)
\]
Dividing both sides by \( (m_1 + m_2) \):
\[
\frac{v^2}{L} = g \cos(\theta)
\]
### Step 6: Substitute Values
Substituting \( v = 5 \, \text{m/s} \), \( L = 0.5 \, \text{m} \), and \( g = 9.8 \, \text{m/s}^2 \):
\[
\frac{5^2}{0.5} = 9.8 \cos(\theta)
\]
Calculating the left side:
\[
\frac{25}{0.5} = 50
\]
So we have:
\[
50 = 9.8 \cos(\theta)
\]
### Step 7: Solve for \( \cos(\theta) \)
Rearranging gives:
\[
\cos(\theta) = \frac{50}{9.8}
\]
Calculating \( \cos(\theta) \):
\[
\cos(\theta) \approx 5.102
\]
Since \( \cos(\theta) \) cannot exceed 1, this implies that the angle \( \theta \) at which the tension becomes zero is not physically possible in this context.
### Conclusion
Since the calculated value for \( \cos(\theta) \) exceeds 1, it indicates that the mass will not swing to a point where the tension is zero under the given conditions. Thus, the maximum angle cannot be determined in this scenario based on the assumptions made.
