The bob of a stationary pendulum is given a sharp hit to impart it a horizontal speed of `sqrt(3gl)`. Find the angle rotated by the string before it becomes slack.

Explanation

To solve the problem of finding the angle rotated by the string of a pendulum before it becomes slack after being given a horizontal speed of \(\sqrt{3gl}\), we can follow these steps:

### Step 1: Understand the Initial Conditions
The pendulum bob is initially stationary and is given a sharp hit to impart a horizontal speed of \(\sqrt{3gl}\). At this point, we need to analyze the motion of the bob as it swings upward.

**Hint:** Identify the initial speed and the forces acting on the bob when it starts moving.

### Step 2: Apply the Work-Energy Theorem
According to the work-energy theorem, the work done by gravity on the bob will equal the change in kinetic energy. The height the bob rises can be expressed in terms of the angle \(\theta\) as:

\[ h = l + l \cos \theta = l(1 + \cos \theta) \]

The work done by gravity is:

\[ W = -mg \cdot h = -mg \cdot l(1 + \cos \theta) \]

**Hint:** Remember that the work done by gravity is negative because it acts downward while the bob moves upward.

### Step 3: Write the Change in Kinetic Energy
The change in kinetic energy can be expressed as:

\[ \Delta KE = \frac{1}{2} m v'^2 - \frac{1}{2} m v^2 \]

Where:
- \(v' = 0\) at the highest point (when the string becomes slack)
- \(v = \sqrt{3gl}\)

Thus,

\[ \Delta KE = 0 - \frac{1}{2} m (3gl) = -\frac{3}{2} mgl \]

**Hint:** The kinetic energy at the highest point is zero since the bob momentarily stops before falling back down.

### Step 4: Set Up the Equation
Equating the work done by gravity to the change in kinetic energy, we have:

\[ -mg l(1 + \cos \theta) = -\frac{3}{2} mgl \]

Cancelling \(mgl\) from both sides gives:

\[ -(1 + \cos \theta) = -\frac{3}{2} \]

**Hint:** Simplify the equation to isolate \(\cos \theta\).

### Step 5: Solve for \(\cos \theta\)
Rearranging the equation yields:

\[ 1 + \cos \theta = \frac{3}{2} \]

Thus,

\[ \cos \theta = \frac{3}{2} - 1 = \frac{1}{2} \]

**Hint:** Use the properties of cosine to find the angle.

### Step 6: Find the Angle \(\theta\)
The angle \(\theta\) can be found using the inverse cosine function:

\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

**Hint:** This angle is the angle from the vertical position.

### Step 7: Calculate the Angle Rotated
The angle rotated by the string before it becomes slack is given by:

\[ \text{Angle rotated} = \pi - \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

**Hint:** Remember that the total angle rotated is measured from the vertical position.

### Final Answer
The angle rotated by the string before it becomes slack is \(\frac{2\pi}{3}\) radians.

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