The maximum tension in the string of a pendulum is two times the minimum tension. Let `theta_(0)` is then what is angular amplitude

Options

A. `1/2`
B. `3/4`
C. `2/3`
D. `3/5`

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Understand the Problem
We have a pendulum where the maximum tension in the string is twice the minimum tension. We need to find the angular amplitude \( \theta_0 \).

### Step 2: Define Tension at Maximum and Minimum Points
1. **Maximum Tension** \( T_{max} \):
- At the highest point of the swing, the tension \( T_{max} \) can be expressed as:
\[ T_{max} = mg + \frac{mv^2}{L} \]
where \( m \) is the mass of the pendulum bob, \( g \) is the acceleration due to gravity, \( v \) is the velocity at the lowest point, and \( L \) is the length of the pendulum.

2. **Minimum Tension** \( T_{min} \):
- At the lowest point of the swing, the tension \( T_{min} \) is given by:
\[ T_{min} = mg \cos \theta \]
where \( \theta \) is the angle at the lowest point.

### Step 3: Set Up the Equation Based on Given Information
According to the problem, we know:
\[ T_{max} = 2 \times T_{min} \]
Substituting the expressions for \( T_{max} \) and \( T_{min} \):
\[ mg + \frac{mv^2}{L} = 2(mg \cos \theta) \]

### Step 4: Simplify the Equation
Rearranging the equation gives:
\[ mg + \frac{mv^2}{L} = 2mg \cos \theta \]
Dividing through by \( mg \) (assuming \( m \neq 0 \)):
\[ 1 + \frac{v^2}{gL} = 2 \cos \theta \]

### Step 5: Express Velocity in Terms of Height
Using the conservation of energy, we can express the velocity \( v \) at the lowest point in terms of the height \( h \):
\[ h = L - L \cos \theta = L(1 - \cos \theta) \]
The potential energy at height \( h \) converts to kinetic energy:
\[ mgh = \frac{1}{2} mv^2 \implies v^2 = 2gh \]
Substituting for \( h \):
\[ v^2 = 2gL(1 - \cos \theta) \]

### Step 6: Substitute Back into the Tension Equation
Substituting \( v^2 \) into the tension equation:
\[ 1 + \frac{2gL(1 - \cos \theta)}{gL} = 2 \cos \theta \]
This simplifies to:
\[ 1 + 2(1 - \cos \theta) = 2 \cos \theta \]
\[ 1 + 2 - 2 \cos \theta = 2 \cos \theta \]
\[ 3 = 4 \cos \theta \]
\[ \cos \theta = \frac{3}{4} \]

### Step 7: Find the Angular Amplitude
Finally, we find the angle \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{3}{4}\right) \]

### Final Answer
The angular amplitude \( \theta_0 \) is:
\[ \theta_0 = \cos^{-1}\left(\frac{3}{4}\right) \]

---