A pendulum consisting of a small sphere of mass m, suspended by a inextensible and massless string of length 1 , is made to swing in a verticle plane. If the breaking strength of the string is 2 mg, then the maximum angular amplitude of the displacement from the verticle can be

Options

A. `90^(@)`
B. `60^(@)`
C. `30^(@)`
D. `0^(@)`

Explanation

To solve the problem of finding the maximum angular amplitude of a pendulum that can swing without breaking the string, we can follow these steps:

### Step-by-step Solution:

1. **Understanding the Forces**:
- When the pendulum swings, two forces act on the mass (m): the gravitational force (mg) acting downwards and the tension (T) in the string acting towards the pivot.
- At the maximum displacement angle (θ), the tension in the string must balance the forces acting on the mass.

2. **Components of Forces**:
- At an angle θ, the weight of the pendulum can be resolved into two components:
- The component along the string: \( mg \cos \theta \)
- The component perpendicular to the string: \( mg \sin \theta \)

3. **Breaking Strength of the String**:
- The problem states that the breaking strength of the string is \( 2mg \). This means that the maximum tension (T) in the string can be \( 2mg \).

4. **Setting Up the Equation**:
- The tension in the string at the maximum angle θ can be expressed as:
\[ T = mg \cos \theta + mg \]
- This is because the tension must support the weight of the pendulum and provide the necessary centripetal force to keep it moving in a circular path.

5. **Substituting the Breaking Strength**:
- We set the maximum tension equal to the breaking strength:
\[ 2mg = mg \cos \theta + mg \]

6. **Simplifying the Equation**:
- Rearranging the equation gives:
\[ 2mg - mg = mg \cos \theta \]
\[ mg = mg \cos \theta \]
- Dividing both sides by \( mg \) (assuming \( m \neq 0 \)):
\[ 1 = \cos \theta \]

7. **Solving for θ**:
- From the equation \( 1 = \cos \theta \), we find:
\[ \cos \theta = \frac{1}{2} \]
- The angle θ that satisfies this equation is:
\[ \theta = 60^\circ \]

8. **Conclusion**:
- Therefore, the maximum angular amplitude of the displacement from the vertical is \( 60^\circ \).

### Final Answer:
The maximum angular amplitude of the displacement from the vertical can be \( 60^\circ \).

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