A particle of mass 100 g tied to a string is rotated along the circle of radius 0.5 m. The breaking tension of the string is 10 N. The maximum speed with which particle can be rotated without breaking the string is

Options

A. `10 " ms"^(-1)`
B. `9.8 "m s"^(-1)`
C. `7.7 " m s"^(-1)`
D. `7.07 " m s"^(-1)`

Answer

To solve the problem, we need to determine the maximum speed at which a particle can be rotated in a circular path without breaking the string. We will use the relationship between the tension in the string, the mass of the particle, and the radius of the circular path.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the particle, \( m = 100 \, \text{g} = 0.1 \, \text{kg} \) (convert grams to kilograms)
- Radius of the circular path, \( r = 0.5 \, \text{m} \)
- Breaking tension of the string, \( T = 10 \, \text{N} \)

2. **Understand the Forces Acting on the Particle:**
- When the particle is rotating in a circle, the tension in the string provides the centripetal force required to keep the particle moving in that circular path.
- The centripetal force \( F_c \) is given by the formula:
\[ F_c = \frac{m v^2}{r} \]
where \( v \) is the speed of the particle.

3. **Set Up the Equation:**
- The maximum tension in the string (which is the breaking tension) must equal the centripetal force required to keep the particle moving in a circle:
\[ T = \frac{m v^2}{r} \]

4. **Rearrange the Equation to Solve for Speed \( v \):**
- Rearranging the equation gives:
\[ v^2 = \frac{T \cdot r}{m} \]
- Taking the square root to find \( v \):
\[ v = \sqrt{\frac{T \cdot r}{m}} \]

5. **Substitute the Known Values:**
- Substitute \( T = 10 \, \text{N} \), \( r = 0.5 \, \text{m} \), and \( m = 0.1 \, \text{kg} \) into the equation:
\[ v = \sqrt{\frac{10 \cdot 0.5}{0.1}} \]

6. **Calculate the Value:**
- Calculate the numerator:
\[ 10 \cdot 0.5 = 5 \]
- Now divide by the mass:
\[ \frac{5}{0.1} = 50 \]
- Finally, take the square root:
\[ v = \sqrt{50} \approx 7.07 \, \text{m/s} \]

### Final Answer:
The maximum speed with which the particle can be rotated without breaking the string is approximately \( 7.07 \, \text{m/s} \).

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