A body of mass m = 10 kg is attached to a wire of length 0.3m. The maximum angular velocity with which it can be rotated in a horizontal circle is (Given, breaking stress of wire `= 4.8 xx 10^(7) Nm^(-2)` and area of cross-section of a wire `= 10^(-6) m^(2)`)

Answer

To find the maximum angular velocity with which a body can be rotated in a horizontal circle while attached to a wire, we will follow these steps:

### Step 1: Calculate the maximum tension in the wire using breaking stress.
The breaking stress (σ) is given by the formula:
\[ \sigma = \frac{F}{A} \]
Where:
- \( F \) is the force (tension in the wire),
- \( A \) is the area of cross-section of the wire.

Given:
- Breaking stress \( \sigma = 4.8 \times 10^7 \, \text{N/m}^2 \)
- Area of cross-section \( A = 10^{-6} \, \text{m}^2 \)

Rearranging the formula to find \( F \):
\[ F = \sigma \times A \]
Substituting the values:
\[ F = (4.8 \times 10^7) \times (10^{-6}) = 48 \, \text{N} \]

### Step 2: Relate the tension in the wire to the centripetal force.
When the body is rotating in a horizontal circle, the tension in the wire provides the centripetal force required for circular motion. The centripetal force \( F_c \) is given by:
\[ F_c = m \omega^2 r \]
Where:
- \( m \) is the mass of the body,
- \( \omega \) is the angular velocity,
- \( r \) is the radius of the circular path (which is equal to the length of the wire).

Given:
- Mass \( m = 10 \, \text{kg} \)
- Length of the wire (which is the radius \( r \)) = 0.3 m

### Step 3: Set the tension equal to the centripetal force.
We know that the maximum tension \( F \) is equal to the centripetal force:
\[ F = m \omega^2 r \]
Substituting the values we have:
\[ 48 = 10 \omega^2 (0.3) \]

### Step 4: Solve for \( \omega^2 \).
Rearranging the equation to solve for \( \omega^2 \):
\[ 48 = 3 \omega^2 \quad (\text{since } 10 \times 0.3 = 3) \]
\[ \omega^2 = \frac{48}{3} = 16 \]

### Step 5: Calculate \( \omega \).
Taking the square root to find \( \omega \):
\[ \omega = \sqrt{16} = 4 \, \text{rad/s} \]

### Final Answer:
The maximum angular velocity with which the body can be rotated in a horizontal circle is \( \omega = 4 \, \text{rad/s} \).

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