A bob of mass 10kg is attached to a wire of 0.3 m long. Its breaking stress is 4.8*10^7 N/m^2. The area of cross section of the wire is 10^-6 m^2. What is the maximm angular velocity with which it can be rotated in a horizontal circle??

a) 8 rad/s b) 4 rad/s c) 2 rad/s d) 1 rad/s.

To determine the maximum angular velocity with which the bob can be rotated in a horizontal circle without breaking the wire, we need to analyze the forces acting on the bob and the properties of the wire itself. Let's break this down step by step.

Understanding the Forces

When the bob is rotated in a horizontal circle, it experiences a centripetal force that is required to keep it moving in that circular path. This force is provided by the tension in the wire. The tension must not exceed the breaking stress of the wire, or else it will snap.

Calculating the Maximum Tension

The maximum tension that the wire can withstand can be calculated using the formula for stress:

• Stress = Force / Area

Rearranging this gives us:

• Force = Stress × Area

Substituting the given values:

• Stress = 4.8 × 10⁷ N/m²
• Area = 10^-6 m²

Now, we can calculate the maximum force (tension) the wire can handle:

Force = 4.8 × 10⁷ N/m² × 10^-6 m² = 48 N

Relating Tension to Centripetal Force

The tension in the wire provides the centripetal force required to keep the bob moving in a circle. The formula for centripetal force (F_c) is:

• F_c = mω²r

Where:

• m = mass of the bob (10 kg)
• ω = angular velocity (rad/s)
• r = radius of the circle (length of the wire, 0.3 m)

Setting the maximum tension equal to the centripetal force gives us:

48 N = 10 kg × ω² × 0.3 m

Solving for Angular Velocity

Now, we can solve for ω:

• 48 N = 3 kg·m × ω²
• ω² = 48 N / 3 kg·m = 16
• ω = √16 = 4 rad/s

Final Answer

The maximum angular velocity with which the bob can be rotated in a horizontal circle without breaking the wire is 4 rad/s. Therefore, the correct option is (b) 4 rad/s.

Approved