Askiitians Tutor Team
Last Activity: 2 Days ago
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:
Rearranging this gives us:
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 (Fc) is:
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.