Question

A mass m is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

• A. the mass is at the highest point
• B. the wire is horizontal
• C. the mass is at the lowest point
• D. inclined at an angle of \(60\degree\) from vertical

Answer 1

The Correct Option is C

When a mass is attached to a thin wire and whirled in a vertical circle, the tension in the wire changes as the mass moves through different points of the circle. The wire is most likely to break when the tension is greatest. Let's explore the physics behind this scenario:

Concept Explanation:

• Vertical Circular Motion: In vertical circular motion, the mass experiences gravitational force and tension from the wire. The net force provides the centripetal force required to keep the mass moving in a circle.
• Forces on the Mass: At any point in the circle, the forces acting on the mass are its weight \(mg\) acting downward and the tension \(T\) in the wire.

Analysis at Different Points:

1. At the Top of the Circle:
   • The gravitational force and tension both act towards the center, so \( T_{top} + mg = \frac{mv^2}{r} \).
   • The tension is minimal at this point because part of the centripetal force is provided by gravity.
2. At the Bottom of the Circle:
   • The gravitational force acts downward, and the required centripetal force is upwards, so \( T_{bottom} - mg = \frac{mv^2}{r} \).
   • Therefore, \( T_{bottom} = mg + \frac{mv^2}{r} \).
   • The tension is maximum at the bottom due to the addition of gravitational force and the centripetal requirement.
3. At a Horizontal Position:
   • Here \( T_{horizontal} \) needs to supply the entire centripetal force, so \( T_{horizontal} = \frac{mv^2}{r} \). The gravitational force is perpendicular and does not add to the tension.
4. At an Angle of \( 60^\circ \) from Vertical:
   • The tension calculation is complex, but it generally does not exceed that at the bottom due to additional energy considerations.

Conclusion:

The wire is most likely to break at the lowest point of the vertical circle because the tension is maximum here. It accounts for both the gravitational force and the centripetal force, making it likely to exceed the wire's strength.