A particle is moving in a vertical circle. The tensions in...

A particle is moving in a vertical circle. The tensions in the string when passing through two positions at angles $30^{\circ}$ and $60^{\circ}$ from vertical (lowest position) are $T_{1}$ and $T_{2}$ respectively, Then (a) $T_{1}=T_{2}$ (b) $T_{1}>T_{2}$ (c) $T_{1}<T_{2}$ (d) $T_{1} \geq T_{2}$

A particle is moving in a vertical circle. The tensions in the string when passing through two positions at angles $30^{\circ}$ and $60^{\circ}$ from vertical (lowest position) are $T_{1}$ and $T_{2}$ respectively, Then
(a) $T_{1}=T_{2}$
(b) $T_{1}>T_{2}$
(c) $T_{1}<T_{2}$
(d) $T_{1} \geq T_{2}$

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Key Concepts

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. In vertical circular motion, this force is provided by the net component of forces acting in the radial direction, and it must be large enough to overcome the effects of gravity at various points in the circle.

Tension Variation in Vertical Circular Motion

In vertical circles, the tension in the string is not constant because it must supply the centripetal force while also counteracting the component of gravitational force that acts along the line of the string. Thus, at different angles from the vertical, the tension adjusts to maintain the circular path, leading to variations such as greater tension at positions where gravity opposes the centripetal requirement.

Conservation of Energy

The conservation of mechanical energy principle explains how the speed of the particle changes as it moves through different positions in the circle. Variations in kinetic and potential energy affect the centripetal acceleration, which in turn influences the tension in the string at different positions.

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