Solution Overview

The problem involves a ball of mass $M$ attached to a string of length $R$, moving in a vertical circle. At the top of the circle (Point $P$), the tension in the string changes with the speed $v$. The problem has three parts: (a) qualitative comparison of tensions, (b) deriving tension $F_{T1}$, and (c) consistency justification.

Part (a): Qualitative Comparison of Tensions

At Point $P$ (top of the circle), two forces act on the ball:

1. Gravitational force: $F_g = Mg$ (downward).
2. Tension force: $F_T$ (towards the center).

The net centripetal force at the top is: $F_T + Mg = \frac{Mv^2}{R}$

From this, the tension can be expressed as: $F_T = \frac{Mv^2}{R} - Mg$

• If $v_2 > v_1$, then $\frac{Mv_2^2}{R} > \frac{Mv_1^2}{R}$.
• Since $Mg$ remains constant, an increase in $v$ increases the tension $F_T$.

Thus: $F_{T2} > F_{T1}.$

Answer: $F_{T1} < F_{T2}$

Part (b): Deriving $F_{T1}$ Using Newton’s Second Law

At the top of the circle, applying Newton's Second Law: $F_T + Mg = \frac{Mv^2}{R}$

Rearranging for the tension $F_T$: $F_{T1} = \frac{Mv_1^2}{R} - Mg$

This is the expression for the tension $F_{T1}$ at the top of the circle when the ball moves at speed $v_1$.

Part (c): Justification of Consistency

In part (a), we qualitatively argued that $F_{T1} < F_{T2}$ because increasing speed increases the tension. The derived expression: $F_T = \frac{Mv^2}{R} - Mg$ clearly shows that $F_T$ depends on $v^2$. As $v$ increases ($v_2 > v_1$), the term $\frac{Mv^2}{R}$ increases, leading to greater tension.

Thus, the derived expression supports the conclusion made in part (a).

Final Answers:

1. (a) $F_{T1} < F_{T2}$
2. (b) $F_{T1} = \frac{Mv_1^2}{R} - Mg$
3. (c) The derived expression is consistent with the claim in part (a).

Related Questions:

1. What happens to the tension if $v$ decreases to the minimum speed required for circular motion?
2. How does the tension change at the bottom of the circle compared to the top?
3. Derive an expression for the minimum speed required at Point $P$ for the string to remain taut.
4. How would the problem change if the ball moves at non-uniform speed in the vertical circle?
5. What role does gravity play in altering the tension at different positions in the vertical circle?

Tip: Always analyze forces acting at different positions in circular motion and use Newton’s Second Law to derive expressions systematically.