Understanding Tension in a String Whirling a Stone

When a stone is whirled in a vertical circle using a string, the tension in the string is not constant. It changes depending on the position of the stone in the circle. This variation in the tension in string is due to the combined effect of gravity and the requirement for centripetal force.

Forces Acting on the Stone

At any point in the vertical circle, two main forces act on the stone:

• Gravitational Force (weight): This force acts vertically downwards and has a magnitude of $mg$, where $m$ is the mass of the stone and $g$ is the acceleration due to gravity.
• Tension Force: This force is exerted by the string and acts along the string towards the center of the circle. The magnitude of this tension in string varies.

Centripetal Force Requirement

For the stone to move in a circular path, there must be a net force acting towards the center of the circle. This net force provides the necessary centripetal force ($F_c = \frac{mv^2}{r}$), where $m$ is the mass, $v$ is the speed of the stone at that point, and $r$ is the radius of the circle (length of the string).

Analyzing Tension at Top and Bottom

Let's consider the forces at the top and bottom of the vertical circle:

• At the Top: Both the tension ($T_{top}$) and the gravitational force ($mg$) act downwards, towards the center of the circle. The net force towards the center is $T_{top} + mg$. This must equal the centripetal force: $$T_{top} + mg = \frac{mv_{top}^2}{r}$$ Therefore, the tension in string at the top is: $$T_{top} = \frac{mv_{top}^2}{r} - mg$$
• At the Bottom: The tension ($T_{bottom}$) acts upwards (towards the center), while the gravitational force ($mg$) acts downwards (away from the center). The net force towards the center is $T_{bottom} - mg$. This must equal the centripetal force: $$T_{bottom} - mg = \frac{mv_{bottom}^2}{r}$$ Therefore, the tension in string at the bottom is: $$T_{bottom} = \frac{mv_{bottom}^2}{r} + mg$$

Velocity Variation in Vertical Circular Motion

Due to gravity, the speed of the stone is not constant. The stone slows down as it moves upwards and speeds up as it moves downwards. The speed is maximum at the bottom of the circle ($v_{bottom}$) and minimum at the top ($v_{top}$). Thus, $v_{bottom} > v_{top}$.

Comparing Tension Values

Comparing the expressions for tension at the top and bottom:

• $T_{top} = \frac{mv_{top}^2}{r} - mg$
• $T_{bottom} = \frac{mv_{bottom}^2}{r} + mg$

Since $v_{bottom} > v_{top}$, the term $\frac{mv_{bottom}^2}{r}$ is greater than $\frac{mv_{top}^2}{r}$. Furthermore, $mg$ is subtracted from $\frac{mv_{top}^2}{r}$ to get $T_{top}$, while $mg$ is added to $\frac{mv_{bottom}^2}{r}$ to get $T_{bottom}$.

This clearly shows that the tension in string is highest at the bottom of the circle and least at the top of the circle.

Conclusion

Based on the analysis of forces and velocity in vertical circular motion, the tension in string developed when whirling a stone in a vertical circle is highest at the bottom of the circle.