Introduction to Vertical Circular Motion
When an object tied to a string revolves in a vertical circle, it is subject to varying forces. Unlike horizontal circular motion, gravity plays a crucial role here because its direction relative to the object changes as it moves around the path.
The Physics Analysis
To find where the tension ($T$) is greatest, we analyze the forces acting on the object at any point. The net centripetal force ($F_c$) required to keep the object in a circular path is provided by the tension and the component of gravity acting along the radius.
Forces at the Bottom Position
At the lowest point of the circle:
- Gravity ($mg$) acts vertically downward.
- Tension ($T_{bottom}$) acts vertically upward toward the center.
- The net force is $T_{bottom} - mg = \frac{mv^2}{r}$.
- Therefore, $T_{bottom} = \frac{mv^2}{r} + mg$.
Forces at the Top Position
At the highest point of the circle:
- Gravity ($mg$) acts downward.
- Tension ($T_{top}$) also acts downward toward the center.
- The net force is $T_{top} + mg = \frac{mv^2}{r}$.
- Therefore, $T_{top} = \frac{mv^2}{r} - mg$.
Conclusion
Comparing the two expressions, it is clear that at the bottom, we add the weight to the centripetal force, whereas at the top, we subtract it. Thus, the tension in the string is greatest at the lowest point of the vertical circle.
Intuitive Summary
At the bottom, the string must support both the weight of the object and provide the centripetal force required for circular motion. At the top, the gravity helps provide some of the required centripetal force, meaning the string doesn't need to pull as hard.