Vertical Circular Motion

Vertical circular motion is more complex than horizontal because the weight of the object has a component along the direction of motion. This means the speed changes throughout the circle (unless energy is continuously supplied), and the forces vary at different positions.

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Forces at Different Points

Consider a ball of mass m on a string of length r being swung in a vertical circle. At any point, there are two forces:

• Weight (mg) — always vertically downward
• Tension (T) — always along the string, towards the centre

The net centripetal force is the resultant of these two forces directed towards the centre.

At the Top of the Circle

At the top, both weight and tension point downward (towards the centre):

$T_{\text{top}} + mg = \frac{mv_{\text{top}}^2}{r}$Ttop​+mg=rmvtop2​​

$T_{\text{top}} = \frac{mv_{\text{top}}^2}{r} - mg$Ttop​=rmvtop2​​−mg

At the Bottom of the Circle

At the bottom, tension points upward (towards the centre) and weight points downward (away from centre):

$T_{\text{bottom}} - mg = \frac{mv_{\text{bottom}}^2}{r}$Tbottom​−mg=rmvbottom2​​

$T_{\text{bottom}} = \frac{mv_{\text{bottom}}^2}{r} + mg$Tbottom​=rmvbottom2​​+mg

Key Insight