Horizontal and vertical circular motion

What you’ll learn

• Resolve real forces towards the centre of a circle.
• Combine radial dynamics with energy and contact conditions.

Understand the idea

Circular motion requires inward radial acceleration, even when speed is constant, because the direction of velocity changes. The required inward resultant is provided by existing forces such as tension, reaction, friction or components of weight. In a conical pendulum or banked-track problem, combine radial motion with vertical equilibrium.

In a vertical circle, speed generally changes as gravitational potential energy changes. Use energy to relate speeds at different heights, then resolve forces radially at the point of interest. A string must have non-negative tension and a contact surface must have a physically possible reaction; reaching zero marks possible slackening or loss of contact.

Key formulae

$$v=r\omega$$

$$a_r=\frac{v^2}{r}=r\omega^2$$

$$\sum F_{\mathrm{inward}}=\frac{mv^2}{r}$$

$$T+mg=\frac{mv_{\mathrm{top}}^2}{r}\quad\text{(string at top of vertical circle)}$$

Worked example

A $0.5\,\mathrm{kg}$ particle moves at $6\,\mathrm{m\,s^{-1}}$ in a horizontal circle of radius $2\,\mathrm m$ on a smooth table, held by a horizontal string. Find the tension.

1. Vertical forces balance, leaving tension as the inward horizontal force.
2. Use $T=mv^2/r$.
3. Substitute $0.5(6^2)/2=9$.

Answer: $T=9\,\mathrm N$.