Horizontal Circular Motion

Now that we understand centripetal force and acceleration, we can analyse specific scenarios where objects move in horizontal circles. The key skill is identifying which real force provides the centripetal force in each situation.

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Car on a Flat Curve

When a car travels around a flat (unbanked) circular curve, friction between the tyres and the road provides the centripetal force.

The forces on the car are:

• Weight (mg) acting downward
• Normal reaction (N) acting upward: N = mg (since there is no vertical acceleration)
• Friction (f) acting horizontally towards the centre of the curve

For the car not to skid: $f \leq \mu N = \mu mg$f≤μN=μmg

The centripetal force required is: $\frac{mv^2}{r} \leq \mu mg$rmv2​≤μmg

Cancelling m: $\frac{v^2}{r} \leq \mu g$rv2​≤μg

Maximum speed on a flat curve: $v_{\max} = \sqrt{\mu g r}$vmax​=μgr​

Worked Example 1

A car enters a flat roundabout of radius 30 m. The coefficient of friction between the tyres and the road is 0.70. What is the maximum safe speed?

$v_{\max} = \sqrt{0.70 \times 9.81 \times 30} = \sqrt{206} = 14.4 \text{ m s}^{-1}$vmax​=0.70×9.81×30​=206​=14.4 m s−1

This is about 52 km/h — consistent with typical roundabout speed limits.