Section 3.5: Circular Motion and Friction
Friction affects objects moving in a circular path by providing or opposing centripetal force. It is crucial in banked roads, flat tracks, or curves where tires must grip the surface to prevent slipping.
Maximum speed without slipping on a flat curve:
\[
v_{\text{max}} = \sqrt{\mu_s r g}
\]
where \(\mu_s\) is the static friction coefficient.
Banked curves: Friction can either increase or decrease the maximum speed, depending on direction of motion.
Key Concepts:
- Friction provides centripetal force if the curve is flat.
- If friction is insufficient, the object will slip outward.
- On banked curves, normal force and friction together provide the centripetal force.
Example 1 — Car on a Flat Curve
A car of mass 1000 kg navigates a flat circular track of radius 50 m. The static friction coefficient is 0.4. Find the maximum speed before slipping.
\(v_{\text{max}} = \sqrt{\mu_s r g} = \sqrt{0.4 \cdot 50 \cdot 9.8} \approx 14 \text{ m/s}\)
Practice Problems
- A car of mass 800 kg travels on a flat curve of radius 30 m with \(\mu_s = 0.5\). Find the maximum speed before slipping.
- Determine the friction force required for a 1200 kg car moving at 20 m/s on a 40 m radius curve.
- A 1000 kg vehicle rounds a banked curve with radius 60 m, bank angle 15°, and \(\mu_s = 0.3\). Compute maximum speed.
- A motorbike goes around a flat circular track of 20 m radius at 10 m/s. Find minimum \(\mu_s\) needed to avoid slipping.
- A car travels at 15 m/s on a flat circular curve of radius 25 m. Find friction force and normal force if \(\mu_s = 0.4\).