Circular Motion Basics
- When an object moves in a circle with uniform speed, it experiences centripetal acceleration directed toward the center:
\(\boxed{a_c = \frac{v^2}{R}}\) - The force causing this acceleration is the centripetal force:
\(\boxed{f_c = \frac{mv^2}{R}}\)
Examples: Tension in a string (for a rotating stone), gravitational force (for planets), or friction (for cars on roads).
Car on a Level Road
- Three forces act: weight (\(mg\)), normal force (\(N\)), and friction (\(f\)). Friction provides the centripetal force.
- Maximum speed without slipping:
\(\boxed{v_{\text{max}} = \sqrt{\mu_s R g}\)
Facts: Speed limit depends on friction coefficient (\(\mu_s\)), radius (\(R\)), and gravity (\(g\)). Mass doesn’t matter!
Car on a Banked Road
- Banking reduces reliance on friction. Forces involved: normal force (\(N\)), friction (\(f\)), and weight (\(mg\)).
- Optimum speed (no friction needed):
\(\boxed{v_o = \sqrt{R g \tan \theta}}\) - Maximum permissible speed (with friction):
\(\boxed{v_{\text{max}} = \sqrt{ Rg \frac{\mu_s + \tan \theta}{1 – \mu_s \tan \theta} }}\)
Examples
- Cyclist on a level road: If \(v^2 > \mu_s R g\), the cyclist slips. For \(v = 5 \, \text{m/s}\), \(R = 3 \, \text{m}\), \(\mu_s = 0.1\): \(25 > 2.94\) → cyclist slips.
- Banked racetrack: At \(\theta = 15°\), \(\mu_s = 0.2\), \(R = 300 \, \text{m}\):
– Optimum speed: \(28.1 \, \text{m/s}\)
– Max speed: \(38.1 \, \text{m/s}\)
Important Concepts for NEET
- Centripetal Force Sources: Identify forces like tension, gravity, or friction acting as centripetal force in different scenarios.
- Maximum Speed on Flat Road: \(v_{\text{max}} = \sqrt{\mu_s R g}\) is a frequently tested formula.
- Banked Roads: Know how to calculate \(v_o\) (no friction) and \(v_{\text{max}}\) (with friction).
- Friction’s Role: Static friction provides centripetal force on flat roads; banking reduces this dependence.
