What is Centripetal Force?
Centripetal force is a center-seeking force that acts on an object moving in a circular path. It is directed toward the center of curvature of the path. Without this force, an object would continue to move in a straight line according to Newton's First Law.
Mathematically, the magnitude of centripetal force $F_c$ is given by: $$F_c = \frac{mv^2}{r}$$ Where:
- $m$ = mass of the object
- $v$ = linear velocity
- $r$ = radius of the path
Derivation for a Bicycle on a Curved Road
When a cyclist turns on a flat, curved road, they must lean inward at an angle $\theta$ with the vertical. This leaning allows the ground to exert a normal force $N$ that provides the necessary centripetal force.
Forces at play:
- Weight ($mg$): Acting vertically downward.
- Normal Reaction ($N$): Acting perpendicular to the ground/bicycle surface.
To balance the forces:
- The vertical component of the normal reaction must balance the weight: $N \cos\theta = mg$
- The horizontal component of the normal reaction provides the centripetal force: $N \sin\theta = \frac{mv^2}{r}$
Solving for the angle of lean:
Dividing the two equations: $$\frac{N \sin\theta}{N \cos\theta} = \frac{mv^2 / r}{mg}$$ $$\tan\theta = \frac{v^2}{rg}$$
This shows that the necessary angle of lean $\theta$ depends on the square of the speed $v$ and the sharpness of the curve $r$.