Centripetal Force

Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, friction between roller skates and a rink floor, a banked roadway’s force on a car, and forces on the tube of a spinning centrifuge.

The inward component of the net force responsible for a curved path is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton’s second law of motion, net force is mass times acceleration: $F_{\text{net}}=ma$. For uniform circular motion, the acceleration is the centripetal acceleration— $a=a_{\mathrm{c}}$. Thus, the magnitude of centripetal force $F_{\text{c}}$ is

By using the expressions for centripetal acceleration $a_{\mathrm{c}}$ from $a_{\mathrm{c}}=\frac{v^2}{r}$, we get an expression for the centripetal force ${\text{F}}_{\text{c}}$ in terms of mass, velocity, and radius of curvature:

Note that if you solve the first expression for $r$, you get

This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve.

In the case of banked curves, where the slope of the road helps you negotiate the curve, some or all of the necessary centripetal force is provided by the normal force. See Figure 2.10.3. The greater the angle $\theta$, the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve”, the angle $\theta$ is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. Conceptually, for ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force $\mathrm{N}$ in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively.

Figure 2.10.3 shows a free body diagram for a car on a frictionless banked curve. If the angle $\theta$ is ideal for the speed and radius, then the net external force will equal the necessary centripetal force. The only two external forces acting on the car are its weight $\mathrm{w}$ and the normal force of the road $\mathrm{N}$. (A frictionless surface can only exert a force perpendicular to the surface—that is, a normal force.) These two forces must add to give a net external force that is horizontal toward the center of curvature and has magnitude $m{v}^2/r$. We omit detailed calculations, which require trigonometry.