第6章ニュートンの法則の応用

Let us now consider banked curves, where the slope of the road helps you negotiate the curve (Figure 6.22). 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. We will derive an expression for $\theta$ for an ideally banked curve and consider an example related to it.

For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions.

Figure 6.22 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 equals the necessary centripetal force. The only two external forces acting on the car are its weight $\overrightarrow{w}$ and the normal force of the road $\overrightarrow{N}.$ (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 $m{v}^2\text{/}r.$ Because this is the crucial force and it is horizontal, we use a coordinate system with vertical and horizontal axes. Only the normal force has a horizontal component, so this must equal the centripetal force, that is,

Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in direction. From Figure 6.22, we see that the vertical component of the normal force is $N\text{cos}\theta ,$ and the only other vertical force is the car’s weight. These must be equal in magnitude; thus,

Now we can combine these two equations to eliminate N and get an expression for $\theta$, as desired. Solving the second equation for $N=mg\text{/}\left(cos\theta \right)$ and substituting this into the first yields

Taking the inverse tangent gives

This expression can be understood by considering how $\theta$ depends on v and r. A large $\theta$ is obtained for a large v and a small r. That is, roads must be steeply banked for high speeds and sharp curves. Friction helps, because it allows you to take the curve at greater or lower speed than if the curve were frictionless. Note that $\theta$ does not depend on the mass of the vehicle.

Airplanes also make turns by banking. The lift force, due to the force of the air on the wing, acts at right angles to the wing. When the airplane banks, the pilot is obtaining greater lift than necessary for level flight. The vertical component of lift balances the airplane’s weight, and the horizontal component accelerates the plane. The banking angle shown in Figure 6.23 is given by $\theta$. We analyze the forces in the same way we treat the case of the car rounding a banked curve.