3. Newton’s Laws and Centripetal Acceleration

Banked Curves

Let us now consider banked curves, where the slope of the road helps you negotiate the curve (Figure). The greater the angle [latex]\theta[/latex], the faster you can take the curve. Racetracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve,” the angle [latex]\theta[/latex] 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 the banking angle, [latex]\theta[/latex], for an ideally banked curve and consider an example related to it.

For ideal banking, the only forces acting on the car are the weight force exerted on the car by the earth and the normal force exerted on the car by the roadway, since we are ignoring any effects of friction in the ideal case.

(Figure) shows a free-body diagram for a car on a frictionless banked curve. If the angle [latex]\theta[/latex] is ideal for the speed and radius, then  only two external forces act on the car: its weight [latex]\stackrel{\to }{w}[/latex] and the normal force of the road [latex]\stackrel{\to }{N}.[/latex] Only the normal force has a component along the radial direction (horizontal in this case). Keeping in mind that the radial component of the normal force is [latex]N_r = N \text{sin}\phantom{\rule{0.2em}{0ex}}\theta[/latex] and designating the [latex]+r[/latex] direction to point toward the center of the circular path around which the car travels:

[latex]\sum{F_r} = ma_r[/latex]

Because the car does not leave the surface of the road, the net vertical force must be zero.  From (Figure), we see that the vertical component of the normal force is [latex]N_y = N\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,[/latex] and the only other vertical force is the car’s weight. These must be equal in magnitude; applying Newton’s Second Law along the vertical axis:

[latex]\sum{F_y} = ma_y[/latex]

[latex]N \text{cos}\theta - w = m(0)[/latex]

Now we can combine these two equations to eliminate N and get an expression for [latex]\theta[/latex]. Solving the second equation for [latex]N,[/latex]:

[latex]N = \frac{mg}{\text{cos}\theta}[/latex]

Substituting this into the first yields

Taking the inverse tangent gives

This expression can be understood by considering how [latex]\theta[/latex] depends on v and r, though again only for ideal banking. A large [latex]\theta[/latex] 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 a certain range of greater or lower speeds without skidding than if the curve were frictionless. Note that the banking angle, [latex]\theta,[/latex] does not depend on the mass of the vehicle.

You may notice that the FBD of the car in this case of a car traveling around a banked curve is the very same as the FBD of an object sliding on a ramp where friction between the object and ramp is negligible. As you go through this example, please note that we analyze this type of problem using horizontal and vertical axes rather than the tilted axis system used in solving problems with an object on an inclined plane.  Why the difference in approaches? The difference in approaches occurs because of the direction of the acceleration of the object in question.  As the car travels around the banked curve, it moves around a horizontal circle.  The direction of the acceleration of the car is toward the center of the circular path since the car is assumed to travel around the curve with a constant speed.  We choose one axis direction to align with the acceleration of the car (horizontal).  In the case of the object sliding along an incline/ramp, the direction of the acceleration of the object was down the ramp.  There, too, we chose one axis to align with the direction of the acceleration of the object (tilted; down the ramp).  We typically use the direction of the acceleration to determine one axis of our system since all of the acceleration of the object is fully along this axis.  We then use axes perpendicular to this first axis to complete our coordinate system, noting that there is no acceleration of the object along these other perpendicular axes.

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) is given by [latex]\theta[/latex]. We analyze the forces in the same way we treat the case of the car rounding a banked curve.

What do taking off in a jet airplane, turning a corner in a car, riding a merry-go-round, and the circular motion of a tropical cyclone have in common? Each exhibits inertial forces—forces that merely seem to arise from motion, because the observer’s frame of reference is accelerating or rotating. When taking off in a jet, most people would agree it feels as if you are being pushed back into the seat as the airplane accelerates down the runway. Yet a physicist would say that you tend to remain stationary while the seat pushes forward on you, consistent with Newton’s First Law. An even more common experience occurs when you make a tight curve in your car—say, to the right (Figure). You feel as if you are thrown (that is, forced) toward the left relative to the car. Again, a physicist would say that this occurs not because you are experiencing a force from the left but rather you are going in a straight line (recall Newton’s first law) while the car moves to the right.

We can reconcile these points of view by examining the frames of reference used. Let us concentrate on people in a car. Passengers instinctively use the accelerating frame of reference of the car as a frame of reference, whereas a physicist might use Earth. The physicist might make this choice because Earth is nearly an inertial frame of reference, in which all forces have an identifiable physical origin. In such a frame of reference, Newton’s laws of motion take the form given in Newton’s Laws of Motion. The car is a noninertial frame of reference because it is an accelerated reference frame since its velocity is changing in time. The force to the left sensed by car passengers is an inertial force having no physical origin (it is due purely to the inertia of the passenger, not to some physical cause such as tension, friction, or gravitation). The car, as well as the driver, is actually accelerating to the right. This inertial force is said to be an inertial force, also called a fictitious force, because it does not have a physical origin, such as gravity.

A physicist will choose whatever reference frame is most convenient for the situation being analyzed. There is no problem to a physicist in including inertial forces and Newton’s second law, as usual, if that is more convenient, for example, on a merry-go-round or on a rotating planet. Noninertial (accelerated) frames of reference are used when it is useful to do so. Different frames of reference must be considered in discussing the motion of an astronaut in a spacecraft traveling at speeds near the speed of light, as you will appreciate in the study of the special theory of relativity.

Let us now take a mental ride on a merry-go-round—specifically, a rapidly rotating playground merry-go-round (Figure). You take the merry-go-round to be your frame of reference because you rotate together. When rotating in that noninertial frame of reference, you feel an inertial force that tends to throw you outward, away from the center of the circular path; this is often referred to as a centrifugal force. Centrifugal force is a commonly used term, but it does not actually exist. You must hang on tightly to counteract your inertia (which people often refer to as centrifugal force). In Earth’s frame of reference, there is no force trying to throw you off; we emphasize that centrifugal force is a fiction. You must hang on to make yourself go in a circle because otherwise you would go in a straight line, right off the merry-go-round, in keeping with Newton’s first law. But the force you exert acts toward the center of the circle.

This inertial effect, carrying you away from the center of rotation, is put to good use in centrifuges (Figure). A centrifuge spins a sample very rapidly, as mentioned earlier in this chapter. Viewed from the rotating frame of reference, the inertial force throws particles outward, hastening their sedimentation. The greater the angular velocity, the greater the centrifugal force. But what really happens is that the inertia of the particles carries them along a line tangent to the circle while the test tube is forced in a circular path by the normal force exerted on the test tube by the surface of the centrifuge.

Let us now consider what happens if something moves in a rotating frame of reference. For example, what if you slide a ball directly away from the center of the merry-go-round, as shown in (Figure)? The ball follows a straight path relative to Earth (assuming negligible friction) and a path curved to the right on the merry-go-round’s surface. A person standing next to the merry-go-round sees the ball moving straight and the merry-go-round rotating underneath it. In the merry-go-round’s frame of reference, we explain the apparent curve to the right by using an inertial force, called the Coriolis force, which causes the ball to curve to the right. The Coriolis force can be used by anyone in that frame of reference to explain why objects follow curved paths and allows us to apply Newton’s laws in noninertial frames of reference.

Up until now, we have considered Earth to be an inertial frame of reference with little or no worry about effects due to its rotation. Yet such effects do exist—in the rotation of weather systems, for example. Most consequences of Earth’s rotation can be qualitatively understood by analogy with the merry-go-round. Viewed from above the North Pole, Earth rotates counterclockwise, as does the merry-go-round in (Figure). As on the merry-go-round, any motion in Earth’s Northern Hemisphere experiences a Coriolis force to the right. Just the opposite occurs in the Southern Hemisphere; there, the force is to the left. Because Earth’s angular velocity is small, the Coriolis force is usually negligible, but for large-scale motions, such as wind patterns, it has substantial effects.

The Coriolis force causes hurricanes in the Northern Hemisphere to rotate in the counterclockwise direction, whereas tropical cyclones in the Southern Hemisphere rotate in the clockwise direction. (The terms hurricane, typhoon, and tropical storm are regionally specific names for cyclones, which are storm systems characterized by low pressure centers, strong winds, and heavy rains.) (Figure) helps show how these rotations take place. Air flows toward any region of low pressure, and tropical cyclones contain particularly low pressures. Thus winds flow toward the center of a tropical cyclone or a low-pressure weather system at the surface. In the Northern Hemisphere, these inward winds are deflected to the right, as shown in the figure, producing a counterclockwise circulation at the surface for low-pressure zones of any type. Low pressure at the surface is associated with rising air, which also produces cooling and cloud formation, making low-pressure patterns quite visible from space. Conversely, wind circulation around high-pressure zones is clockwise in the Southern Hemisphere but is less visible because high pressure is associated with sinking air, producing clear skies.

The rotation of tropical cyclones and the path of a ball on a merry-go-round can just as well be explained by inertia and the rotation of the system underneath. When noninertial frames are used, inertial forces, such as the Coriolis force, must be invented to explain the curved path. There is no identifiable physical source for these inertial forces. In an inertial frame, inertia explains the path, and no force is found to be without an identifiable source. Either view allows us to describe nature, but a view in an inertial frame is the simplest in the sense that all forces have origins and explanations.