# Uniform Circular Motion and Gravitation

## Centripetal Force


### Learning Objectives

By the end of this section, you will be able to:
1. Calculate coefficient of friction on a car tire.
2. Calculate ideal speed and angle of a car on a turn.
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.
Any net force causing uniform circular motion 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: net . For uniform circular motion, the acceleration is the centripetal acceleration— . Thus, the magnitude of centripetal force  is
By using the expressions for centripetal acceleration  from , we get two expressions for the centripetal force  in terms of mass, velocity, angular velocity, and radius of curvature:
You may use whichever expression for centripetal force is more convenient. Centripetal force  is always perpendicular to the path and pointing to the center of curvature, because  is perpendicular to the velocity and pointing to the center of curvature.
Note that if you solve the first expression for , 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.

      
Let us now consider banked curves, where the slope of the road helps you negotiate the curve. See . The greater the angle , 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  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  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.
 shows a free body diagram for a car on a frictionless banked curve. If the angle  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  and the normal force of the road . (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 . 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, and 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 the figure, we see that the vertical component of the normal force is , and the only other vertical force is the car’s weight. These must be equal in magnitude; thus,
Now we can combine the last two equations to eliminate  and get an expression for , as desired. Solving the second equation for , and substituting this into the first yields
Taking the inverse tangent gives
This expression can be understood by considering how  depends on  and . A large  will be obtained for a large  and a small . 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 is frictionless. Note that  does not depend on the mass of the vehicle.

### Section Summary

1. Centripetal force  is any force causing uniform circular motion. It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity  and has magnitude
    which can also be expressed as
    

### Conceptual Questions


### Problems Exercise
