Centripetal Force and Banked Curves Chapter 5 Lesson 2
Circular Motion When an object moves in a circle at constant speed, we describe it as undergoing uniform circular motion. Its speed is constant, but its velocity is not because velocity includes direction and the object’s direction is clearly changing.
Circular Motion A changing velocity means acceleration. The pull on the string is always directed perpendicular to the velocity. The pull accelerates the ball into a circular path, even though the ball does not speed up or slow down. The pull changes only the direction of the velocity, not the magnitude.
Centripetal Acceleration The acceleration arising from the change in direction of the velocity vector is called the centripetal acceleration and is determined mathematically by:
Centripetal Acceleration
Centripetal Acceleration Centripetal means center-seeking. Centripetal acceleration is always directed toward the center of the circle of motion. It is this centripetal acceleration that is responsible for the change in the direction of the velocity; the magnitude of the velocity remains constant.
Centripetal Force The centripetal force always points toward the center of the circle about which the object moves with uniform speed. "Centripetal" means "center-seeking", and "centrifugal" means "outwards-seeking" or, more literally, "center-fleeing".
Directions in centripetal force problems: Positive direction is inwards toward center of circle. Negative direction is outward away from center of circle.
The centrifugal force do not belong in a free-body diagram. The force F in the picture would provide the centripetal force needed to maintain the circular path.
Radius “R” is the distance from the center of the mass to the axis of rotation.
Motion On A Flat Curve On a flat, level curve, the friction between the tires and the road supplies the centripetal force. If the tires are worn smooth or the road is icy or oily, this friction force will not be available. The car will not be able to move in a circle, it will keep going in a straight line and therefore go off the road. Notice that the
Motion On A Flat Curve Accelerations: Equation: Fc = FF ; ay = 0 m/s2 ax = ac = Equation: Fc = FF ;
Motion On A Banked Curve Some curves are banked to compensate for slippery conditions. In addition to any friction forces that may or may not be present, the road exerts a normal force perpendicular to its surface. The downward force of the car’s weight is also present. These two forces add as vectors to provide a net force Fnet that points toward the center of the circle; this is the centripetal force. The centripetal force is directed toward the center of the circle, it is not parallel to the banked road.
Motion On A Banked Curve The effect of banking is to tilt the normal force Fn toward the center of curvature of the road so that the inward radial component FNsin can supply the required centripetal force. Vehicles can make a sharp turn more safely if the road is banked. If the vehicle maintains the speed for which the curve is designed, no frictional force is needed to keep the vehicle on the road.
5-3 Highway Curves, Banked and Unbanked When a car goes around a curve, there must be a net force towards the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.
5-3 Highway Curves, Banked and Unbanked If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show.
5-3 Highway Curves, Banked and Unbanked As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways: The kinetic frictional force is smaller than the static. The static frictional force can point towards the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.
5-3 Highway Curves, Banked and Unbanked Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required. This occurs when:
Example 5-6 A 1000. kg car rounds a curve on a flat road of radius 50. m at a speed of 50. km/h (14 m/s). Will the car follow the curve, or will it skid? Assume (a) the pavement is dry and the coefficient of static friction is s=0.60; (b) the pavement is icy and s=0.25.
Example 5-7 (a) For a car traveling with speed v (14 m/s) around a curve of radius r, determine a formula for the angle at which a road should be banked so that no friction is required. (b) What is the angle for an expressway off-ramp curve of 50 m at a design speed of 50 km/h?
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