Centripetal force is a force applied perpendicular to the motion of an object over a period of time, causing the object to move

in a curve path.Depending on the way centripetal force is

applied, the path of the object may be a slight curve to a circle or other conic section. The Law of Inertia causes a centrifugal inertia force, which is equal and opposite to the

centripetal force.

Car going around a curveWhen an automobile moves along a road, it will tend to move on a straight line, due to its inertia.

Tires provide centripetal force for car going around a curve.The friction between the front tires and the road create a force that is perpendicular to the direction of motion.

v = at = (F/m)dt

v

v

v

However, if it comes to a curve in the road, the driver turns the steering wheel to aim the front wheels in a direction following the curve in the road.

That friction force is the centripetal force, causing the automobile to go on a curved path.

Strings and Flat Surfaces• Suppose that a mass is

tied to the end of a string and is being whirled in a circle along the top of a frictionless table as shown in the diagram below.

• •

• A freebody diagram of the forces on the mass would show

The tension is the unbalanced central force: T = Fc = mac, it is supplying the centripetal force necessary to keep the block moving in its circular path.

Conical Pendulums • An object on the end of string

as shown below. • A freebody diagram of the

mass on the end of the pendulum would show the following forces.

Notice, that its path also tracks out a horizontal circle in which gravity is always perpendicular to the object's path.

T cos θ is balanced by the object's weight, mg. It is T sin θ that is the unbalanced central force that is supplying the centripetal force necessary to keep the block moving in its circular path: T sin θ = Fc = mac.

Flat Curves • In this case friction is the source

of the centripetal force. Suppose that a car is traveling through a curve along a flat, level road.

• A freebody diagram of this situation would look very much like that of the block on the end of a string, except that friction would replace tension.

Friction is the unbalanced central force that is supplying the centripetal force necessary to keep the car moving along its horizontal circular path: f = Fc = mac. Since f = μN and N = mg on this horizontal surface, most problems usually ask you to solve for the minimum coefficient of friction required to keep the car on the road.

Banked Curves

• “Bank” a turn so that normal force exerted by the road provides the centripetal force

• To calculate the angle to bank at a set speed:tan θ = v²/gr

• As long as you aren’t going over the recommended velocity, you should never slip off a banked road (even if the surface is wet)

frictionless

mg

N

Fy 0 N cos mg

Fr ma N sin mv2 / R

tan v2gR

Rv2g tan

Ex: A car of mass, m, is traveling at a constant speed, v, along a curve that is now banked and has a radius, R. What bank angle, , makes reliance on friction unnecessary?

The Anik F1 satellite has a mass of 3021 kg. How high above the equatormust the satellite be in order to maintain geosynchronous Earth orbit?

Earth’s period is 23 hours, 56 minutes, and 4 seconds.

v=480 km/hr

L

L

W

Fr 0 2L sin 40 Mv2

r

Fy 0 2L cos40 Mg

2Mg

2 cos40sin 40 M

v2

r

r v2

g tan 40

L Mg

2 cos40

An airplane is flying in a horizontal circle with a speed of 480 km/hr. If the wings of the plane are tilted 40o to the horizontal, what is the radius of the circle in which the plane is flying? (Assume that the required force is provided entirely by an “aerodynamic lift” that is perpendicular to the wing surface.)