UNIFORM CIRCULAR MOTION.

Specific objectives.

• Define circular motion.
• Define angular quantities.
• Describe simple experiment to illustrate centripetal force.
• Explain the applications of simple circular motions.
• Solve numerical problems involving circular motion.

                        Circular motion.

• Is the motion along a curved path or along the circumference of a circle.
• Is determined or forced by forces e.g. centripetal force.

                                    Angular quantities.

• These are;

1. Angular displacement.
2. Angular belocity.

• Angular acceleration.

1. Angular displacement.

• Is the angle swept by a radius of a circle when a body moves from point A to point B along the circumference of a circle.
• Is measured in radiance.

                                                Radian measure.

• Angle subtended at the center of a circle by an arc length equal to the radius of the circle.
• When the arc length is equal to radius then = 1 radiance.
• When the arc length is equal to the circumference of a radius then

1. Angular velocity.

• Is the rate of change of angular displacement.
• It is noted by the vector omega ().

            V = linear velocity

                        V = r                                    SI unit = radiance/second²

• Angular acceleration ().
• The rate of change of angular velocity.
• Is determined by the letter alpha (.

            Angular acceleration   =

                                     = ²r                                              SI unit = radiance/second²

                                     =

• The acceleration is directed towards the center of a circle and is called centripetal acceleration.
• It experiences a force called centripetal force.

                                                Centripetal force  F.

• Is the force that forces a body to accelerate towards the center of a curved part of a circle.

            F  change of momentum

            F

            F =

            F = M²r

                                    Factors affecting centripetal force.

1. Radius of the curved path.
2. Angular velocity.
3. Mass of the object.
4. Linear velocity.

• The factors show that the centripetal force requires to keep a body in a circular path increases with;

1. Increase in the mass of the body.
2. Increase in linear velocity of the object.
3. Decrease in radius of the circular path.

• For the case of angular velocity

1. An increase in angular velocity.
2. An increase in radius.

Centrifugal force.

• Is the force that makes an object fly away from the circular path or from the center of a curved path.

                                                Inertia.

• Is the natural tendancy of a body to remain at rest or uniform motion in a straight line if there is external force acting on it.

                                                Motion in a vertical circle.

• Consider a particle of mass M tied to one end of a string moving with uniform speed in a vertical circle of radius R.

• Two forces act on the particle at any instance.

1. Weight acting downwards.
2. Tension acting towards the center of the circle.

• This force, tension, changes in magnitude at different positions of the particle.
• The two forces provide the centripetal force that maintains the body in a circular motion.
• At position A;

            F = W + T

             = Mg + T

            T =  – Mg

• At position B, centripetal force = tension (T).
• At position C

            Centripetal force F = T –W

                                     = T – Mg

                                    T =  + Mg

• At D is the same as B.
• The tension in the string is maximum when the object is at the lowest point and the string is most likely to cut or snap.
• The tension T is minimum at the highest at the highest point and at this point, a certain minimum speed must be maintained in order to keep the string tight or the weight does not fall.
• The minimum velocity or speed.

             = Mg

             = g

            V² = rg

            V =

                                    Areas where centripetal force is used.

1. A matatu rounding a bend.Frictional force from the road on tires provide centripetal force.
2. Electrons orbit round an atomic nucleus. Electrostatic force provide the centripetal force.
3. Moon in their orbit and planet on their orbit, gravitational force provide the centripetal force.
4. Satellite or space shuttle in an orbit round the earth at a certain height, gravitational force provides the centripetal force.
5. A car rounding a sharp bend reduces its speed. When a car is speeding at a bend, its linear motion is being transformed into circular motion. Centrifugal force and centripetal force are produced. When a car is at high speed, very little time is available for balancing the two forces. The car then slows down to allow for the two forces to balance.
6. Explain why the road on a steep bend is made sloppy with its inner side a little lower than the outer. When a vehicle is negotiating a bend, it acquires circular motion thereby producing centrifugal force which tries to make the vehicle fly away from the track and gets overturned. To overcome centrifugal force, enough centripetal force has to be produced. This is achieved by making the road sloppy towards the inside so that the vehicle is slightly tilted and its weight force more on the inside. This reduces the tendency of the vehicle to fly away from the track.