Uniform Circular Motion

When a body moves in a circular path with a constant speed, it is said to undergo uniform circular motion.

Although the speed is constant, velocity is continually changing, since it is constantly changing its direction of motion.

Centripetal

V

V

ac

ac

Acceleration is directed towards the centre of the circle and is therefore called “centripetal acceleration.” ac =v^2r

ac =v^2r

If T is the time taken for one revolution then: V = 2πrT

ac =v^2r

= (2πrT)^2r

= 4π^2r^2T^2r

= 4π^2rT^2

Let us work with ac against T^2 on graphs.

Note:

Frequency is the number of revolutions completed per seconds. Cycles/seconds

Centripetal Force

… is the force responsible for centripetal acceleration. Hence the centripetal force may be defined as the force directed towards the centre of the circular path responsible for restraining the object to undergo uniform circular motion.

Fc= m.v^2r

Example

A car of mass 1.20X10^3 kg is rounding a curve of radius 1.55X10^3m at a speed of 25.0m/s/ Find: 1. The centripetal acceleration.

2. the centripetal force

ac =v^2r

Non-Uniform Circular Motion

Vertical Circle

Resultant force toward the centre is

FNET = FC= mv2r

Top: mv2r = mg + Tension

Bottom: mv2r = tension - mg

Universal Law of Gravitation

Everybody in the universe attracts every other body in the universe with a force that is proportional to the product of their masses, and inversely proportional to the square of the distance between them. This force acts along the line joining the two masses.

G is called the universal gravitational constant, and its value is 6.67 X 10-11

Eg.

What happens to the forces of attraction two identical masses if the distance between their centre’s half?

Gravitational Field Strength

It is the gravitational force that acts on each kilogram of a mass in the field. g= Fkg therefore the gravitational force strength of Earth is 9.8Nkg. Nkg = kgms^2kg

= kgms2X kg

= ms2

Another name therefore is the acceleration due to gravity.

Calculate the Earth’s gravitational field strength at the height of 1000m above the surface of the Earth.

Satellite Orbitals

A satellite in an orbit that is always the same height above the earth moves with uniform circular motion. ∴ GMeMsr2=MsVs2r2

⇒ GMeMs = MsVs2

⇒ GMeMsMs=V2

⇒ Vs=GMeMsMs

A satellite is orbiting the Earth 100km above the surface. Calculate the orbital velocity of the satellite if; Raidus of earth (re) = 6.37 x 106m

Mass of earth (me) = 5.98 x 1024kg

G = 6.67 x 10-11 NM2/kg2

Work

The product of the force exerted on an object and the distance the object moves in the direction of the force when the force is constant.

W = Fd

F; the size of the force

d; the size of the displacement in direction force

work is therefore a scalar quantity

units: Joule

1 Joule = 1neton x mass

NB: work is only done when the force and displacement are in the same direction.

Eg.

A person lifts a box that weighs 200N. The box is lifted through 0.15m How much work does the person do?

W=Fd

W= 200 x 0.75

W=150J

If a force is exerted perpendicular to the motion then no work is done.

Energy

Forms of energy:

Kinetic energy: the energy due to the motion of an object

Potential energy: the energy stored in an object due to its position or state

Electrical, mechanical energy and heat are forms of energy. All are measured in Joules

Energy may be transformed from one form into another.

When potential energy decreases, the kinetic energy increases in proportion to the decrease of the potential energy.

Any object which is in motion is said to have kinetic energy (KE), the size of which is related to its speed. KE=12mv2

Therefore, when a body accelerates and its speed increases, its KE also increases. Because acceleration is produced by a net force then the application of a net force to a body causes a change in...