2.4 circular motion

2.4 Circular Motion

Topic 2 - Mechanics

Acceleration

Recall that when a nett force acts on an object it causes an acceleration.This is the basis of Newton's second law

This acceleration will cause a change in velocity of the objectIt will change speed,

It will change direction

It will change speed and direction

Centripetal Acceleration

Consider an object that is forced to go in a circle by a string tied at the centre of that circle.

IF the string is cut, the object will move off with a linear velocity.

IF the object is moving at constant speed then;There can be no component of acceleration in the direction of the velocity.

There must still be an acceleration towards the centre of the circle because the string is there.

The direction of the velocity must change by this radial acceleration


Acceleration is the rate of change of velocity.

Assume that the object moves from A to B in time t.

The change in velocity can be found graphically by v u

u

v

r

r

u

v

s


Consider that the displacement triangle and the velocity triangle are mathematically similar if the speed and radius of the circle are constant.

u

v

r

r

u

v

s


The average velocity is defined as:

Substituting this in the previous equation gives:

Which rearranges to:

u

v

r

r

u

v

s


The object travels through a total angle of 2 radians in one orbit in a time Period of T

The angular speed is therefore

The angle turned in time t is therefore (distance = speed x time):

Degrees can be converted into radians by:

u

v

r

r

u

v

s


Using the angular speed the orbital speed is thus:

Which makes the acceleration equation:

u

v

r

r

u

v

s


Mathematically, the position vector of a particle at time t is given by:

Differentiating for v gives:

Differentiating again gives the acceleration:

xy

r

r sin r cos

What causes this Acceleration?

As has been shown, the uniform circular acceleration occurs along a radius (radially) as shown by the r in the equation and towards the centre of the circle, as shown by the -.

Any force that acts in this way will cause a centripetal acceleration.e.g. the gravitational pull of the Earth on the Moon causes the acceleration that keeps the Moon in orbit.

The sideways friction force (towards the centre) that keeps a car's tyres from slipping outwards causes the acceleration to make the car turn the corner.

The electrostatic attraction of a positive nucleus pulling on a negative (particle-like) electron keeps it orbiting in a shell.

Centrifugal Acceleration

A centrifugal (centre-fleeing) acceleration does not really exist.

It is a consequence of Newton's first law that states that objects with inertia will continue in a straight line unless an external force acts.When we go around a corner in a car, we feel that we are being thrown outwards.

In reality our inertia causes us to try to go straight.

The forces acting on the car tyres makes the car accelerate (turn) across us.

We collide with the side of the car, which exerts a sideways force on us (Newton's 3rd), that causes us to accelerate around the corner with the car.

We feel flung-out when in reality we are pushed round

Questions

The Moon orbits the Earth at a radius of 3.2x106m in a time period of 28.5 days. What is the centripetal acceleration of the Moon?

The Earth orbits the Sun at a radius of 150 million km in a year. What is the centripetal acceleration of the Earth?

A formula one car travels around a bend of radius 25m at 110kmh-1. What is its centripetal acceleration?

Questions

The tyres of a car of mass 1500kg can exert a maximum sideways force of 500N when cornering. What is the maximum speed that this car can travel around a bend of radius 8m and radius 24m?

A formula 1 car has a mass of 625kg. The driver wants to take a corner at no less than 90kmh-1 using tyres that can exert a maximum sideways force of 35kN. What is the minimum radius corner that the car can cope with before slipping?

2.4 circular motion

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