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MECHANICS

 

2D Motion

 

Circular Motion 1

 

position vector

velocity

acceleration

vertical circle

 

 

 

Summary of equations

 

 

 

 

Describing the circle - position vector R

 

 

circle position vector R

 

 

i & j are unit vectors along the x and y-axis respectively.

 

The position vector R of a particle at P from O, at time t, is given by:

 

 

circle - position vector equationo

 

 

As the position vector R rotates anti-clockwise, the particle at P traces out a circle of radius r .

 

 

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The velocity vector V

 

The velocity vector V at an instant is given by differentiating the position vector R with respect to t .

 

Here the unit vectors i & j, parallel to the x and y-axes, are centred on the particle at P.

 

 

vector equation for velocity

 

 

 

circle vector V

 

 

 

the magnitude of the velocity is given by:

 

 

circle velocity equation

 

 

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The acceleration vector A

 

The acceleration of the particle at P is given by differentiating V with respect to t .

 

 

circle - the acceleration vector

 

 

circle - equation for acceleration

 

 

Example

 

A satellite is moving at 2000 ms-1 in a circular orbit around a distant moon.

 

If the radius of the circle followed by the satellite is 1000 km, find:


i) the acceleration of the satellite
ii) the time for the satellite to complete one full orbit of the moon in minutes(2d.p.).

 

 

circle problem#01

 

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Non-uniform circular motion(vertical circle)

 

A more in-depth treatment of motion in a vertical circle is to be found in 'kinetics/more circular motion'.

 

Here we look at the more general case of the acceleration component along the circle and the component towards the centre varying.

 

circle - non-uniform acceleration

 

 

Example

 

A particle starts to move in a circular direction with an angular speed of 5 rad s-1 .
The radius of the circle of motion is 4 m, and the angular speed at time t is given by,

 

circle - non-uniform acceleration problem #01

 

What is,

 

i) the linear speed of the particle 6 secs. after it starts moving?

ii) the resultant particle acceleration?

(answers to 1 d.p.)

 

circle - non-uniform acceleration problem #01b

 

 

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