MECHANICS

2D Motion

Circular Motion 1

 position vector velocity acceleration vertical circle

Summary of equations

Describing the 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:

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

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.

the magnitude of the velocity is given by:

The acceleration vector A

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

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.).

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.

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,

What is,

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

ii) the resultant particle acceleration?

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