The locus of a point moving in a circle

Basic Principles

A locus is a path defined by certain properties that a point follows.
In the context of a circle, the locus would be any point that moves along the circumference of the circle.
When a point is moving in a circle, it constantly changes direction but the distance from the centre of the circle remains constant. This constant distance is the circle’s radius.

Properties of the Circle

A circle is a set of points in a plane that are equidistant from a fixed point called the centre.
The radius of the circle is the constant distance from the locus to the centre of the circle.
The circumference, the edge of the circle, can be calculated using the formula 2πr, where r is the radius. This is the path followed by the locus.

Locus Motion

As the locus moves around the circumference, its coordinates change.
Using Cartesian coordinates, the position of the locus at any point in its motion can be determined by the equations x = rcos(θ) and y = rsin(θ), where θ is the angle subtended at the centre.
In term of complex numbers, the motion can be expressed as z = reiθ, where z is the position of the locus.

Determining Locus Position

Given the radius and the angle subtended at the centre from a reference line (like the x-axis), you can determine the position of the locus.
You could also determine the locus position at a particular time given its speed and direction of rotation.

Path Tracing

Plotting the positions of the locus as it moves around the circle gives a path that forms the circle.
Depending on the problem, you may have to consider the path taken before the present point or the path that will be travelled after.

Applications

Understanding the locus of a point moving in a circle has applications in physics, engineering, and computer graphics.
You can use this knowledge to describe planetary motion, analyse waveforms in electronics, or create dynamic animations.

Remember reviewing illustrative examples and practising problem-solving is key to internalising these concepts. Make sure to utilise resources such as past papers and online tutorials to reinforce learning and clarify any areas of difficulty.