Concept of Rest and Motion

Rest and motion are relative, not absolute. A passenger sitting in a moving bus is at rest with respect to the seat beside them, yet in motion with respect to someone standing on the pavement watching the bus go by. Neither description is 'more correct' than the other — both are correct once you specify with respect to what. That reference is called a frame of reference: a coordinate system against which the position of an object is measured, at any instant, by its coordinates $(x, y, z)$. Only after a frame of reference is fixed does it make sense to say whether something is moving.

To say precisely where an object is, we need a coordinate system. The one physics almost always uses is the Cartesian coordinate system: three mutually perpendicular axes, $x$, $y$ and $z$, meeting at a common origin $O$. Any point in space is then labelled uniquely by its three coordinates $(x, y, z)$ — the perpendicular distances of the point from the three coordinate planes.

There is more than one way to arrange three perpendicular axes, and physics settles on one convention: the right-handed coordinate system. Curl the fingers of your right hand starting from the positive $x$-axis and sweeping towards the positive $y$-axis; your thumb then points along the positive $z$-axis. Equivalently, rotating the $x$-axis onto the $y$-axis (through the smaller angle) appears anticlockwise when viewed from the positive $z$-axis. A coordinate system built this way is called right-handed; if the same rotation instead looks clockwise from the positive $z$-axis (or if $x,y,z$ are otherwise permuted), the system is left-handed. Left-handed systems are mathematically just as valid, but physics fixes on the right-handed convention throughout, so that formulas like the vector cross product always come out with a consistent sign.

Point mass. Real objects have size and shape, which makes tracking every part of them separately hopeless for most problems. Physics gets around this with the idealisation of a point mass — the entire mass of the object imagined concentrated at a single point, with zero physical extent. A point mass has no shape or size, only mass and position. Two things are worth stressing: (i) mathematically a point mass has finite mass squeezed into zero volume, which cannot literally exist, but it is an excellent approximation whenever the object's own size is negligible compared to the distances involved; and (ii) 'point mass' is a relative idea — the Earth, huge as it is, can be treated as a point mass when studying its orbit around the Sun (because the Earth-Sun distance dwarfs the Earth's own radius), while the very same Earth obviously cannot be treated as a point mass when studying earthquakes on its surface. Likewise a small stone thrown through the air can be treated as a point mass because the stone's size is tiny compared with the distance it travels.

Types of motion observed in everyday life:

• Linear (rectilinear) motion — motion along a straight line, e.g. an athlete sprinting down a straight track, or a particle falling straight down under gravity.
• Circular motion — motion along a circular path, e.g. a stone whirled on the end of a string, or a satellite orbiting the Earth on a (nearly) circular path.
• Rotational motion — the object spins about an axis passing through (or near) itself, so that every point of the object except those actually on the axis traces its own circle around that axis, e.g. a disc spinning about an axle through its centre, or the Earth spinning once a day about its own polar axis.
• Vibratory (oscillatory) motion — the object repeats a to-and-fro motion about a fixed equilibrium point, e.g. a plucked guitar string, or a swing rocking back and forth.

Motion in one, two and three dimensions. Take the position of a particle to be given by rectangular coordinates $(x, y, z)$. As time passes, the particle is 'in motion' provided any of these coordinates changes — it is not necessary for all three to change together. …