Why spinning needs its own language
Watch a bicycle wheel turn. A point on the rim races around while the hub barely moves, yet you feel that the whole wheel turns as one solid object. If you tried to describe the wheel with ordinary velocity, you would need a different number for every point — hopeless. Rotation asks for a smarter description: one that captures how fast the whole thing turns, regardless of how far a given point sits from the center.
The trick that makes rotation simple is to measure motion by angle turned instead of distance travelled. Every point of a rigid wheel sweeps through the same angle in the same time, even though the outer points cover more distance. That shared angle is the key that unlocks the whole subject.
The radian: the natural unit of angle
Degrees are a human convention — 360 is just a number the Babylonians liked. Physics prefers the radian, defined by the geometry itself: mark an arc on a circle whose length equals the radius, and the angle it subtends is one radian. In general the angle is the arc length divided by the radius.
An angle in radians is arc length s divided by radius r — a pure ratio, so radians are dimensionless.
Because a radian is a ratio of two lengths it is dimensionless: it carries no metres. That is exactly why radians make formulas clean — the bridge v = r\omega below has no hidden conversion factor, which is false if you insist on degrees.
How fast and how hard: angular velocity and acceleration
The angular displacement \theta is how far the object has turned. Its rate of change is the angular velocity \omega (Greek omega), measured in radians per second — literally how many radians the object sweeps each second. If \omega itself changes, the object has an angular acceleration \alpha (alpha), in radians per second squared.
Angular velocity is the rate of turning; angular acceleration is the rate at which the turning rate changes — perfect echoes of v and a.
These three — \theta, \omega, \alpha — are the rotational twins of position, velocity and acceleration. A record player spinning steadily has constant \omega and zero \alpha; a drill starting up has both. Everything you learned about straight-line motion is about to return in this new dress.
The bridge back to straight lines
Angular quantities describe the whole object, but a point at radius r still traces a real path with a real speed. The link is simple and worth memorizing: multiply the angular quantity by the radius to get the linear one along the rim.
The tangential (rim) speed and tangential acceleration of a point are its radius times the angular velocity and angular acceleration.
This is why the rim of a wheel moves faster than the middle even though every point shares the same \omega: larger r, larger v. It is also why a longer wrench, a bigger gear, or a farther seat on a merry-go-round all feel different — same rotation, different radius.
Which way does it spin? The axis and the right-hand rule
A spin has not just a rate but a sense — clockwise or counter-clockwise — and an axis it turns about. Physicists package both into a single arrow that points along the axis, using the right-hand rule: curl the fingers of your right hand the way the object turns, and your thumb points the way the angular-velocity vector \vec{\omega} points.