Linear and Angular Speed

Trigonometry · Axiom Academy

LESSON Linear and Angular Speed How fast you spin and how fast you travel are two different things — tied together by one clean relationship, . 1. Angular Speed: How Fast the Angle Turns Angular speed (the Greek letter omega) measures how fast an angle is swept out — radians of turn per unit of time. Sweep an angle in time t and the angular speed is just the angle divided by the time. Angular speed = radians of turn per unit time One full revolution is radians 2. Arc Length: How Far the Point Travels As the radius sweeps, the point on the rim traces a curved path — an arc . When the angle is measured in radians , the length s of that arc is beautifully simple: the radius times the angle. Why does it work? A full turn is , and it traces the whole circumference . A smaller angle traces the same fraction of the circle, so scales right down with it. This is exactly why radians are the natural unit here — the factor is just r , with no extra conversion constant. A point from the center sweeps through radians. It travels along the arc. Linear speed v is the everyday kind of speed: distance over time. The distance along the rim is the arc length , so dividing by time and recognizing collapses everything into one relationship. Same spin, bigger radius, faster point The two points on the disk above share the same , but the outer point sits at twice the radius — so it sweeps twice the arc in the same time and moves twice as fast . That is made visible. With and , the linear speed is .

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