← Angular velocity and the speed on the circle

Angular velocity is not speed

Astrid and Bruno ride the same carousel. Astrid's horse is 1.0 m from the center. Bruno's horse is 3.0 m out, on the outer row.

Each second, both horses sweep 20°.

Do the two horses move at the same speed?

Both horses sweep 20° each second, so their angular velocities are equal.

You have seen speed: the distance an object covers each second.

In one second, each horse covers the arc its 20° cuts from its own circle.

A swept angle counts radius-lengths of arc, and Bruno's radius-length is three times Astrid's.

20°center1.0 m3.0 mAstrid's horseBruno's horsein one second, both horses sweep the same 20°
One second of the ride: both horses sweep the same 20°.

So the same 20° cuts an arc three times as long on Bruno's circle.

centerabout 0.35 mabout 1.05 mthe same angle cuts a three-times-longer arc from the three-times-bigger circle
The same 20° cuts 0.35 m from the inner circle and 1.05 m from the outer: more path each second on the outside.

Bruno's horse covers more distance each second: its speed is greater.

It is tempting to say that sweeping the same angle each second means moving at the same speed. The horses show that is not so: same angular velocity, different speeds.

Angular velocity counts the angle swept each second. Speed counts the path distance covered each second. They are two different quantities.
Two quantities, told apartquantitywhat it counts each secondangular velocitythe angle sweptspeedthe path distance covered
Angular velocity counts angle; speed counts path distance.

A wheel sweeps 25° each second. Is that a statement of its angular velocity or of its speed?

A cart on a circular track covers 6.0 m of path each second. Is that a statement of its angular velocity or of its speed?

A ride's seat covers 4.0 m of path each second. Is that a statement of its angular velocity or of its speed?

The sentence counts meters of path each second, not degrees or radians.

It states the seat's speed.

Two seats on one spinning platform sweep equal angles each second. The outer seat rides the bigger circle. Which seat moves at the greater speed?

Equal angles each second mean equal angular velocities.

The same angle cuts a longer arc from the bigger circle.

The outer seat covers more path each second.

The outer seat moves at the greater speed.

Dagny says: 'The outer rider on a carousel sweeps the same angle each second as the inner rider, but moves at a greater speed.' Is Dagny correct?

Your turn

Two seats ride one spinning platform, one near the center and one at the edge. Which statement is correct?

Which quantity states the path distance an object covers each second?

Two fans spin side by side, and each blade sweeps 360° each second. The first fan's blade tip rides a circle of radius 0.2 m; the second fan's tip rides a circle of radius 0.4 m. Which tip moves at the greater speed?