Circular motion starts with measuring angle in radians and rate of turn as angular speed. From there a single relation, v = rω, ties the rate of turning to how fast a point actually moves.
For an object moving in a circle, the angle in radians is arc divided by radius, the angular speed is ω = 2π / T = 2πf, and the linear speed is v = rω.
A point on a spinning disc sweeps out equal angles in equal times. Watch the angle, the period and the linear speed update together, then notice that the rim moves faster than a point near the axis even though both turn at the same rate.
One radian is the angle whose arc length equals the radius, so a whole turn is 2π rad. The angular speed is ω = Δθ / Δt = 2π / T = 2πf, where T is the period and f the frequency. A point a distance r from the axis moves along its circle at v = rω, so for a rigid body every point shares one ω while the linear speed grows with r.
Four quick checks on radians, angular speed and v = rω. Each correct answer earns XP and lights this skill on your star map.
One radian is the angle subtended at the centre of a circle by an arc whose length equals:
A wheel turns at 2.0 revolutions per second. Its angular speed ω = 2πf is:
A point 0.15 m from the axis of a disc turning at ω = 8.0 rad s⁻¹ has a linear speed v = rω of:
On a rigid rotating disc, compared with a point near the axis, a point near the rim has:
Radians, not degrees. The relations ω = 2π/T and v = rω require the angle in radians, so keep your calculator in radian mode and remember 2π rad = 360°. And although every point of a rigid rotating body shares one angular speed ω, the linear speed v = rω is not shared: it grows with distance from the axis.
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