Sometimes the angle, or the time, is the unknown instead of the angular velocity. A ride turns steadily at 40°/s. What angle does it sweep in 6.0 s?
Each second of turning adds 40° to the sweep.
In 6.0 s, the sweep reaches 40 × 6.0 = 240°.
One rearrangement of ω = Δθ/Δt says that multiply in general, and a second rearrangement finds the time.
Start from the equation you know: ω = Δθ/Δt.
Make Δθ the subject: multiply both sides by Δt.
That gives Δθ = ωΔt.
Make Δt the subject: divide Δθ by ω.
That gives Δt = Δθ/ω.
The forward direction works as before: 6π radians swept in 3 s gives ω = 2π rad/s.
The questions below mix all three directions; every value arrives already in the unit the answer wants.
A ride turns steadily at 40°/s for 6.0 s. Calculate the angle it sweeps.
Write down the values:
ω = 40°/s, Δt = 6.0 s, Δθ = ?
Write down the equation:
Δθ = ωΔt
Substitute in the values, and calculate:
Δθ = 40 × 6.0 = 240
The ride sweeps 240°.
A wheel turns steadily at 2.0 rad/s. Calculate how long it takes to sweep 9.0 radians.
Write down the values:
ω = 2.0 rad/s, Δθ = 9.0 rad, Δt = ?
Write down the equation:
Δt = Δθ/ω
Substitute in the values, and calculate:
Δt = 9.0 / 2.0 = 4.5
The wheel takes 4.5 s.
A gear sweeps 150° in 2.5 s, turning steadily. Enter its angular velocity in °/s.
A turbine turns steadily at 3.0 rad/s for 8.0 s. Enter the angle it sweeps in radians.
A carousel turns steadily at 18°/s. Enter the time it takes to sweep 90°, in s.
A cement mixer drum sweeps 7.2 radians in 4.0 s, turning steadily. Enter its angular velocity in rad/s.
A revolving door turns steadily at 2.4 rad/s as people walk through. Enter the time it takes to sweep 18 radians, in s.
A rotating sign turns steadily at 45°/s for 7.0 s. Enter the angle it sweeps in degrees.
A rotating restaurant platform turns steadily at 0.1°/s all evening. Enter the angle it sweeps in 350 s, in degrees.
During a slow-speed test, a fan blade sweeps 66 radians in 30 s, turning steadily. Enter its angular velocity in rad/s.
A telescope mount tracks a star by turning steadily at 15°/s during a repositioning move. Enter the time it takes to sweep 345°, in s.