Angular Velocity from Revolutions per Time

Angular velocity $\omega$ is the rate of change of angle with respect to time. It's the rotational analog of linear velocity. The basic conversion: one full revolution = $2\pi$ radians.

Three Equivalent Ways to Express Rotation Speed

• Angular velocity $\omega$ (rad/s): radians per second
• Frequency $f$ (Hz): revolutions per second
• Period $T$ (s): seconds per revolution

The relationships are:

$\omega = 2\pi f \qquad T = \dfrac{1}{f} = \dfrac{2\pi}{\omega}$

Step-by-Step Solution

Given: 5 revolutions in 2 seconds.

Find: $\omega$ in rad/s, $f$ in Hz, and $T$ in seconds.

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Step 1 — Find the frequency.

Frequency = revolutions per second:

$f = \dfrac{5\;\text{rev}}{2\;\text{s}} = 2.5\;\text{rev/s} = 2.5\;\text{Hz}$

Step 2 — Find the period.

$T = \dfrac{1}{f} = \dfrac{1}{2.5} = 0.4\;\text{s}$

So one full revolution takes 0.4 seconds.

Step 3 — Find the angular velocity.

Each revolution sweeps $2\pi$ radians. Total angle in 2 seconds:

$\theta = 5 \times 2\pi = 10\pi\;\text{rad}$

Divide by elapsed time:

$\omega = \dfrac{\theta}{t} = \dfrac{10\pi}{2} = 5\pi\;\text{rad/s}$

$\omega \approx 15.708\;\text{rad/s}$

Step 4 — Verify $\omega = 2\pi f$.

$2\pi f = 2\pi(2.5) = 5\pi \approx 15.708 \;\;\checkmark$

Same answer.

Step 5 — Convert to rpm.

Revolutions per minute = $f \times 60$:

$\text{rpm} = 2.5 \times 60 = 150\;\text{rpm}$

For comparison: a typical washing machine spins at 1200–1600 rpm; a Formula 1 engine revs to 15{,}000+ rpm; the Earth rotates at about 0.000694 rpm (24-hour period).

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Answer:

• $f = 2.5\;\text{Hz}$
• $T = 0.4\;\text{s}$
• $\boxed{\;\omega = 5\pi \approx 15.708\;\text{rad/s}\;}$
• Equivalent: $150\;\text{rpm}$

The wheel rotates 2.5 full turns every second, or one full turn every 0.4 seconds, or $5\pi$ radians per second.

Try It

• Adjust the revolutions and time sliders.
• Watch the wheel rotate at the computed angular velocity.
• The HUD shows all four equivalent representations: $\omega$, $f$, $T$, and rpm.
• Try setting time = 1 second — the angular velocity in rad/s equals $2\pi$ × (revolutions).