v–w Calculator (Linear & Angular Velocity)
Use this tool to calculate v (linear velocity), w/ω (angular velocity), or r (radius) from the relationship:
where v is in m/s, ω (w) is in rad/s, and r is in meters.
Tip: Fill any two values and choose the third value to compute.
What is a v w calculator?
A v w calculator helps you solve a core circular-motion equation: v = ωr. In many textbooks and engineering notes, angular velocity is shown as the Greek letter omega (ω), but it is often typed as w. This tool treats w and ω as the same value.
The equation links how fast something rotates to how fast a point on it moves in a straight-line sense. If you know any two variables, you can solve for the third:
- v = ωr (find linear speed)
- ω = v / r (find angular speed)
- r = v / ω (find radius)
When to use this calculator
This calculator is useful in physics, robotics, mechanical design, and everyday problem-solving. Common use cases include:
- Estimating tire edge speed from wheel rotation
- Calculating turntable or fan blade tip speed
- Checking rotating shaft specifications in machines
- Solving homework and lab questions in rotational kinematics
Units and conversions that matter
1) Linear velocity (v)
Linear velocity is measured in distance per time, usually m/s. It tells you how quickly a point travels along its path.
2) Angular velocity (ω or w)
Angular velocity is measured in rad/s (radians per second). If your data is in RPM, convert first:
- ω (rad/s) = RPM × 2π / 60
3) Radius (r)
Radius is the distance from the center of rotation to the point of interest. Use meters for consistent SI calculations.
Step-by-step examples
Example A: Find v
Suppose a rotating arm has ω = 8 rad/s and r = 0.5 m.
v = ωr = 8 × 0.5 = 4 m/s.
Example B: Find w (ω)
A wheel edge moves at v = 15 m/s with radius r = 0.3 m.
ω = v/r = 15 ÷ 0.3 = 50 rad/s.
Example C: Find r
If v = 20 m/s and ω = 10 rad/s:
r = v/ω = 20 ÷ 10 = 2 m.
Common mistakes to avoid
- Mixing RPM and rad/s without conversion
- Using diameter instead of radius
- Entering only one value and expecting a unique result
- Ignoring zero-division cases when solving for w or r
Practical interpretation
The formula shows a simple but powerful idea: for the same angular velocity, points farther from the center move faster linearly. That is why the edge of a spinning disk travels faster than points near the hub.
Conversely, if you need the same edge speed with a smaller radius, you must spin faster (higher ω). This tradeoff appears in motor sizing, wheel design, centrifuges, and conveyor systems.
Quick FAQ
Is w the same as omega?
Yes. In plain text, people often use “w” to represent the symbol ω.
Can I use centimeters instead of meters?
Yes, but keep units consistent. If v is in m/s, convert radius to meters first.
Does this work for non-circular motion?
This exact relation applies to rotational/circular motion around a fixed center. For general curved paths, additional kinematic relationships may be needed.
Final thoughts
A good v w calculator saves time and reduces unit errors. Whether you are doing physics homework, tuning a machine, or validating a design estimate, the equation v = ωr is one of the most useful rotational formulas you can keep handy.