About

In mechanical engineering and physics, relating the speed of a vehicle or belt (linear velocity) to the rotation speed of its wheels or pulleys (angular velocity) is a fundamental task. This conversion is impossible without defining the radius of rotation, as smaller wheels must spin faster to cover the same distance as larger wheels. Engineers dealing with conveyor belts, vehicle transmissions, and robotics use this calculation to synchronize motors with driven elements. This tool enforces strict unit definitions to prevent common scaling errors, such as mixing centimeters and meters, which can lead to calculations being off by a factor of 100.

Formulas

The relationship between linear velocity (v), angular velocity (ω), and radius (r) is governed by:

v = ω ⋅ r → ω = vr

To convert from Radians per Second to Revolutions Per Minute (RPM):

RPM = ω × 602π

Reference Data

Linear Speed	Radius	Angular (rad/s)	Angular (RPM)
1 m/s	0.1 m	10.0	95.5
1 m/s	0.5 m	2.0	19.1
10 m/s	0.3 m (Car Tire)	33.3	318.3
27.8 m/s (100 km/h)	0.3 m	92.6	884.2
340 m/s (Sound)	1.0 m	340.0	3246.8
5 m/s	0.05 m (Pulley)	100.0	954.9
0.5 m/s	0.2 m	2.5	23.9
15 m/s	0.35 m	42.9	409.2

Frequently Asked Questions

No. This is a kinematic calculation, meaning it describes motion without regard to forces or mass. Weight affects acceleration and momentum (dynamics), but not the direct mathematical relationship between speed and rotation rate.

Angular velocity measures rotation angle per time, while linear velocity measures distance per time. The radius acts as the "lever arm" that translates rotation into distance. A point on the rim of a large wheel travels a longer distance per rotation than a point on a small wheel.

Yes, but you must divide the diameter by 2 before using the standard formula. This tool includes a toggle or simply requires you to input the radius, which is half of the diameter.

Radians per second is the standard SI unit for rotational speed. One revolution is equal to 2π radians (approximately 6.28 radians). It is preferred in physics formulas because it eliminates arbitrary constants.