The formula to convert angular velocity expressed in revolutions per minute [rpm] to linear speed expressed in meters per second [ $m.s^{-1}$ ] is given by:

$$v = \frac {2 \pi \times r}{60} \times N$$

Considering the units:

$$v_{(m.s^{-1})} = \frac {2 \pi \times r_{(m)}}{60} \times N_{(rpm)}$$

$$v_{(m.s^{-1})} \approx 0.1047 \times r_{(m)} \times N_{(rpm)}$$

where:

• $v$ is the linear velocity expressed in $m.s^{-1}$
• $N$ i s the angular velocity expressed in $rpm$
• $r$ s the radius expressed in $m$

Here is an online converter that convert rpm to m/s.

To convert revolutions per minute (rpm) to meters per second (m/s), you need to know the circumference of the circle that the rotating object is tracing out. Here's a step-by-step guide on how to do the conversion:

1. Determine the radius (r) of the circle: The radius should be in meters.

2. Calculate the circumference (C) of the circle:

   $$C = 2 \pi r$$

3. Convert rpm to revolutions per second (rps):

   $$\text{rps} = \frac{\text{rpm}}{60}$$

4. Calculate the linear speed in meters per second (m/s):

   $$\text{linear speed (m/s)} = \text{rps} \times C$$

Putting it all together, the formula to convert rpm to m/s is:

$$\text{linear speed (m/s)} = \left( \frac{\text{rpm}}{60} \right) \times (2 \pi r)$$

Example Calculation

Let's take the example of a vehicle with 0.5m diameter wheels turning at 120 rpm.

1. Calculate the circumference:

$$C = 2 \pi \times 0.5 = \pi \approx 3.1416 \, \text{meters}$$

1. Convert rpm to rps:

$$\text{rps} = \frac{120}{60} = 2 \, \text{revolutions per second}$$

2. Calculate the linear speed:

$$\text{linear speed} = 2 \times 3.1416 \approx 6.2832 \, \text{m/s}$$

So, a wheel with a radius of 0.5 meters rotating at 120 rpm has a linear speed of approximately 6.2832 meters per second.