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

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

Considering the units:

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

$$N_{(rpm)} \approx 9.55 \times \frac{v_{(m.s^{-1})}}{r}$$

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 $m$

To convert meters per second (m/s) to revolutions per minute (rpm), you need to know the circumference of the circular path the object is moving along. Here are the steps to convert m/s to rpm:

1. Find the circumference: The circumference $C$ of a circle is given by the formula $C = 2 \pi r$, where $r$ is the radius of the circular path.

2. Convert m/s to meters per minute: Since there are 60 seconds in a minute, multiply the speed in m/s by 60 to get the speed in meters per minute (m/min).

3. Calculate the number of revolutions per minute: Divide the speed in meters per minute by the circumference of the circle to get the number of revolutions per minute.

Let's denote:

• $v$ as the speed in m/s,
• $r$ as the radius of the circle in meters.

The conversion steps are as follows:

1. Calculate the circumference:
   $$C = 2 \pi r$$

2. Convert the speed to meters per minute:
   $$v_{\text{min}} = v \times 60$$

3. Calculate the revolutions per minute:
   $$\text{rpm} = \frac{v_{\text{min}}}{C} = \frac{v \times 60}{2 \pi r}$$

Simplified, the formula to convert m/s to rpm is:
$$\text{rpm} = \frac{v \times 60}{2 \pi r} = \frac{v \times 30}{\pi r}$$

Example Calculation

For example, if a vehicle with wheels of radius 10 cm is travelling at 2 m/s, the rotational speed of the wheels is given by :

$$N_{(rpm)} = \frac{60}{2\pi \times r} \times v_{(m.s^{-1})} = \frac{60}{2\pi \times 0.1} \times 2 \approx 190.99 ~ rpm$$