The formula to convert a linear velocity expressed in kilometres per hour [km/h] to an angular velocity expressed in revolutions per minute [rpm] is given by:

$$N_{(rpm)} = \frac {25} { 3 .\pi . r} v_{(km/h)}$$

Where:

• $v$ is the linear velocity expressed in kilometres per hour (km/h)
• $N$ is the angular velocity expressed in revolution per minute (rpm)
• $r$ is the radius expressed in meters (m)

Converting kilometers per hour (km/h) to revolutions per minute (rpm) requires knowing the circumference of the rotating object, such as a wheel or a shaft. Here’s how you can do it step-by-step:

Steps to Convert km/h to rpm:

1. Determine the Circumference:

   • Measure or calculate the circumference $C$ of the rotating object in kilometers.

   • If you know the radius $r$ of the object, you can find the circumference using the formula:

     $$C = 2\pi \times r$$

     Ensure $r$ is in meters before converting to kilometers (divide by 1000).

2. Convert km/h to m/s:

   • First, convert the speed from kilometers per hour to meters per second (m/s) using the conversion factor $\frac{1000}{3600}$:

     $$\text{Speed in m/s} = \frac{\text{Speed in km/h} \times 1000}{3600}$$

3. Calculate RPM:

   • Once you have the speed in meters per second and the circumference in kilometers, use the following formula to calculate rpm:

     $$\text{RPM} = \frac{\text{Speed in m/s} \times 60}{C}$$

     Where $C$ is the circumference of the rotating object in kilometers.

   • Replace $C$ by $2\pi r$ :

     $$N_{rpm} = \frac{1000}{3600} \times \frac{60}{2\pi \times r} \times v = \frac {25} { 3 \times \pi \times r} v_{(km/h)}$$

Example Calculation:

Let's say you have a wheel with a radius of 0.35 meters (which gives a circumference $C \approx 2.199$ meters or $0.002199$ kilometers) and it is moving at a speed of 50 km/h.

1. Convert km/h to m/s:

   $$\text{Speed in m/s} = \frac{50 \times 1000}{3600} = \frac{50000}{3600} \approx 13.89 \text{ m/s}$$

2. Calculate RPM:

   $$\text{RPM} = \frac{13.89 \times 60}{0.002199} \approx 379 \text{ rpm}$$

Therefore, a wheel with a diameter of 0.35 meters (circumference approximately 2.199 meters or 0.002199 kilometers), moving at a speed of 50 km/h, would be rotating at approximately 379 revolutions per minute (rpm).

Here is an online unit converter that convert kilometers per hour [km/h] to revolutions per minute [rpm] and vice-versa.