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How to convert a linear velocity expressed in kilometres per hour [km/h] to an angular velocity expressed in revolutions per minute [rpm] ?

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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:

**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).

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

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

**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 $2\pi r$ :

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

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.

**Convert km/h to m/s**:$\text{Speed in m/s} = \frac{50 \times 1000}{3600} = \frac{50000}{3600} \approx 13.89 \text{ m/s}$

**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.