The formula to convert kilometers per hour to radians per second is given by:

$$\omega_{(rad.s^{-1})} = \dfrac { \text{v}_{ (km.h^{-1}) }}{3.6 \times r }$$

Where :

• $\omega_{(rad.s^{-1})}$ is the angular velocity expressed in radians per seconde (rad/s)
• $\text{v}_{ (km.h^{-1}) }$ the linear velocity expressed in kilometers per hour (km/h)
• $r$ is the radius expressed in meters (m)

The following explains how to convert speed from kilometers per hour ( $\text{km/h}$ ) to radians per second ( $\text{rad/s}$ ) :

Formula:

$$\omega \, (\text{rad/s}) = \frac{\text{v} \, (\text{km/h}) \times 1000}{3600 \times r}$$

Where:

• $\text{v}$ is the linear speed in $\text{km/h}$,
• $r$ is the radius of the circular path in meters.

Explanation:

1. Convert $\text{km/h}$ to $\text{m/s}$ :
   Since $1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} = 5/18 \, \text{m/s}$ , multiply the speed by $\frac{5}{18}$ .

2. Find angular speed ( $\text{rad/s}$ ):
   The relationship between linear velocity ( $v$ ) and angular velocity ( $\omega$ ) is:
   $$\omega = \frac{v}{r}$$
   where $r$ is the radius of the circular motion in meters.

Example:

Suppose a vehicle is moving at $72 \, \text{km/h}$ and the radius of its circular path is $10 \, \text{m}$.

1. Convert $72 \, \text{km/h}$ to $\text{m/s}$:
   $$v = 72 \times \frac{5}{18} = 20 \, \text{m/s}$$

2. Calculate $\omega$:
   $$\omega = \frac{v}{r} = \frac{20}{10} = 2 \, \text{rad/s}$$

Shortcut:

To directly convert $\text{km/h}$ to $\text{rad/s}$, use:
$$\omega (\text{rad/s}) = \frac{\text{km/h} \times 5}{18 \times r}$$

This approach keeps the conversion steps compact.