The formula to convert miles per hour [mph] to radians per second [rad/s] is given by:

$$\text{rad/s} = \frac { \text{mph} \times 0.44704 }{r}$$

or

$$\omega = \dfrac{1609.344}{3600} \times \frac{v}{r}$$

With the units:

$$\omega_{(rad/s)}= \dfrac{1609.344}{3600} \times \frac{v_{ (mph) }}{r_{(m)}}$$

where:

• $v_{(mph)}$ is the linear velocity expressed in miles per hour [mph]
• $\omega_{(rad/s)}$ is the angular velocity expressed in radians per second [rad/s]
• $r_{(m)}$ is the radius expressed in meters [m]

To convert miles per hour (mph) to radians per second (rad/s), you can use the following formula:

$$\text{rad/s} = \text{mph} \times \frac{1}{60 \times 60} \times \frac{1609.34}{r}$$

Where:

• $\text{mph}$ is the speed in miles per hour.
• $1609.34$ is the exact conversion factor from miles to meters (since 1 mile = 1609.34 meters).
• $60 \times 60$ is the ratio between hours and seconds
• $r$ is the radius in meters, which is necessary to convert linear velocity to angular velocity.

The steps are as follows:

1. Convert mph to meters per second (m/s):

Convert mph to meters per second by multiplying by $\frac{1609.34}{3600}$.

$$v_{m.s^{-1}} = \frac{1609.34}{3600} \times v_{ (mph) }$$

2. Convert linear speed to angular speed:

Divide by the radius $r$ to convert to angular velocity in radians per second. Angular speed $\omega$ in radians per second is given by $\omega = \frac{v}{r}$, where $v$ is the linear speed in m/s and $r$ is the radius in meters.

$$\omega_{(rad.s^{-1})} = \frac{v_{m.s^{-1}}}{r} = \frac{1609.34}{3600} \times \frac{v_{ (mph) }}{r}$$

Thus, the full formula is:

$$\omega_{(rad/s)}= \dfrac{1609.344}{3600} \times \frac{v_{ (mph) }}{r_{(m)}}$$

So, combining these steps:

$$\text{rad/s} = \frac{\text{mph} \times 0.44704}{r}$$