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What is the formula to convert miles per hour [mph] to radians per second [rad/s] ?

alphonsio

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:

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

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

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

**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:

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