Source: High school physics (Chinese)
Problem
The relationship between linear velocity $v$, angular velocity $\omega$, and radius $r$ is given by a standard formula.
- If the angular velocity $\omega$ is measured in units of "degrees/second", is the formula $v = \omega r$ still valid?
- Explain why or why not.
[Q1] No.
[Q2] The formula $v=\omega r$ is derived from the arc length formula $s=r\theta$, which is only valid when the angle $\theta$ is in radians. Consequently, for the velocity relationship to hold, the angular velocity $\omega = \Delta\theta/\Delta t$ must be expressed in radians per unit time (e.g., rad/s). The correct formula for $\omega$ in degrees/second is $v = r \omega_{\text{deg/s}} (\frac{\pi}{180})$.
[Q2] The relationship between linear velocity $v$ and angular velocity $\omega$ stems from the definition of arc length, $s$, for a given angle $\theta$ and radius $r$.
The formula for arc length is:
$$s = r\theta$$This fundamental geometric relationship is valid only when the angle $\theta$ is measured in radians. The definition of a radian ($\theta_{\text{rad}} = s/r$) directly leads to this formula.
Linear velocity is the rate of change of arc length, $v = \Delta s / \Delta t$. Angular velocity is the rate of change of angle, $\omega = \Delta \theta / \Delta t$. By taking the rate of change of the arc length equation, we derive the velocity relationship.
Let's assume the angle is in radians ($\theta_{\text{rad}}$) and the angular velocity is in radians per second ($\omega_{\text{rad/s}}$). Starting with the valid arc length formula:
$$s = r\theta_{\text{rad}}$$Over a small time interval $\Delta t$, the change in arc length $\Delta s$ corresponding to a change in angle $\Delta \theta_{\text{rad}}$ is:
$$\Delta s = r (\Delta \theta_{\text{rad}})$$Dividing both sides by $\Delta t$:
$$\frac{\Delta s}{\Delta t} = r \frac{\Delta \theta_{\text{rad}}}{\Delta t}$$Substituting the definitions of linear and angular velocity:
$$v = r \omega_{\text{rad/s}}$$This derivation shows that the formula $v = \omega r$ requires the angular velocity to be in radians per unit time.
If $\omega$ is given in degrees per second ($\omega_{\text{deg/s}}$), we must convert it to radians per second using the conversion factor $\pi \text{ rad} = 180^\circ$.
$$\omega_{\text{rad/s}} = \omega_{\text{deg/s}} \times \frac{\pi}{180}$$Substituting this into the valid velocity formula gives the correct relationship when using degrees:
$$v = r \left( \omega_{\text{deg/s}} \frac{\pi}{180} \right)$$This modified formula is not the same as $v = \omega r$. Therefore, the original formula is invalid for angular velocity in degrees/second.
[Q1] No, the formula $v = \omega r$ is not valid if the angular velocity $\omega$ is measured in units of "degrees/second".