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Kinematics of Rotation

Explore how rotating objects are described mathematically.

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A Miracle at the Car Maintenance

What dictates the reading of the car speedometer?

License: CC BY-NC 3.0

A car undergoing maintenance. [Figure1]

Remember how you had a series of kinematic equations to analyze linear motion? Well, if you replace them with angular, you can now analyze rotational motion too.

Linear Kinematic Equation

Rotational Kinematic Equivalent

x = xo + vt

\begin{align*}\theta\end{align*}= \begin{align*}\theta\end{align*}o + \begin{align*}\omega\end{align*}t

v = vo + at

\begin{align*}\omega\end{align*}= \begin{align*}\omega\end{align*}o + \begin{align*}\alpha\end{align*}t

x = xo + v0t + (1/2)at2

\begin{align*}\theta\end{align*}= \begin{align*}\theta\end{align*}o + \begin{align*}\omega\end{align*}ot + (1/2)\begin{align*}\alpha\end{align*}t2

v2 = vo2 + 2a(x - xo)

\begin{align*}\omega\end{align*}2 = \begin{align*}\omega\end{align*}o2 + 2\begin{align*}\alpha\end{align*}(\begin{align*}\theta\end{align*}- \begin{align*}\theta\end{align*}o)

However, the rotational kinematic equations, like their linear counterparts, are only applicable under one condition: constant acceleration. This means that  has to be the same throughout.

In cars, the linear speed is often based off the angular speed of the tires. When you take your car to maintenance, you would notice that even when the car is lifted off the surface, the speedometer moves as you step on the gas pedal. The car itself is not moving, but the wheels move, causing the speedometer to show.

Creative Applications

1. Let’s say that your car is lifted off the ground at maintenance and you are trying to see what the speedometer will read when you step on the gas midair. You know the angular acceleration of the vehicle and you are trying to find the angular speed at a certain time. Which kinematic equation would you use?

2. For the last scenario, the angular acceleration of the vehicle is 5 rad/sec2. What is the angular speed of the vehicle by 5 seconds?

3. When wheels don’t slip, there is a relation between linear velocity and angular velocity, given by the following formula: V = r\begin{align*}\omega\end{align*}. R is the radius of the wheel. What is the significance of finding the linear velocity in our scenario?

4. Suppose that your car tire has a radius of 0.6 m. What will your speedometer read when you accelerate the car for 5 seconds? The angular acceleration of your vehicle is 10 rad/sec2.

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  1. [1]^ License: CC BY-NC 3.0

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