Analogy between linear and angular movement

Uniform circular movement refers to a body moving in a circular path without angular acceleration (circular activity is impossible without centripetal acceleration). Angular acceleration is the rate of readjust of angular velocity, and also angular velocity is the price of change of angular displacement. In short, any angular amount is the very same as its straight quantity, except it defines the angle between the axis the rotation and also the position of object, rather than the distance-based quantities.

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Analogy in between linear and angular motion

As defined before, rotational motion have the right to be interpreted by analogy to straight motion. This section will list the analogs and then describe why the analogy works.

For accelerated linear motion, we had actually 3 quantities: velocity, acceleration and also displacement. This angular analogs and their signs are noted below:

Linear/Planar quantity Angular/Rotational quantity
Displacement (\(\overrightarrows \)) Angular displacement (\( \overrightarrow\theta \))
Velocity (\( \overrightarrowv \)) Angular velocity (\( \overrightarrow\omega \))
Acceleration (\( \overrightarrowa \)) Angular acceleration (\( \overrightarrow\alpha \))

Recall our three kinematic equations. For rotational motion, the exact same equation apply. The only difference is that we substitute in the angular analog of the matching quantities. The equations are displayed below:

Linear/Planar equation Angular/Rotational equation
\(\overrightarrows = \overrightarrowu t + \tfrac12 \overrightarrowa t^2 \) \(\overrightarrow\theta = \overrightarrow\omega t + \tfrac12 \overrightarrow\alpha t^2 \)
\(\overrightarrowv = \overrightarrowu + \overrightarrowat \) \(\overrightarrow\omega = \overrightarrow\omega_o + \overrightarrow\alphat \)
\( v^2 - u^2 = 2as \) \(\omega^2 - \omega_0^2 = 2\alpha \theta \)

Why go this analogy work?

It could seem strange that we have the right to swap the end the worths this way, yet it is actually quite reasonable to do so. Once we devised this equations, us didn"t clearly use any property the vectors. So, this equations just define the relation in between the size of these numbers, i.e. Lock are just relations in between quantities changing a specific way. However, note that we have used the arrowhead denoting a vector in the equations top top both sides. This means that this equations hold for vector quantities as well.

This is due to the fact that the angular amounts too kind a vector similar to direct motion. This is actually a pseudovector and also is the result of taking the cross-product(coming soon) that the radius that rotation and the angular quantity. Effectively, we can now law rotational activity the very same as linear motion.

However, the is necessary to keep in mind that us cannot add/subtract vectors and also pseudovectors. We first have to settle the difficulty as a difficulty of straight motion in ~ a tiny scale, and then calculation the pseudovector. Pseudovectors are past the scope of this book, however we have the right to utilize them for our problem-solving if we keep this allude in mind.

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