The Four Basic Equations in Kinematics (Rotational Motion)

Why is there always something wrong with my equations in Physics, Mechanics, or Kinematics? — Have you ever thought about this? Then this blog account is for you. It will explain — in detail — everything you are doing wrong. If you want to make sure you’re not doing any silly mistakes, follow me at MechanicallyChallenged on medium.com

The Stuff we already know

By now, you’ve probably already learned how to use the four basic kinematic’s formulas for linear motion. Like this when the acceleration is constant:

The four basic Kinematic equations when acceleration is constant

Or maybe, you’re a little more advanced and calculate for varying acceleration.

Seeing the connection

Dealing with rotational movement may be scary at first because there are all these new symbols. The secret is that we’ve already learned the formulas.

Formulas for constant acceleration

For constant acceleration, we have the equations depicted above. There is some change from linear motion, but the formulas are mostly the same. What’s different?

• Instead of displacement \Delta x , we now have angular stretch \Delta \thetaWhat’s the difference? Displacement \Delta xworks in a straight line, while rotational motion \Delta \thetaworks in an arc.
• Instead of linear velocity v , we now have angular velocity . Angular velocity is the velocity an object has along an arc. The reason we don’t use linear velocity for objects going in circles is that the linear velocity for an object moving in an arc would be constantly changing direction.

A step further (When acceleration varies)

Let’s say we’re given a problem where the acceleration varies. Then the formulas in the last picture won’t work. What can we do? — Well, we already know how to deal with varying acceleration in linear motion, so let’s see if we can make another connection.

We can take the basic kinematics function equations and replace displacement with angular stretch, and linear velocity with angular velocity. This gives us three new equations we can use for rotational motion when the acceleration is not constant.

If you’ve discovered that we are now down on three formulas instead of four, there is a good reason for that. The missing formula is not dependent on acceleration.

Converting between the two systems

What do you do when you have either linear values or rotational values and want to convert them?

The equation for converting between linear and rotational motion

By manipulating the formula above, we can easily change between linear velocity and angular velocity after our desire. If you want a thorough explanation of how this formula is made, the following video is one of many that explains it well:

Relationship between angular velocity and speed (video) | Khan Academy

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