According to NASA, we use the same physics in making planes fly to building race cars. And It all boils down to angular acceleration.

    In this guide, we're going to explain angular acceleration and why both NASA and NASCAR think it's so important. If you need to convert units, use this handy calculator to go from rad/s2 to rpm2 and more.


    Why convert angular acceleration?

    Typically, rad/s2 is the international standard for angular acceleration.

    Occasionally you'll see rpm2, which is "revolutions per minute per minute squared". This can be more useful in certain circumstances, like car engines that turn with impeccable speed.

    A lot of students prefer this in general because radians seem counter-intuitive. Degrees are the common unit.

    Radians, however, tend to be much more useful, since they are more logical (mathematically) and directly related to the portions of a circle.

    And having a good conversion calculator to help you with the math reduces your margin of error.

    What is angular acceleration?

    By definition, angular acceleration is the change in velocity during a rotation per unit of time.

    Angular Acceleration Explained

    The best way to understand angular acceleration is to break it down to the basics. It starts with an understanding of position and builds up to concepts like speed and angular velocity.

    1) Position

    If you start small and think about a resting toy car, it stays in one place. There's no speeding up or slowing down.

    The only way it'd move is if it was pushed hard enough. And if that happens, the car's position changes.

    Sure, you could see how big the distance between the two positions is. But, how would you know how fast the toy car was going?

    2) Speed

    Speed isn't just about distance. You could measure how far something rolled, but that doesn't answer how quickly it went.

    It's the factor of time that creates speed.


      Let's say the toy car turns out to have gone 4 feet in 2 seconds. What's the speed?

      4 feet/2 seconds

      = 2 ft/s

      We might be tempted to convert that into mph, but it's good the way it is.

    3) Velocity

    Sometimes it's not enough to know how fast something is going alone. It's important to factor in the direction something is traveling in.

    Turns out velocity is as simple as this. It's speed with the concept of "direction" factored into it.


      Consider a drag car with a parachute. It's definitely going in one direction, but when the parachute releases, there's a force acting upon it in the opposite direction.

      So it's useful not only to know how strong forces are but also where they're pointing to.

    4) Acceleration

    If you think about it, the original speed question is really asking, "At what rate is our position changing?" Velocity is how quickly that particular change happened.

    But, for better or for worse, the world is more complicated than this. If there were only velocity, how would anything start moving in the first place?

    That is, if a car only ever moves at a constant rate (e.g., 70 mph or 2 ft/sec) or not at all, then how is it that NASCARs and airplanes don't go on moving forever?

    When they start up their engines, they don't go full speed right off the bat. It's a slow curve upward.

    This is because there's a rate of change for not only position, but also velocity itself.

    And acceleration is the rate of change of velocity, or, "How fast is something speeding up or slowing down?"

    5) Angular Velocity

    Now, think about when a race car turns. Its entire body is shifting at an angle.

    It was at one angle coming into the turn, and went to another angle coming out of it. The distance is the angle of change.

    There was an average speed at which that all happened, called angular velocity.

    6) Angular Acceleration

    While that race car is turning, it's speed varies throughout the turn.

    Initially, its momentum is big and positive, but as the turn comes to an end the car's acceleration will die down as it's no longer needed.

    This is the essence of angular acceleration. It's the rate of change in angular velocity, or, "How fast is something speeding up or slowing down during a change of rotation?"

    What is the formula for angular acceleration?

    Once we have the rate at which an angle is changing (angular velocity, Δ ω), we simply divide it by the change in time (Δt).

    Don't let the Δ (delta) scare you; it just means "change in."

    So the formula is:

    Angular Acceleration = Δ ω / Δ t


    Angular acceleration = (ω final - ω initial) / (t final - t initial)

    This comes from the idea of obtaining the rate of change in velocity per time unit.

    Equation Example

    A driver lost control of her steering and spun out of control. She turned at a final speed of 2 radians/second within 1.5 seconds.

    To find the angular acceleration, we'd simply plug in the numbers:

    Δ ω / Δ t

    = (2 rad/sec) / (1.5 seconds)

    = 4/3 rad/s2

    The angular acceleration of her rotation was 4/3 radians per second squared.

    How do you convert between different units?

    You'll probably only have to do conversions between different units of time or between rpm2 and rad/s2. Below is a handy conversion table for you.

    The easiest way to do these conversions is with a conversion calculator, but we'll illustrate an example.

    Time Conversions

    The time unit in angular acceleration is always squared since velocity already has time factored in.

    For example, since there are 60 seconds in one minute, here's how you'd go about converting 4/3 rad/s2 to rad/min2:

    4/3 radians / sec2 · (60 seconds / 1 minute)2

    4/3 radians / sec2 · (60 seconds / 1 minute)2

    = 4/3 radians / sec2 · (3600 sec2 / min2)

    = 4800 radians / min2

    Again, a conversion calculator is the best way to learn and check your answers related to angular acceleration. Try ours today!

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