The angular displacement of the wheel is measured by calculating the angle that it makes with the ground. There are many different ways to calculate this, but one of them is as follows: Angular Displacementsin (theta) radius

So what should you do? Well, for starters, we need to find out what time interval we are looking at. So let’s take a look at our data and see if there is any information in there about when these measurements were taken.Let’s say that the data is taken from a five-second interval. The angular displacement at t 15 s would be: sin (theta) radius 0.0074 radians, and there are 360 degrees in a circle so this works out to about 14° or 30′ depending on what unit of measurement you use for angles.

At t 12s, it would be sin (theta) radius0.0135 radians; which means an angle of around 45° or 60′.Different intervals will have different values, but they should all increase as time goes by since the wheel rotates more quickly when it gets closer to its apex than if it were near ground level.

Angular velocity is a scalar quantity that has a magnitude equal to the angular displacement and a direction given by the right-hand rule. Angular velocity is measured in radians per second, or revolutions per minute (rpm).

The instantaneous rate of change of an angle quantity with respect to time is called its angular acceleration; it too has a magnitude (sometimes known as “rate”) that can be calculated from these two scalars and also a direction determined by applying the right-hand rule both ways.

We’ve learned how to calculate linear velocities but what about rotational ones? Well, when dealing with rotating bodies such as wheels on axles we use what’s known as tangential rotation instead of linear velocity because there are no forces being applied perpendicular to the axis only the torque of the axle.

#### So what is the angular displacement of a wheel between t 15 s and t 30 s? Is it 0, or 180 degrees?

The answer is neither! The linear velocity would be zero because there’s no motion to speak of (since we’re working with tangent rotation). And if you were to measure the angle at that point in time using protractors, protractor corners or trigonometry then your answer would also most likely be “0” radians. But this number isn’t particularly helpful when calculating an average rate of change for instance – since it requires both sides to have some sort of magnitude. A more useful quantity here is actually 45° which can be calculated by multiplying the starting angle by 15.What about between t 30 s and 45 s?

In this case, it’s 60°. This is because in this timeframe (t 30s to 45s) we have both a linear velocity of 20 meters per second and an angular displacement of “30” degrees which are multiplied together when calculating the average rate of change: So what would be different if we only had 25 seconds instead? Well then the equation for our average rate of change would be: or simply 0 m/sec² since there’s no motion at all!

For example purposes, I’m going to use some hypothetical values below, but don’t forget that these come with errors from not being the exact values. What is the Angular Displacement of a Wheel Between 0 Seconds and 15 Seconds?

For these examples, my linear velocity would be 20 meters per second and I’m going to show what our angular displacement is for both periods of time t 0s (or “0”) to 15 seconds (“15”). When we take this equation of average rate of change: , or simply 0 m/sec² since there’s no motion at all! What is the Angular Displacement of a Wheel Between 30 Seconds and 45 Seconds?

This period has an angular displacement that’s about 18°. Notice how it dips below zero before reaching its peak position. This makes sense because in this timeframe from t30s to t45s) we have a linear velocity still going in the same direction, but it is slowing down. What is the Angular Displacement of a Wheel Between 60 Seconds and 75 Seconds?

Notice how it dips below zero before reaching its peak position. This makes sense because in this timeframe from t=60s to t75s) we have a linear velocity still going in the opposite direction, but it is speeding up. What is the Angular Displacement of a Wheel Between 90 Seconds and 105 Seconds? At last! Now our linear velocity doesn’t change throughout these 45 seconds (t 90105). We can see what happens when there isn’t any variation.