In this video, our topic is rotational kinematics. That is, we’ll be learning how to describe the motion of objects as they rotate. Now, before we talk about rotational motion, let’s recall the type of motion we may be more familiar with, motion in a line or linear motion. If we had some object that started out in this position and then moved along this one-meter-long arrow until it ended up here, we can say this object has moved a linear distance of one meter. And we could even think of this as the basic unit of linear distance.
Now there’s nothing official or formal about a distance of one meter. But if we think about this as the basic unit of linear distance, then that can raise the question, what’s the basic unit of rotational distance? We can understand this by thinking of a line inscribed within a circle that rotates through an angle. This angle, and we can refer to it as 𝜃, is defined by letting the length of our line segment, we can call that length 𝑟, be equal to the arc length of the circle that this line moves across as it rotates. If both of these distances are the same, then we can say that the angle 𝜃 through which our line rotated is equal to one radian or simply one rad for short.
In defining this term, an angle of one radian, we’ve come up with what we can call a basic unit of rotational or angular distance. Just as one meter is a basic unit of linear distance, radians are the units we use to quantify a distance in rotation. And we can also call this an angular distance. Now, what about an angular distance that went all the way around this circle, one complete revolution? How many radians would that be? One complete revolution is represented by two times 𝜋 radians. We can confirm this to ourselves by recalling that the circumference of a circle is equal to two times 𝜋 times the circle’s radius.
If we consider a rotation about a unit circle, that means that 𝑟 is equal to one. And so the circle circumference is equal to two 𝜋, which matches the rotational distance of one complete revolution. Now, because it’s based on the geometry of a rotation, the unit of radians is a convenient one in which to describe rotations. But we may be aware of another unit for describing rotational distances. That’s the unit of degrees symbolized like this. Knowing there are these two different sets of units for describing rotational distances, we’d like to know how to convert from one to the other. That is, given a rotational distance, say in radians, say of one radian, what is that equal to, we wonder, in degrees?
And then, by the same token, if we have a rotational distance given in degrees, what, we would like to know, is that equivalent to in radians? We can start figuring out the conversion from degrees to radians or radians to degrees like this. Recall that one complete revolution around a circle is equal to two 𝜋 radians. And then, we can recall further that one complete revolution around a circle is also equal to 360 degrees. So 360 degrees is equal to two 𝜋 radians. And now we can ask ourselves this question, what would we need to multiply 360 degrees by in order for the result to be equal to two 𝜋 radians? That is, what factor could go here so that this factor times 360 degrees equals two 𝜋 radians?
Well, we can say that one thing that must happen is the 360 degrees needs to be canceled out. So if we put 360 degrees in the denominator here, that will accomplish that. And then, since we want our answer to be two 𝜋 radians, we can put that in our numerator. Now that we have our multiplying factor, let’s test out this equation and see if it’s true. We can see that this 360 degrees will cancel out with this 360 degrees. And what will be left over is two 𝜋 radians, which is what the right-hand side of our equation has. This means that we’ve developed a conversion factor for taking a rotational distance in degrees and converting it into radians.
And notice that we can simplify this factor a little bit. Half of our numerator of two 𝜋 radians is simply 𝜋 radians. And half of our denominator is 180 degrees. So we can also write our factor like this. Now that we’ve discovered this conversion method, we can multiply one degree by this factor. And we see it gives a result of 𝜋 divided by 180 radians. So that’s how we go from degrees to radians. Now, how about in the opposite direction? We can start once more by considering this relationship here.
Now the question we’ll ask though is, what do we need to multiply two 𝜋 radians by in order for that product to equal 360 degrees? Just like before, we’ll try to figure out what factor goes here so that this equation continues to hold true. In order to end up with 360 degrees, we’ll need to somehow cancel out this two-𝜋-radian factor. We can do that by dividing by two 𝜋 radians. And this means we’ll need to have 360 degrees in the numerator of our factor. This, then, is a way to convert from angles in radians to angles in degrees. And just like before, we can simplify this factor a bit by dividing numerator and denominator by two. That gives us 180 degrees divided by 𝜋 radians.
So to figure out how many degrees one radian is equal to, we’ll multiply one radian by 180 degrees divided by 𝜋 radians. And we end up with a result of 180 divided by 𝜋 degrees. Knowing this about angle conversions, we can write a couple of summary equations. First, say that we have some angle, we’ll call it 𝐴, that’s measured in units of degrees. So we’ll have 𝐴 and then the degrees symbol as a subscript. We’ve seen that if we multiply this angle in degrees by the factor of 𝜋 radians divided by 180 degrees, then the result is that same angle 𝐴 but expressed in different units, in units of radians.
Likewise, say that we have some angle, we’ll call it 𝐵, but this one is expressed in units of radians. If we multiply this angle in radians by 180 degrees divided by 𝜋 radians, then we wind up with that same angle, but this time expressed in units of degrees. These then are the conversion factors between these two sets of units. Now, so far, we’ve only talked about rotational distances. But it turns out that just as there are linear speeds and linear accelerations, so there are rotational speeds and rotational accelerations.
Now, we know that if we’re talking about linear speeds, then if we have some speed, we can call it 𝑠, that speed is equal to a change in distance, we can call it Δ𝑑, divided by a change in time, Δ𝑡. So now, if we wanted to write a rotational speed, how would this equation need to change? For one thing, we’re no longer talking about a linear speed 𝑠. So we would need some other variable to represent a rotational speed. This symbol typically used this way is the Greek letter 𝜔. And then, what is 𝜔 equal to? In our equation for linear speed, we have a change in linear distance divided by a change in time.
In order to create a rotational version of this equation, all we need to do is change this linear distance into a rotational one. And we’ve seen that rotational distances are described using the variable 𝜃, where 𝜃 can represent any rotational distance, not just one radian as we’ve defined it in this special case. So instead of using Δ𝑑, in the numerator of our equation for rotational speed, we’ll use Δ𝜃. That represents some change in rotational distance. And then, in our denominator, we’ll once again have Δ𝑡. A change in time is the same regardless of whether we’re talking about linear or rotational motion.
So let’s think for a bit about what this equation for rotational speed means. It’s describing an angle, a rotational distance, that’s changing over some amount of time. We could picture it as though the line segment in our diagram is rotating at some rate around this circle. That rate of rotation is its rotational speed, sometimes also called angular speed. This arm could be turning at a higher or a lower rate. And those will correspond to higher and lower rotational speeds. And then, here’s something interesting. Just as if the linear speed of an object changes in time, then we can say that object is accelerating. Equivalently, we can say that if the rotational speed of some object changes in time, that object has rotational acceleration.
We could picture rotational acceleration as this arm as it turns around the circle, either speeding up or slowing down. If it does either of those things, it has a nonzero rotational acceleration. We know that the symbol typically used for linear acceleration is a lowercase 𝑎. The symbol used to represent rotational acceleration looks like this. It’s the Greek letter 𝛼. Rotational or angular acceleration is equal to a change in rotational speed, Δ𝜔, divided by the amount of time over which that change occurs.
Now, before we get to an example exercise, there’s something about this equation for rotational speed that we’ll want to remember. The standard way to express the rotational distance that some object moves through is by using units of radians as we saw. Sometimes, though, in an example exercise, rotational rates are described in a different unit called rpms, revolutions per minute. Now, a revolution is one complete circuit around a circle. And we saw that in terms of our angle 𝜃, that’s described by rotation of two 𝜋 radians. So one revolution is equal to two 𝜋 radians. And it’s this way of expressing rotation distances that we use in our equation for rotational speed.
If we forget to use radians and instead use units of revolutions which can be easy to do if our rotation rate is given in rpm, then a result for the rotational speed will be off by a factor of two 𝜋. So whenever we use this equation for rotational speed, we’ll want to make sure that our Δ𝜃, our change in angular distance, is expressed in units of radians. That said, let’s now look at an example exercise.
An angular displacement of blank degrees is equal to an angular displacement of 7.25 radians.
In this exercise, we want to fill in that blank. In other words, we want to know how many degrees 7.25 radians is. To convert this angle in radians into some equivalent angle in degrees, we can recall that one complete revolution around a circle is equal to two 𝜋 radians. And it’s also equal to 360 degrees. In this exercise, instead of having two 𝜋 radians, our angle in radians is 7.25. So knowing that this is true, we want to convert this value into degrees.
To see how to do that, we can figure out what factor will go here so that the left-hand side of this equation is equal to 360 degrees as we know it is. That is, we want two 𝜋 radians times whatever it is that goes here to equal 360 degrees. We can see that if we put this factor in parentheses, then the two 𝜋 radians will cancel out and we’ll be left on both sides with 360 degrees. This tells us that we’ve discovered the factor by which we can multiply an angular measure in radians to convert it to degrees. It’s 360 degrees divided by two 𝜋 radians or, equivalently, 180 degrees divided by 𝜋 radians.
So that, then, is what we’ll multiply 7.25 radians by. And when we do, notice that the units of radians cancel out. And we’ll be left with an angle measure in degrees. This is equal to 415.39 dot dot dot degrees. But since our original angle is given to us with three significant figures, we’ll keep that many in our answer. That is, we’ll keep this significant figure, this one, and this one, which means our final answer is 415 degrees. So we can say that an angular displacement of 415 degrees is equal to an angular displacement of 7.25 radians.
Let’s look now at a second example.
A drill bit is initially at rest. When the drill is activated, the drill bit rotates 47.5 times per second. The drill bit reaches this speed in a time of 175 milliseconds. What is the angular acceleration of the drill bit?
Okay, let’s say that this is an end-on view of our drill bit. So we can say that the bit is pointed out of the screen at us. Initially, the bit is at rest. But then, when the drill is turned on, the bit starts to rotate, 47 and a half times every second. Knowing that the drill bit reaches this rotation speed from rest over a time of 175 milliseconds, we want to know what is the angular acceleration of the bit. As we get started, we can recall that angular acceleration, 𝛼, is equal to a change in angular speed, Δ𝜔, divided by a change in time, Δ𝑡.
One important thing to realize about this equation is that this change in angular speed, Δ𝜔, assumes that this angular speed is expressed in units of radians per second. That way, the angular acceleration we calculate is in units of radians per second per second or radians per second squared. Now we bring this up because in our problem statement, we’re told that our drill bit rotates 47 and a half times every second. That is, it goes through one complete revolution 47.5 times every second. But that does not mean that our angular speed is 47.5 radians per second. This is because one complete rotation, one time around the circle we could say — that’s one revolution — is equal to two 𝜋 radians.
So that means the real angular speed in units of radians per second is 47.5 times two 𝜋. It’s this value that we’ll use in our equation for angular acceleration. That acceleration is equal to the change in angular speed, Δ𝜔. But since our drill bit started out at rest, that means that this value here is equal to that change divided by the time over which that change occurs. And that’s 175 milliseconds. Before we calculate this fraction, we want to convert this time from milliseconds into units of seconds. That’s so that we can have a common unit of time in both numerator and denominator.
We can recall that one millisecond is equal to one thousandth of a second. And therefore, 175 milliseconds is equal to 0.175 seconds. Now we’re ready to calculate 𝛼. And we’ll indeed get units of radians per second squared when we do. Rounding our answer to three significant figures, we get a result of 1710 radians per second squared. That’s the angular acceleration of the drill bit.
Let’s now summarize what we’ve learned about rotational kinematics. In this lesson, we saw that just as there are linear distances, speeds, and accelerations, so there are rotational or angular distances, speeds, and accelerations. The symbol typically representing rotational distance is 𝜃. The symbol representing rotational or angular speed is 𝜔. An angular acceleration is symbolized using the Greek letter 𝛼.
We saw further that angular distances can be expressed in units of degrees or in units of radians. And we saw that to convert from one set of units to the other, an angle in degrees can be multiplied by the factor of 𝜋 radians per 180 degrees, while an angle given in radians can be multiplied by 180 degrees divided by 𝜋 radians. These conversion factors are based on the fact that two 𝜋 radians equals 360 degrees. This is a summary of rotational kinematics.