hey everyone in this video we're going to talk about the tangential and angular descriptions of motion and how exactly they tie into circular and rotational motion this topic can get a little confusing so let's review how everything is related and introduce a few more things first we'll review the difference between circular and rotational motion then we'll talk about the circular motion descriptions the rotational motion descriptions and then converting between the tangential and angular descriptions at the end we'll do a summary of the main points before we start what do we mean by description of motion let's look at this card driving around in a circle the motion is what the car is doing what's happening in the real physical world the description of that motion is how we're choosing to describe measure or study the motion of the car the description of motion includes the axis the units and the positive and negative directions that we choose so that we can describe this Motion in a way that's useful based on the description that we choose we get some variables and equations that we can use to study that motion do some math and solve problems the car doesn't care what description we choose it's going to do the same thing either way the reason we're distinguishing between these two things is because there's more than one way to describe a given motion depending on the situation or the problem we're trying to solve we might choose one description or another and we might need to convert from one description to a different one now let's quickly review the difference between circular and rotational motion we're talking about an object's motion here not the description of motion with circular motion an object follows a circular path centered on a point that is some distance away from the object with rotational motion an object is rotating about its own Center the point that lies on the object itself we could say that the car on the left is translating through two-dimensional space while the record on the right is not translating it's only rotating the car on the left is in circular motion but if we could look at one of the wheels and ignore the car we would see that the wheel is in rotational motion we've seen that a record is in rotational motion while it's being played but imagine a small fly is sitting on the record the fly traces out a circular path so the fly is in circular motion while the record is in rotational motion the Earth orbits the sun once a year and from this point of view the Earth is in circular motion around the Sun at the same time the Earth rotates once a day around its own axis so if we only look at the Earth it's in rotational motion so those are a few examples the terms circular and rotational are only referring to the type of real physical motion now let's talk about the descriptions of motion that can be used for each one we'll start with the circular motion descriptions circular motion typically uses the tangential description of motion that's what we used in the video on circular motion the tangential description means we're focusing on the path of the object around the circumference of a circle and the direction of that motion is always tangent to the circle these are the variables and equations that we use and the units represent length now imagine we draw a line between the center of this circle and the car which follows the car as it moves when we look at it this way we find that circular motion can also use the angular description of motion because the object sweeps out an angle as it moves so even though the car is an object in circular motion it's possible to describe its motion using angles and we can use the units axis variables and equations from the angular description of motion here's the two descriptions side by side the car is doing the same circular motion in each case but using the angular description on the left we'd say the car moved to an angular position of 240 degrees and using the tangential description on the right we'd say the car moved to a position of 80 meters based on that axis you might be wondering why anyone would want to describe the motion of this car using angles even though this car is moving around on its own it's more common for things that travel in a circle to be attached to something that's rotating like the fly sitting on a record also this will be useful when we learn about periodic motion later on here's another example that highlights the difference between these descriptions the yellow car and the red car are in circular motion on two different tracks the yellow car is on a track with a radius of 20 meters and the red car is on a track with a radius of 40 meters if we use a line to follow the motion of the cars we can see that both cars have the same angular motion they both sweep out the same angle in the same amount of time so they have the same angular velocity we could also say they both travel a full Revolution 360 degrees in the same time however the cars have different tangential velocities so the drivers in each car are seeing different speeds on their dashboard the red car is on a track with a bigger radius so its circumference is bigger and the road is longer if both cars travel one revolution in the same amount of time and the red car covers more distance then the red car has a greater tangential velocity on the flip side here's an example where the cars have the same tangential velocity but different angular velocities the driver in the yellow car is driving the same speed as the driver in the red car but since the yellow car is driving around a smaller circumference it drives around its track 360 degrees in less time so it has a higher angular velocity even though the tracks have different lengths or different tangential distances they both cover the same angle of 360 degrees or in SI units 2 pi radians this is why cars want to take the inside path or hug the inside of a curve during a race if the two cars are driving at the same speed the yellow car will complete a lap in less time because it's taking a shorter path it's following the path of a circle with a smaller circumference so those are some of the ways we can describe circular motion now let's move on to rotational motion descriptions rotational motion typically uses the angular description of motion like we did in the rotational motion video with the angular description we're placing an imaginary line on the object and watching how the angle of that line changes over time these are the variables and equations that we use for the angular description of motion and the units represent angles like we mentioned earlier what if a small fly is Sitting on This Record while it's playing the record is in rotational motion but the fly is in circular motion because it's following a circular path from the fly's perspective it's like driving a car around a circular track we're using a small fly as an example because the fly represents a point on the record so we could say that a point on a rotating object can use the tangential description of motion because that point is in circular motion that means we can describe the motion of the fly using the variables equations and units of the tangential description of motion we can measure the distance the Phi travels using a unit of length like meters or centimeters we already knew that circular motion uses the tangential description so why is this important well since the motion of the fly is linked to the motion of the record we can say that a rotating object and a point on that object will both have the same angular descriptions of motion if the record goes through one revolution so does the fly in fact all points on a rotating object have the same angular motion this is why we could place the line anywhere on the record because all points on the record rotate together as an example now we have three flies on this record because the Flies are all on the same rotating object they all have the same angular velocity the same angular velocity as the record itself Phi A and B also have the same tangential velocity as each other but fly C has a slower tangential velocity why is that it's like the example with the cars fly C is traveling along a circular path with a smaller circumference so it's traveling less distance in the same amount of time but fly A and B both travel along a path with the same circumference so they have the same tangential velocity it might seem weird that two points on the same object could be traveling at different speeds their angular speeds are the same but because they follow different circular paths in the same time their tangential speeds are different this is an important point so take a moment to think about it it's like the cars on different tracks that happen to be traveling at the same angular speed they have different tangential speeds and the car on the outside is traveling faster it just seems weird that the flies have different tangential speeds because they're on the same object so we're finding that an object's distance from the center of rotation makes a difference we call the distance between the circumference and the center of a circle the radius and it turns out that we can convert between an object's angular motion and its tangential motion by using the radius with that in mind let's move on to the final topic converting between tangential and angular descriptions let's think about how we might do this using an example this car is in circular motion during a period of time the car travels both in angular displacement Delta Theta and a tangential displacement Delta s if the angular displacement is pi over 2 radians what's the tangential displacement put another way what is the arc length that corresponds to an angle of pi over two radians we can see that the car traveled one quarter or one-fourth of the way around the circle if we did the math we would also find that pi over two radians is one-fourth of two Pi radians the number of radians in a full circle so if the angle is one-fourth of a circle then the Arc Length would also be one-fourth of a circle if we knew the circumference we could find one-fourth of that to get the arc length so to solve this we're going to need the radius of the circle if the radius is 5 meters then the circumference would be 2 times pi times 5 meters which is 10 pi meters usually we can't tell what fraction of a circle we're dealing with so let's calculate that first if we start with the angle Theta which is pi over 2 radians and we multiply that by the relationship of one Circle equals two Pi radians then the radians will cancel out and we find that the angle is one-fourth of a circle now we can take that fraction and multiply it by the circumference but let's continue using unit relationships so we'll multiply one-fourth of a circle times the relationship of one circumference per one Circle we just found the circumference which is 10 pi meters now the unit of circles will cancel out and we get 10 pi over four meters that's about 7.85 meters in decimal form so that's the arc length or tangential displacement Delta s that we're looking for to recap we found what fraction of a circle the angle Theta represents and then we multiply that fraction by the circumference to get the Arc Length because the angle in the Arc Length represent the same fraction of a circle but what if we were given a different angle and radius and we wanted to do the same thing let's simplify the work that we did to find something that applies to any angle in any radius first instead of calculating the fraction of the circle and then multiplying it by the circumference we could have just multiplied the first part by the circumference radians and circles would cancel out and we'd get the same answer next instead of multiplying the angle pi over 2 radians by the relationship one Circle per two Pi radians and then one circumference per Circle we could combine those into one relationship one circumference corresponds to 2 pi radians because they both correspond to one Circle so that's a lot less stuff to write now let's replace the values with variables using this general formula we can plug in an angle in radians a radius in meters and we'll get an arc length in meters if we remove the units we get this simple equation and in fact this is the equation for Arc Length although we usually write it like this what would happen if we plug in one radian for the angle Theta R times 1 is just R so the Arc Length equals the radius that's actually the definition of a radian it's the angle swept out by an arc length that's equal to the radius what if we plug in 2 pi radians for Theta then we get an arc length of R times 2 pi which is the equation for the circumference of a circle so this works for angular and tangential displacements what if we wanted to convert from an angular velocity to a tangential velocity let's say this car is driving around the same circle with a radius of 5 meters and an angular velocity of pi over 2 radians per second what's the car's tangential velocity we could come up with an equation the same way we did before if we take an angular velocity Omega in radians per second and multiply it by the relationship of one circumference two pi r meters which corresponds to an angle of two Pi radians then the 2 pi would cancel out the radians would cancel out and we'd be left with a tangential velocity in meters per second so the radius times the angular velocity gives us the tangential velocity this will be our conversion equation for velocity Let's test it if we plug in 5 meters for the radius and pi over 2 radians per second for the angular velocity the equation tells us the tangential velocity will be 5 pi over 2 in meters per second why does this equation also work with velocity let's think about it this way we know from before that an angular displacement of pi over 2 radians corresponds to a tangential displacement of 5 pi over 2 meters if the car travels those displacements in one second then we're just dividing the displacement by seconds to get the velocities this works for angular and tangential acceleration too now let's look at everything together on the left we have the equations for the tangential description of motion and on the right is the angular description now we have equations that we can use to convert between the two if we take the angular variable and multiply it by the radius of the circular path we get the tangential variable it's important to remember that for this to work the angular variable must use radians so if it uses degrees or revolutions we have to convert to radians radians per second or radians per second squared before plugging it in as long as the tangential variable and the radius use the same length unit things will still work but it might be good practice to only use the SI units to wrap up this section let's look at some examples with a car driving around a circle with a radius of 20 meters the tangential and angular values are linked and the relationship depends on the radius if the car is angular position at a certain time is 2 radians then the tangential position would be 2 times 20 or 40 meters if the car travels a tangential displacement of 60 meters it also travels in angular displacement of three radians in this case we take Delta s and divide it by R to get Delta Theta if the car is angular velocity is one radian per second the tangential velocity would be 20 meters per second if the car speeds up both velocities would increase and if the car slows down both velocities decrease if the car's velocity is constant then both the tangential and angular acceleration are zero if the car speeds up and its tangential acceleration is 10 meters per second squared then its angular acceleration would be 0.5 radians per second squared and if the car slows down the tangential and angular acceleration would both be negative again the tangential and angular motions are linked together if they describe the same object the difference is that tangential describes the motion around the circumference in terms of length and angular describes the motion around a circle in terms of angles so that's everything we're going to cover what should we be expected to know from this video motion and the description of that motion are two separate things the motion is what the object is doing in the physical world and the description is how we choose to study that motion the terms circular and rotational refer to two different types of physical motion circular motion is when an object follows a circular path and rotational motion is when an object rotates about its own Center circular motion typically uses the tangential description of motion however circular motion can also use the angular description of motion because the object sweeps out an angle as it moves on the other hand rotational motion typically uses the angular description of motion but a point on a rotating object can use the tangential description of motion because the point is in circular motion also all points on the same rotating object will have the same angular motion however those points will have different tangential motions if the points are different distances from the center finally if we want to convert between the tangential and angular descriptions of motion we can use these equations which are based on the relationship between an arc length and the angle swept out by that Arc if we multiply the angular value using radians times the radius of the circular path we get the corresponding tangential value thanks for watching and I'll see you in the next video