Okay, class. In this lecture video, we're going to talk about how we can kind of relate the angular kinematics that we're talking about, like radius, angular velocity, angular acceleration, into the linear range. So we can put a lot of this angular displacement and angular velocity into the linear fashion, like linear velocity or linear displacement.
Using a few different equations. So we can first look at the relationship between linear and angular displacement. So angular displacement, again, measures how far something is, you know, kind of like rotated, like the angle we see of rotation.
So think of that as like how far along a baseball swing, while the angle of the bat moves through the angle of displacement, while the distance of the tip the bat covers is the... linear displacement we're actually seeing. So we're applying kind of two different things here.
We can also apply this to linear and angular velocity, and then also with acceleration as well, which we can, you know, we'll talk about a little bit more throughout this lecture. So question of the day, what is common in these upper extremity movements of these sports? So thing we want to talk about is we're trying to achieve either it's spiking a ball or throwing a baseball we want to achieve maximal linear speed so the ball is going in a straight line we want that to be optimized so like the volleyball it's going straight down it's not curving it's not angular well can be that's that's due to drag and air resistance but we're not going to get into that what we really want to talk about is more of how we can translate these rotational aspects of like the arms moving, the joints moving, the ball is moving rotationally until it's released and then it goes in a linear fashion.
So how do these things relate and how can we quantify these? So first we can look at this graph right here where we have linear and angular displacement. So again, when we're talking about linear or angular displacement, we are looking at the angle right here in radians, which is equal to the radius, which is a linear aspect in meters, like how far from the point of axis is the motion occurring. And then also this arc length, which is the length of the length of this segment. So from point from this angle right here, how much distance is traveled across the circumference of this circuit.
a circle. So keeping that in mind, we can say that radians is kind of going to be a value that kind of interconnects these two. So S and R are both in meters. They're both distances in meters that we're looking at here.
So radius R is how far from the center to the outside, and the arc length is how far along this circle are we traveling to get to the next point. Okay, so the equation we use is angle and radians equals arc length over radius. So again, the magnitude we're looking at here is linear distance. So it's going to equal radius times angular distance, which we'll kind of talk about in this next slide.
Oh, no, I don't know. There we go. Okay. So.
So we can look at this example right here. So hitting a baseball at different locations along a bat. So this is kind of going to work on the inner working here.
So it's of like angular arc length versus angle and radians versus radius. So as we increase the radius, we increase the arc length, while the angle can actually stay the same if we keep the angle similar. So starting maybe right here.
So if we think about this as a baseball, I don't know. anyone played baseball here or softball you know you can feel when you're hitting a ball well when you get a lot of contact and it's going to go far versus hitting it not so well and depending on where the ball hits on the bat the location is going to determine how far it's going to go so if you kind of hit towards the base of the bat it's not going to feel too great you're not going to get a whole lot of distance out of that while if you hit it towards the end of it We'll get a little bit more. We'll get into exactly why we see that in the next slide.
But the thing I want to cover right here is that if we keep the angle the same. If we increase radius one to radius two, the arc length is going to get larger. So as we go out from this, we increase this radius. We're going to see the angle stay the same while the arc length also gets bigger. OK, which makes sense because if we look at the equation, we were right here.
We can actually look at the right here. So if we increase D1. So if we go from D1, we increase this, we're going to have to increase R as well.
So D1 to D2 is an increase in distance, which means the radius is also going to increase as well, or vice versa. And the angle is going to stay the same. So we can kind of interwork this equation in multiple different ways.
So since D2 is greater than D1, R2 is going to be greater than R1, and both are going to be interconnected. So how does this apply to different sports? So when we think about throwing or striking motions, depending on the sport, like baseball, tennis, javelin, we really focus on fully extending the joints of the upper limb. If anyone's ever done any baseball or pitching or throwing, you want to have... a really extended long upper limb in order to get maximum amount of speed or velocity out of that.
So we can break this down like the radius of rotation is the distance from the axis of rotation. So if it's your arm, it's your shoulder to, say, the ball in your hand. That's what we're kind of looking at.
So if we fully extend that, we're going to increase that radius. It's going to also increase the arc length as well, while the angle that you're rotating it through probably won't. change much if you're doing a short a pitch with it not fully extended versus fully extended you're still going through the same range of motion it's just you're getting more uh arc length more extension from that arm so and this as we'll look in the next uh slide actually increases the velocity that the ball or whatever it is we're throwing is going to travel at the end of it so increasing radius or increasing arc length is going to increase the velocity that the object is going to travel on the other side.
So mainly the radius is what we're going to focus on here. So we see here, we see a person throwing it. You can see how he's kind of fully extending his arm.
So that takes us into this slide right here, where linear and angular velocity are going to be interrelated. So as we talked about, that arc length equals radius times the angle and radians. We can also kind of rework this to look at the linear fashion. So where r in meters is equal to the velocity in radians per second times the radius that's traveling.
So the key concept here is the magnitude of linear velocity is directly proportional to the radius of and the angular velocity that's going, which makes sense with angular velocity. The more angular velocity, the quicker you're moving. you're going to see a faster release of the ball.
But also, if you increase that radius, we're going to also see an increase in overall linear velocity when you release the ball. Because if you think about it, if you're rotating something, as soon as you let it go, it's going to go straight. Like as we see here, this velocity is going to kind of go straight out when we release.
So at different points, we're going to see different velocities as well. So again, in terms of application, we can think about slinging rotating objects. So if the radius of the sling is larger, the objects at the end of the sling is going to have a greater linear velocity, assuming the angular velocity is going to remain constant.
And conversely, reducing the radius is going to decrease the linear velocity. So it's not going to go as far. We're going to see as much speed after release. So demonstrating this with slings of different radii can prove a clear illustration of the concepts we're talking about here.
So that's perfect. Okay, I have a video. It's not going to probably show.
So this slide features kind of an example pitch or batting motion. I can't really show it. It shows both, but it's kind of showing the relationship between the angular and linear velocity.
So. In movements like pitching and batting, the ultimate goal is to maximize the speed of the ball coming off the bat or off the hand. So whether it's throwing or hitting, you really want it to go as... far as possible. So this is directly tied to the mechanical objective of increasing a ball's linear velocity.
So as the pitcher batter swings, their arm and bat undergo angular motion. So you're rotating through the arm initially bent, you know, extends fully during the release or contact phase, giving it a larger radius. So this extension at the end maximizes the radius of rotation.
And as we know that with the equation we just talked about, V equals R, you know, W where R is the radius, velocity is, angular velocity is W. Increasing the radius while maintaining or increasing even the angular velocity will lead to a higher linear velocity at the end of it. So the speed of the ball is maximized just before release of contact. So we kind of see he's... As you can see in this picture a little bit, you see how he's kind of like bringing the ball in.
He's kind of increasing how much angular velocity he's gaining. And then he extends out, keeping the angular velocity going while also getting a radius bump as well. We also have a velocity of a volleyball strike.
Again, I can't show these videos. But again, like it's similar to the baseball pitch example. The goal of volleyball spike is to hit the ball.
With as much speed as possible. So in this case, you see that the player will start coming up. He's going to rotate through. He's not going to fully extend yet, but right before he reaches contact with the ball, he's going to fully extend the arm, increasing his radius, while also maintaining that angular velocity, which is going to increase performance and optimize how quickly that ball is going to leave his hand when he hits it. Okay.
Another example, you can see javelin throw. When you go through this motion, you see that he's going to try to keep his arm fully extended towards the end. So he's focusing on, you know, rotating as quickly as possible while also, you know, extending at the end to get that radius aspect.
So, okay, let's move on to linear and angular acceleration. So one second. All right, so now we're going to talk about linear and angular acceleration. So tangential acceleration and radial acceleration are the two types of acceleration that we're going to really focus on when we're talking about something rotating around a circle. So we'll talk about tangential first.
This is like the one that kind of goes similar to like linear or like linear velocity coming off of a like spinning. like something that's spinning, like when you throw something, we see that as we're rotating around, we see a acceleration force that's kind of in line with velocity going forward. You know, we have this kind of like going, like wanting, a force wanting to kind of keep an object moving forward. But as we know, we're rotating, it's maybe attached to something, attached to a string, or you're rotating through a ball. You know, we see that it wants to keep going forward.
acceleration is directed away from the path of the rotating body. So it's acting along the tangent to the rotational path at any point. So this tangential acceleration is the result of a change in magnitude of angular velocity over a period of time.
Don't need to get into that quite yet, but just, you know, know that tangential acceleration is the force kind of going out as we're rotating around. And depending on where it is, so say it's down here in the arc, it's going to be going straight out like this. So depending on where we are in the arc, it's going to still keep going. It's going to be a tangent to wherever it is, like wherever we are rotating around.
The other thing I want to talk about is centripetal force or radial acceleration. So radial acceleration acts towards the center of whatever is rotating. It's responsible for keeping the rotating body on its circular path.
The cause of this arises from the body's angular velocity as it rotates. So even if the speed of the rotation doesn't change, radial acceleration is still present to keep the object moving in a circular path rather than flying off tangentially. So the radial acceleration always kind of points inward along the radius of the object's motion. So...
We can kind of move into how this relates in different... equations. So here are the first two we just talked about.
This is arc length, which is almost like angular displacement. Angular displacement is just going to be the angle in radians. Typically we're going to use radians now when we're doing any of these calculations.
Arc length is the distance traveled, like the, like the, you can consider almost like a linear distance because it's just like the distance around the outside of a circle. And then we can kind of relate that down into velocity, which is radius, which is how far away from the center of rotation, times angular velocity. And again, note here that we have radius in both of these equations.
So again, radius really influences a lot of these equations that we're getting when we're transferring linear and angular mechanics. Again, when we're looking at the acceleration component, let's start with a tangential. So tangential is the one that wants to keep going out.
The equation is radius times angular acceleration. Radius of rotation times the angular acceleration. And we can calculate that in a couple different ways. So this is one way of calculating tangential. But we can also calculate it right here with...
velocity final minus velocity initial over time. So this is looking at it as almost like a linear fashion. We can use both depending on what we're given.
So knowing that the tangential acceleration indicates, really it's just indicating how fast the linear velocity of an object is changing. So when we apply a force to this object to increase or decrease the angular speed of rotation of the object, we are also affecting its tangential acceleration. So for example, when you swing a bat or a golf club, the tangential acceleration is what causes the club head or the bat to speed up as it follows its circular path. So it's why we get a lot more speed when we go through this circular path. We can now kind of move on to...
radial acceleration so which is another important concept in rotational motion so rotate radial acceleration as we just talked about is the concept of the acceleration that is directed towards the center of a circular path so tangential is kind of away from the circular path kind of perpendicular to it while the radial acceleration is what's kind of moving it inwards so it's also referred to as centripetal acceleration you guys have heard of this especially if you go on a ride that spins you around You're feeling like almost an urge to fly for its centripetal force that you're kind of feeling. So the force that is keeping an object from moving in a curved path rather than flying out in the straight line following the tangential line. So radial acceleration is responsible for constantly changing the direction of an object's motion as it rotates.
Even though the object might not... be speeding up or slowing down in terms of linear speed the change in direction because it's a vector linear velocity is a vector we actually see a acceleration acting on it so for example when you spin a ball on a string the string applies the centripetal force pulling the ball inward that's why you feel tension like it wants to keep going outward the force kind of coming inwards which you're kind of causing you're creating an accelerator centripetal force to keep it from flying out. If that force were to disappear, the ball would just go flying off in a straight line following the tangential direction, because that's the kind of direction the acceleration that it's going to follow through with. And this is how we can solve for this.
You can solve for this in two different ways. The first way I want to talk about is this one. So radial acceleration equals velocity, linear velocity. So the velocity would just go if it were to keep going straight at that very uh that kind of time point squared over the overarching radius again radius is really important if you increase the radius you're going to actually decrease the angular uh the radial acceleration and you kind of notice that when you're like say you're spinning something on a string you're letting it spin around your finger as it gets close you're losing slack the ball gets closer and closer to your finger that you're spinning it around it's going to get faster and faster. So you're decreasing that radius, you're actually increasing that velocity, the linear velocity.
But sometimes this won't work. Sometimes we're not going to give you the linear velocity it has at that very moment. So what we might give you is the angular velocity.
And so using some algebra and using what we know about linear velocity, so linear velocity equals radius times angular velocity. We know that we just can square the radial velocity and the angular velocity. We can cross out these two radiuses and that leaves us with radius equals. angular velocity squared. Okay.
So depending on what we give you, you can use either of these two things. Okay. Again, velocity is that tangential velocity that we were talking about. And radius is the radius of rotation that we know is just how far away is the thing rotating from the center of axis.
So again, that it's a direction like the rotational force, rotational acceleration going towards the center. And when you apply acceleration to a object with mass, it turns into a force. Okay. Last concept we want to talk about with this is resultant acceleration. So this refers to the overall linear acceleration of a rotating body.
It is important to remember that the linear acceleration of such a body is not constant because it consists of two components. It's a tangential acceleration and radial acceleration. So these two types of acceleration combine to produce a resultant acceleration, which can vary both in magnitude and the direction it's traveling. So while tangential acceleration affects the speed along the path of motion, the radial acceleration is responsible for changing the direction of the motion that it's following, so creating this curved trajectory. So when we combine these, we can actually understand how they relate.
So we can get this resultant force because there's two accelerations acting on an object that's rotating. So when we combine those two together, we get a resultant acceleration. So we can look at this equation when we're trying to relate the resultant acceleration into the tangential and the radial.
And if this looks familiar to you guys, it should. It's pretty similar to a squared plus b squared equals c squared, where c squared is the resultant aspect. So what's happening here is we're seeing we create this kind of right triangle here. Hard to kind of do, but we can rate this right triangle between these where this new a is the new kind of hypotenuse that we're seeing. Just note that the direction it's traveling is going to change, but C squared equals B squared, A squared.
It doesn't matter. These can be flip-flopped. But again, what we're seeing here is that this resultant acceleration is going to be kind of like the net acceleration we're seeing if we consider the tangential and the radial. This is where we're going to see the net go.
And it makes sense because it's going to keep kind of rotating this object around the circle. And so depending on where it is, we're going to see bigger changes as we go around the circle. So this thing is kind of always changing.
So this linear acceleration of a rotating body is like never constant because it's constantly rotating around. So when we look at this overarching like resultant acceleration, which is a linear aspect, it's going to keep changing depending on where we are in the circle. So.
Keep that in mind as we go forward. So here's an example question that you might see. A golf club is swung at an average angular acceleration of 10.5 radians per second squared. Again, note radians. That's what we're going to be using for a lot of this.
When we're looking at degrees, it's going to be in radians. The angle is going to typically be in radians. So what is the linear velocity of the club head? as it strikes the ball at the end of a 0.8 second swing.
The radius of this is going to be 1.5 meters. Okay, so what is the known? Well, we know how much time passed. So that's going to be, change in time is going to be 0.8 seconds.
We know the acceleration is going to be 10.5 radians per second squared. So what do we need to try to calculate? So the unknown is the angular acceleration and the tangential velocity, so the linear velocity that we get at.
the, you know, point at the end of this or whatever point we're looking at, which is probably the end of the swing. So again, how can we do this, the formula is going to be, why is it not loading? Okay, there we go.
The acceleration is going to be equal to change in time over change of angular displacement. So we're looking at angular velocity, angular velocity is going to equal change of angular angular velocity. We're looking at angular acceleration.
Angular acceleration is going to equal change of angular velocity, not top, over change of time. And then we can kind of rework this into velocity equals radius times angular velocity. Okay. So let's try and use both these equations in order to find the...
new linear velocity at the end of this. So first let's calculate for what the change or what that final velocity is going to be of the angular velocity. So let's see.
We have this equation right here where we know what the angular velocity is. We know that's going to be 10.5 radians per second squared. We know the time.
It's going to be 0.5. 8 seconds. And when we multiply those together, again, that's just, you know, multiplying.
We're just doing some algebra, multiplying this up here to here. That's what this equation is down here. And what we're given is 8.4 radians per second.
That's the final. If we're considering the initial to be zero, the change is going to be equal to the final. So now that we have this final angular velocity, We can plug this into this other equation over here that we know to get linear velocity, radius times angular velocity.
So at this final velocity, what is going to be the linear velocity? And we just plug in what we know. We know the radius is 1.5 meters.
We know the angular velocity final, the one we just calculated, is going to be 8.4, which is going to give us a velocity in the linear aspect, linear velocity. tangential velocity that we're calculating of 12.6 meters per second so you see how what we did here is we took the angular aspect we were given we found an angular aspect that we can like use in order to calculate linear aspects and we just kind of worked down we had to do two parts here okay uh here's another example we can look at so the throwing phase of a baseball pitch is uh 0.15 seconds. Shoulder flexion is at 135 degrees and elbow extension is at 95 degrees. If the arm length is 0.42 meters and the forearm length is 0.44 meters, what is the baseball's velocity at the time of release?
Okay, so this might look pretty daunting, especially with all the angles here, but we can work this. out where we're going to try to find the linear velocity. So that's what they're looking for was the baseball's velocity was that tangential velocity right at point of release.
And how are we going to solve that? Well, we know some different angles here, we have the angle of the shoulder and we have the angle of the elbow. And both of these are going to kind of look at, you know, we're going to use these in order to figure out what's uh how much radius we're seeing um we also see that the radius is of the shoulder is 0.42 meters raise the elbow is 0.44 meters and we know the time or change of time is going to be 0.15 so what equations can we use here so we have the angular velocity we know that angular velocity is equal to change in angle or the angle that we're looking at times or over time.
And then we also know that velocity is equal to radius times angular velocity. So again, we know a couple of these things right here. So now we can kind of start trying to plug these different aspects here.
So we know that the change of the shoulder joint is going to be 135 degrees. So 135 degrees, that's how much it's going to... change.
That's the angle it's going through when we see it flexing. Okay. When we're flexing, we're kind of bringing it inward.
So when we're seeing this flexion occur, we're seeing it's moving through 135 degrees in 0.15 seconds. It's not looking at the angle it is at that very time. It's actually looking at the change.
That range of motion is the change in angle we're seeing. And that's going to equal 15.71 radians per second when we try to calculate out using the time and the change in angle. And we have to convert that change in angle in degrees to radians.
Going to equal 15.71. We also have to calculate the elbow. There's two things rotating here. We have the shoulder rotating and the elbow is rotating.
Okay, and we're seeing the elbow extend out. So we see 95 degrees of extension occur in the elbow by the time we see from like start to the end of the pitch. So we're seeing 95 degrees occur in 0.15 seconds. We're rotating that.
in 0.15 seconds. So we have to divide that in order to figure out the angular velocity, which is going to be 633 degrees per second. When we convert that to radians, which we always do, it's going to be 11.05 radians per second.
Okay. Now what we need to do is add up the velocity we would see at the elbow and add the velocity we'd see at the, I guess we're calling this the wrist, I guess. So we're looking at, you know, two different aspects of an arm.
So we have the shoulder here, we see the elbow here, and we see the hand here. So we're seeing a rotation right here. That's what we're calculating, this rotation here, and we're seeing a rotation here. So it's like there's two different joints rotating at the same time. So we have to calculate both of these.
Okay, so if we basically find the velocity of the elbow, velocity of the wrist, and we do that by calculating the radius, which we're given. So the radius of the arm length is going to be 0.42. So this length right here is 0.42.
This length right here is 0.44. Multiply by the angle we see it rotating through, the velocity, the angular velocity. we can calculate the linear velocity of both these points.
Okay, so we have 6.6 meters per second and 4.86 meters per second for the wrist. So the elbow is rotating at a speed, it has a linear speed of 6.6, and the wrist has a speed of 4.86. At the time at the end, we just have to add those two together in order to get the total net. velocity of the baseball. So when you're given something with a multi-joint or multiple things rotating at the same time, you have to calculate those individually and then combine them together in order to get the net velocity we're going to see at the end of it.
Okay, hopefully that wasn't too confusing. So these concepts we were just talking about in the example questions that we were just going over kind of show how we can... mechanically analyze athletes you know using these values and these concepts so uh two things we want to talk about here qualitative versus quantitative qualitative is more of an objective uh like assessment which is not say it's wrong like to objectively assess something so it's like if you say i feel better like hitting it like this or you know i am seeing that i'm you you know, better form or like it looks more smooth. These are qualitative measurements. You're kind of objectifying or you're not, you're putting words to these values and like using those to assess performance, which is great.
But when we get into the pros, we want more, you know, rigid things. We want numbers. We want quantitative analysis.
So we can look at, you know, a professional player's maybe pitch and say, okay, we're not getting as much. angular velocity, we're not seeing as much extension, which would lead to more angular velocity during this section of the pitch. And maybe you cue that and you start trying to train that in order to try to get better performance. And again, we can use quantitative data in order to do that. So this quantitative data that we're looking at right here, these angular velocities, these changes of angle measurements are going to come into play.
you know when you're maybe assessing like an elite athlete most elite athletes have trainers that use some of this data and like manipulate in order to figure out where maybe we can see a tweak occur or how we can tweak someone's performance. Okay, let's move on to here. I'm going to pause real quick so I can switch to my iPad.
All right, so for some reason, my iPad is not going to work. So we're going to just try our best. I have like the answers kind of like written out.
So I'll just talk it through instead of actually drawing it out, unfortunately. I'll draw out more of these practice problems in class on Monday. So these are just problems you might see on an exam.
So like 235 degrees is equal to how many radians? So this is basically, you probably won't see this on the exam, but how do we calculate this? I'm going to give you, if you guys want to pause and see if you can try to figure it out, you can. But I'm going to move to this. So we know that one radian equals 57.3 degrees.
Okay. So if we have 235 degrees, we just have to multiply by one radian equals 57.3 degrees. So that's the conversion here. If we divide this by 57.3, we can use that as something to convert different things into radians. So how we can do this, we just multiply.
divide by 57. These degrees will cross out and you will be left with 4.1 radians. Okay. So again, make sure you understand how to calculate this, how to kind of work this.
So whatever the answer is in degrees, just multiply by one over 57.3 and you should get the correct answer in radians. Again, you're going to definitely need to be able to convert. I don't think we have that in your equation sheet.
So. Something you have to memorize. If the absolute angle of the upper arm is relative to the right horizontal is 100 degrees and the absolute angle of the forearm is 145 degrees, what is the relative angle of the elbow? So we need to draw this out first of all. So we have basically two absolute angles.
So those are segment angles. And we're looking for the joint angle or the relative angle of the elbow. So we can draw it out like this, where we have 100 degrees up here for this one, 145 degrees for this bottom one.
And what we want is this angle right here in the middle. Whoa, it's not coming out where I want it to. So we can call that, I don't know, just a theta. Okay, so how do we do this?
Well, we want to create this imaginary, you know, a continuation line, this beta value, this beta angle right here. And to calculate that beta angle, we know is degree of the bottom one, 145 degrees, subtracted by the degree of the top one, 100 degrees. And that is going to leave us with 45 degrees.
That's not your answer. That's just the angle of this beta right here. But we are looking for this one right here.
So what we need to do is subtract 180 degrees by 45. Again, because this whole thing is 180. We know this B portion of it. The remaining part is the angle we're looking for, or the alpha angle, or the theta, whatever it is, whatever you label it. And that's going to be 135 degrees in this case. 180 minus 45 is 135. Okay, next practice problem. If the thigh segment has a coordinate endpoint of 2, 7, and 5, 3, what is the segment's absolute angle relative to the right horizontal?
Okay, so what we're given is a segment coordinate. We have 2, 7, 5, 3. So we can draw that out. And what we're looking for is this angle right here.
So how can we calculate that? Well, there's a couple of ways. There's not just one.
One way we can do it is by looking for this angle right here, making a triangle all the way through to try to find this angle right here. And the way we can do that is kind of have it extend down. So the angle you see up here is going to be the same angle down here, okay, relative to the right horizontal.
So what we can do, because we don't know this middle point, what we do know is this bottom point right here. So what we can do is create this imaginary dot right here and find the lengths of each of the sides. We kind of did shorthand of this.
We kind of like manipulated this equation already. So we know that this side length is going to be X2 minus X1. This side length is going to be Y.
2 minus y1. And if we know that this is the opposite, this is the adjacent, we can put this into this equation right here. So opposite tangent, because again, we're looking for the theta, we have to get rid of this tangent, put it on this side, and that's just using opposite function.
You have to take the tangent function of 7 minus 3 over 2 minus 5. And now those are going to give you these lengths. Like we just shortened this down. So we know that this side is going to be four and we know that this side is going to be, you know, three.
So we just kind of did a shorthand of this. And what we're seeing here is a degree of 53.1, negative 53.1, which again, we can basically calculate that into 180. So subtract 180 by this 53.1, and you'll be able to get 126.9, because we know this full thing, this whole, why is it not, there we go, kind of, there we go. This whole thing is 180. If we calculate this 53.1 as this angle right here, I can't draw that out.
This angle right here is 53.1. We know that the remaining side is what we're looking for, and that is going to equal 126.9. Okay, so again, that's kind of manipulating in order to find absolute angle here.
Okay, last one we're going to cover today is what is the angular velocity in a joint which has an initial angle of 35 and a final angle of 175 degrees, changing angles over 0.3 seconds. Okay, so what are we given? We're given the initial angle and the final angle. So we can calculate the change in angle.
And if we want to find the angular velocity, we just have to divide that by the overarching time, which is 0.3 seconds. That is going to give us 175 minus 35 over 0.3 seconds, which is 466.7 degrees change per second. And we can convert that to radians by dividing, or...
basically multiplying by 1 over 57.3. Okay? That gives us 8.14 radians per second.
And if they gave us the radius of the segment, we can calculate that into linear velocity. We know linear velocity equals to angular velocity times radius. But they didn't give us that. They just wanted the... This step right here.
Oh my God, I can't draw an R with this thing. There we go. We just wanted this angular velocity here.
But I know there's a lot of information to cover. There's a lot of conceptually difficult things to understand. We're going to cover this. We're going to work on this throughout this section of the class, this exam.
We're going to cover this a bit in lab this week. So again, I recommend if you're confused on any of this, make sure you come to Monday's lecture. We're going to go over this. practice a little bit more of some practice problems.