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Radians In order to further investigate circular motion, we need to establish a different means of representing an angle. Up until now, all angles have been measured in degrees. This is the most recognizable unit of measure. However, for the purposes of handling physics equations not yet discussed, we need a different way of expressing an angle. Consider the picture below.  The radius of the circle is drawn at two positions. The distance between the two radii on the curve of the circle, often referred to as the arc length, is labeled as d, and the angle between them is labeled as theta. There is an interesting relationship between the radius r, d, and theta. It turns out that the distance or arc length divided by the radius equals the angle, but it is not an angle in degrees.  This angle represents what is called radians. A radian is the ratio between the arc length and the radius of a circle. In fact, when the angle is equal to one radian, the arc length is exactly equal to the radius of the circle because the ratio has to equal one. As a result of this fact, one radian equals approximately 57.3 degrees. Carrying this relationship further, it turns out that 360 degrees equals two times pi radians, where pi is 3.141592654... and so on forever. This relationship can be written as,  Which can be reduced for a more efficent conversion rate as,  So, this means that the arc length of a half circle divided by the radius of that circle equals pi.  This relationship is actually illustrated in the equation for the circumference of a circle.  If you divide that equation out by r, you get circumference, which is the arc length of an entire circle, divided by the radius. What does that equal? It equals 2 pi radians! You might be saying to your self: I never knew the 2pi actually had units called the radian. The reason: the radian is not actually a unit! If you notice, I never once referred to radians as a unit above. It is another means of expressing an angle, but is not an actuall unit of measure. Why? Look at the equation it comes from. Notice how the radian equals arc length, measured in meters, divided by the radius, also in meters. The meters cancel leaving you with a unitless number. This fact allows us to use radians in equations that would not work with degrees. More on that later. First, let's take a look at some of the common angles in radians and how they relate to degrees. Radians and Degrees Consider a half circle which equals 180 degrees or pi radians.  If we wanted to convert 30 degrees into radians, we would use the conversion rate from above and multiply 30 degrees by pi over 180 degrees.  Since degrees are on the top and bottom of the equation, they cancel leaving you with just radians, and because 30 goes into 180 six times, the fraction reduces nicely to pi over 6 radians. Now, let's see this represented on our half circle. If you were to cut the half circle into 30 degree pieces, you would have six pieces.  Since a full half circle equals pi radians, and a 30 degree slice equals one sixth of the whole, the angle in radians is pi/6. Let's do this with 60 degrees. Using the conversion equation it would be,  This makes sense because 60 is two times bigger than 30, which means the slices will be two times bigger. As a result, it will take half as many to make the whole half circle.  Using the visual representation, you can easily determine what some angles in degrees would equal in radians. For example, what would 120 degrees be in radians? Since 120 is two times bigger than 60, and 60 degrees equals pi/3 rad, 120 would be 2pi/3 rad.  Another way of looking at it would be to realize that it takes two slices out of three to make 120 degrees. Therefore, it encompasses 2/3 of the half circle, or 2/3pi rad. Let's try another angle. What would 150 degrees equal in radians? First, you would have to see that 150 is divisible by 30. It would take five 30 degree slices to equal a 150 degree slice. Therefore, the angle in radians would be 5pi/6, since pi/6 is one 30 degree slice.  Again, notice how it takes 5 out of the six slices to make 150 degrees. Therefore, the angle in radians is 5/6pi rad. You can get the same result by using the conversion equation.  Mathematically, both 150 and 180 are divisible by 30. So, the 150 reduces to 5 since 5 times 30 is 150, and the 180 reduces to 6 for the same reason. Let's consider one more that is not so obvious: 315 degrees. Neither of the two previous angles can be multiplied by a whole number to make 315. However, 45 degrees can. It would take seven 45 degree slices to make 315 degrees. What would that be in radians? First, we need to determine what 45 degrees is in radians. Without using a visual, it should be obvious that it takes four 45 degree slices to make 180 degrees, or pi radians. Therefore, the angle in radians would be pi/4 rad. This means 315 degrees would be seven of those slices, or 7pi/4 rad.  The conversion equation is the best method to use every time because sometimes the angles are not nice and neat. The purpose of showing the visual representations was to give you a better understanding of how radians work. From here on out, we will only use the conversion equation. What if you wanted to go from radians to degrees? You would use the same conversion equation, only this time you would multiply by 180 degrees and divide by pi rad.  Here, the pi's cancel, and the fraction 3/4 reduces the 180 to 135. Angular Speed Angular speed is defined as the angle of rotation divided by the time it took to make that rotation. It is just like speed except instead of distance travled, it is the angle traveled. Angular speed is represented by the greek letter omega, so the equation looks like,  Omega can be measured in degrees per second, revolutions per second, rotations per second, or radians per second (or any other unit of time). It does not matter which units you use when all you need is the angular speed. However, if you want to use it in conjuction with another equation, it has to be in radians per second! I repeat: angular speed has to be in radians per second if you use it with any other equation!. The conversion factor for revolutions or rotations to radians would be,  For example, if a car engine is at 1200 rpm's, we can find that angular speed in radians per second. The term rpm means rotations per minute. So, we need to convert the rotations to radians, and then convert the minutes to seconds. For the rotations to radians we have,  and the minutes to seconds we have,  It is convenient to leave the pi in the answer for two reasons: it looks better than a decimal, and it is an exact answer because the pi represents an infinitely long decimal number. You can multiply the answer out and round to the appropriate significant digits. In this case, the answer would be 63 rad/s. Angular Velocity Angular velocity is just like velocity in the fact that it has a magnitude, which is the angular speed, and a direction. Thus the equation becomes,  This means that the angle is now a vector, and it is very similar to displacement in that it represents the final angle the object ends up relative to the starting position. The direction of angular velocity is not as obvious as the direction of linear velocity, because the object is rotating and vectors have to be straight lines that point in one direction. They cannot point around like the picture below shows.  So, what direction does the angular velocity vector point? It would seem plausible that it would point like the picture shows below.  The only problem with that logic is that this vector represent the linear velocity, which is completely different from angular velocity. So, that will not work. The best way of making the angular velocity vector different from linear velocity and a straight line is to represent it with a line that points away from the rotating object along the axis of rotation.  The picture above illustrates the correct angular velocity vector. The axis of rotation is an imaginary line that cuts through the center of the object perpindicular to its rotation. Now the question becomes which direction should it point: right or left? To answer this, we need the Right-Hand Rule. The Right-Hand Rule In order to find the direction of the vector, take your right hand and point your fingers in the direction of rotation. The thumb will be pointing in the direction of the angular velocity. So, if you look at the picture above, you will notice that the disc is rotating in to the left of the paper. Therefore, the thumb would be pointing out of the page as opposed to into the page. If the disc were to rotate the other direction, the thumb would be pointing into the page as shown below.  The positive and negative signs still apply here. It is just like with any other vector. If you make out of the page positive, into the page would be a negative angular velocity. This convention may not seem very logical, but the next illustration should help it make some sense. Imagine a screw with right-handed threads. This means when the screw is turned to the right with a screw-driver, it will sink into the material, which is exactly the same direction your thumb would be pointing if your fingers were in the direction of rotation. If you turn the screw to the left, it will come out of the material, which again is the direction your thumb would be pointing. So, the angular velocity will always point in the direction a screw would go depending upon the direction of rotation.  Angular Acceleration It should be clear by now that for every linear quantity such as velocity and acceleration, there is an angular quantity. Since linear acceleration is change in velocity over change in time, angular acceleration is change in angular velocity over change in time. Angular acceleration is represented by the greek letter alpha, which looks like a fish.  The same rules apply for angular acceleration. If omega is positive and alpha is negative, the object's angular velocity is decreasing. If omega and alpha are both either positive or negative, the object's angular velocity is increasing. Also, alpha points along the same axis as omega, which can sum up the last two sentences by saying: if alpha and omega are pointing in the same direction regardless the sign, the object's rotation is increasing, and if alpha and omega are pointing in the opposite directions regardless the sign, the object's rotation is decreasing. It is just like linear velocity and acceleration. We will deal more with alpha later. Angular Speed and Linear Speed Recall from chapter six that the linear speed of say a car traveling in a circle is the speed displayed on the speedometer, and it is the speed the car would have if it were to suddenly go in a straight line. So, objects traveling in a circle have both a linear speed and an angular speed. It turns out that there is a very interesting relationship between the two speeds. The following narrative outlines the mathematical steps that shows this relationship. Since omega equals the angle traveled over time, and the angle is equal to the arc length divided by the radius,  we can substitute the d/r for the theta and get,  The r moves to the bottom which gives,  Since distance equals velocity times time, from the v=x/t equation, we can substitute vt for d and get,  The t's cancel out of the equation which leaves,  Solving for the linear velocity finally gives,  This simple equation means that the linear velocity of an object going in a circle depends on the rate of rotation, omega, and how far the object is from the center of rotation, which is the radius. This may not seem very interesting, but consider the implications this has on a rigid rotating body such as a record. Consider a point some distance away from the center called A.  Point A on the record has to travel a certain distance, which is the circumference of its circle. If omega is 2pi rad/s, point A will travel around that circle and return to the starting position in one second. Now consider point B which is twice as far from the center as point A.  In order for point B to make it around its circle in the same time as point A, it has to have a faster linear speed, because it has a bigger circle to traverse. So, two points on the same object actually have different linear speeds! Since the speed is directly proportional to the radius, if radius increases, the speed increases the same amount and vice versa. So, pretend you were standing at point A holding onto a rope while the record was spinning at a constant angular speed. (assume this record is very large) You would have a certain linear speed. If you walked to point B, which is twice as far, your linear speed would double! If you walked to a point three times as far, your speed would triple and so on. It may seem odd that different points on a solid one piece object can be traveling at different linear speeds. Consider this though: if point B did not have twice the linear speed as point A, B would lag behind and not complete its circle in the same amount of time as A, which would mean the object would not be solid. So, the very fact that the object is solid means point B has to have a faster linear speed than A. Let's consider the centripetal force in this situation. How much more force would you need to hold onto the rope? It is not four times as much, but two times! Remember, centripetal force is directly proportional to the square of the linear speed. So, if it doubles, the force quadruples. But, in this example we are also doubling the radius. Since force is inversely proportional to the radius, doubling the radius results in making the force decrease by a half. Putting it all together, doubling the speed increases the force by four, and doubling the radius decreases it by two, which results in an overall increase of two. Let's put some numbers to this problem to make it more concrete. Point A is 5 meters from the center, point B is 10 meters, and omega is 3 rad/s, which means the record almost makes one complete revolution a second. (One revolution a second would be pi rad/s or 3.14...rad/s. I used 3 to make round numbers). A person's linear speed at A would be, The force required to hold on to the rope would be, Now, moving to point B, the speed would be, Which is double the original speed. The force would be, Which again is double the original. Notice in the v = (r)(omega) equation, that you have radius, which is measured in meters, multiplied by omega, which is measured in rad/s. We know that the unit for velocity is m/s. This is where it is critical that omega's units are rad/s. Since the radian is not a true unit, meters times rad/s just equals m/s.  If omega was degrees per second, the equation would be meters*degrees per second, which is definitely not the units for velocity. Let's try another example. A person is sitting on the edge of a merry-go-round that has a radius of 4.0 meters. If its angular velocity is 25 rpm's, find the linear speed of the person, and determine whether or not a quarter setting next to the person would stay on the ride if the coefficient of friction is .75. This should seem familiar because we learned how to do this problem in chapter six. We are going to do it again, only this time we will use the v = (r)(omega) equation instead. We must first convert 25 rpm's to rad/s.  Plugging into the v and omega equation gives, Since mass is not a factor here, the quarter has the exact same linear speed. Recall from chapter six that in order to determine whether or not the quarter would stay, the following inequality would have to be true.  This is again because the left side represents the friction force, which is the available force that supplies the needed centripetal force, the right side, to go in a circle.  All we need to do is plug in the numbers and evaluate the answer.  Since it is not true, the quarter would fly off. |