WEBVTT physics/ap-physics-1-2/fullerton
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Hi everyone and welcome back to Educator.com.
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I am Dan Fullerton and today we are talking about rotational kinematics.
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Our objectives are going to be to understand the analogy between translational and rotational kinematics, to use the right-hand rule to associate angular velocity with a rotating object, and to apply equations of translational and rotational motion to solve a variety of problems.
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Let us start by talking about radians and degrees.
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In degrees, one time around a circle is 360 degrees.
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In radians though, once around a circle is 2π, where a radian measures the distance around an arc equal to the length of an arc's radius.
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So distance around a circle -- oftentimes written ΔS -- is the circumference, which is 2π radians or it would be 360 degrees if you are looking at an angular measurement.
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Let us convert 90 degrees to radians.
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If we start off with 90 degrees -- if we want to convert that to radians, we are going to multiply this by -- well we want degrees to go away, so 360 degrees = 2π radians.
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So the degrees will cancel out and I will be left with 90/360, that is 1/4 and that is going to be π/2 radians or 1.57 radians.
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Let us convert 6 radians to degrees, going the other way.
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We have 6 radians, and we are going to multiply that -- we want radians to go away -- I know there are 2π radians in a complete circle and 360 degrees in a complete circle.
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So we will be left with 6/2π × 360 or 344 degrees.
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As we do this, let us talk for a few minutes about linear versus angular displacement.
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Linear position displacement, is given by ΔR ΔS.
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If we talk about angular position or displacement though, we can talk about how much this angle changes.
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That is given by Δ θ, and there is a conversion between these.
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The linear distance is equal to R × θ, or ΔS = R × Δθ.
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Multiply the angular displacement, Δθ by the radius to get a linear displacement.
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We can also look at this for linear versus angular velocity.
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Linear speed or velocity is given by the symbol V.
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Angular speed or velocity is given by ω, kind of a curly W.
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Now whereas velocity was the change in displacement over time, angular velocity is the change in angular displacement over time.
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Dθ, DT, or Δθ with respect to T.
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If we want to take a look at the direction of angular velocity, we use the right-hand rule.
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And the way we do that is you wrap the fingers of your right hand in the direction of the angular velocity -- your thumb will point in the direction of that vector.
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Having a typical vector is not going to work because angular velocity, the direction linearly, is constantly changing, so you have to define it with something perpendicular.
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Wrap the fingers of your right hand around the circle -- your thumb will point in the direction of the angular velocity vector.
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In this case, as we have here on the screen, angular velocity is around this way, so as I wrap the fingers of my right hand around that direction, my thumb points out toward me.
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I show that by showing a dot coming toward me, almost as if there is an arrow being pointed toward me -- that is what I would see.
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So that is out of the plane of the board.
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If it was into the plane of the board, the way I would draw it would be an x like I am looking at the fletchings of an arrow as it is moving away from me, so that would be into the plane.
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In this case though, angular velocity points out of the plane.
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Angular velocity is the cause of counterclockwise rotations, typically referred to as positive, and those that cause clockwise, negative.
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How do we convert linear to angular velocity?
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Well, linear velocity is just equal to angular velocity times the radius or angular velocity equals linear velocity divided by the radius.
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Let us take an example.
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Let us find the magnitude of Earth's angular velocity in terms of radians per second (rad/s).
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Angular velocity is going to be a change in angular displacement divided by the time.
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The Earth goes once around on its axis or 2π radians every 24 hours, once a day.
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And let us multiply that to get radians per second -- let us convert hours into seconds.
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One hour is 3,600 s, so my hours make a ratio of 1 and I am left with ω = 2π/24/3,600 or 7.27 × 10^-5 rad/s.
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As we look at linear versus angular acceleration, linear acceleration is given by A, and angular acceleration is given by the symbol α and it too is a vector.
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Just like linear acceleration is change in velocity over time, angular acceleration is change in ω over time.
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The rate of change of the angular velocity with respect to time, or we can write that as Δ ω/ΔT.
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The conversions between them are pretty straightforward as well -- A = Rα or α = A/R.
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Another example -- angular acceleration.
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A clown rides a unicycle. If the unicycle wheel begins at rest and accelerates uniformly in a counterclockwise direction to an angular velocity of 15 rpms in a time of 6 s, find the angular acceleration of the unicycle wheel.
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Let us start by converting this 15 rpms to radiants per second. We have 15 rpms or revolutions per minute.
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We need minutes to go away, so I will put minutes on the top and I want seconds here, so I know 1 minute is 60 seconds and now I have revolutions per second (rps).
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So, I also need to multiply to make the revolutions go away.
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One revolution is 2πradians.
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Unit conversions then -- minutes make a ratio of 1, revolutions make a ratio of 1 and I am left with 15 × 2π/60, or 1.57 rad/s.
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Similarly, the angular acceleration -- now I can find as change in angular velocity divided by time.
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That is going to be final angular velocity minus initial angular velocity over time or 1.57 - 0/6s, which is 0.26 rad/s².
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So let us put this all together to talk about kinematic variable parallels.
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We talked about displacement in the translational or linear world.
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Displacement -- we are writing as ΔS or D, or ΔX or we would even have it as R.
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In the angular world, it is Δθ.
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Velocity is V. Angular velocity is ω, acceleration is A, angular acceleration is α, and time is the same translationally and in the angular world.
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And there are more parallels we can draw, such as kinematic variables -- we can convert them.
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Displacement is S = Rθ or if we want the angular version we have θ = S/R.
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Velocity or V = Rω, angular velocity is ω = V/R, acceleration linear is A = Rα, angular is α = A/R, and same as before, time equals time.
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So it is very easy to translate back and forth to these variables.
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We even have parallels with the kinematic equations -- translational kinematic equations, V final = V initial + AT.
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In the rotational world, we have kinematic equations too -- we just replace the variables with their angular equivalents.
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So ω = ωinitial + αt and for translational, ΔX = V initial T + 1/2 AT².
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For rotational, we have Δθ = ωinitial T + 1/2αT² and final velocity² = initial velocity² + 2 × acceleration × ΔX.
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The rotational equivalent -- final angular velocity² = initial angular velocity² + 2 α Δθ.
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So really there is not a whole lot new to learn here. It is just using different variables to cover the rotational kinematics as opposed to just the linear kinematics.
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Let us take an example of a medieval flail.
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A knights swings a flail of radius 1 m in 2 complete revolutions. What is the translational displacement of the flail?
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Well S = R θ, so R is going to be 1 m, θ is 4π radians, twice around the circle.
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So that is just going to be 4π × 1 or 12.6 m.
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Or let us look at a CD player.
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A compact disc player is designed to vary the disc's rotational velocity so that the point being read by the laser moves at a constant linear velocity of 1.25 m/s.
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What is the CD's rotational velocity in revolutions per second when the laser is reading information on an inner portion of the disc when the radius is 0.03 m?
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Angular velocity is linear velocity divided by radius, so that is going to be 1.25 m/s over the radius of 0.03 m which is 41.7 rad/s.
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We want that in revolutions per second so let us convert it.
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41.7 rad/s times -- there are 2π radians in one revolution, so radians make a ratio of 1, 41.7 × 1/2π -- I get 6.63 rps.
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We can even look at a carousel.
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A carousel accelerates from rest to an angular velocity of 0.3 rps in 10 s. What is its angular acceleration?
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Well, just like we did in kinematics, we can make a table -- ω initial = 0, ω final is 0.3 rad/s -- Δθ, α and we know t is 10 s.
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What is its angular acceleration? We can use our kinematic equations.
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α = ω - ω initial/t or 0.3 rad/s - its initial 0/10 s for an angular acceleration of 0.03 rad/s².
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What is the linear acceleration for a point at the outer edge of the carousel 2.5 m from the axis of rotation?
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Well to do that we just need to find the linear acceleration from the angular acceleration.
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A = Rα, where R = 2.5 m and our α -- we just determined, 0.03 rad/s² gives me a linear acceleration of 0.075 m/s².
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Or a circular saw example -- a carpenter cuts a piece of wood with a high powered circular saw.
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The saw blade accelerates from rest with an angular acceleration of 14 rad/s² to a maximum speed of 15,000 rpms.
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What is the maximum speed of the saw in rad/s?
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Well 15,000 rpms or revolutions per minute -- let us convert those minutes to seconds -- 1 min is 60 seconds, and instead of revolutions, we need this in radians.
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So we have 2π radians per revolution -- revolutions make a ratio of 1, minutes make a ratio of 1 and I come up with 1,570 rad/s.
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How long does it take the saw to reach its maximum speed? Well, that is a kinematics problem.
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ω initial = 0 -- final, its maximum speed is 1,570 ras/s, Δθ, α -- which we said was 14 rad/s² and time.
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If we are looking for how long -- we are looking for time -- I will use the formula ω = ω0 + αT and rearrange this for the time.
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Time = ω - ω0/α or 1,570 - 0/14 rad/s², which gives me a time of 112s.
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Hopefully that gets you started with rotational kinematics.
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It is really similar to what we did with linear kinematics, it is just we have some slightly different variables dealing with objects going around the circle.
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Once you know the change -- the parallels with the variables -- and you know the equations already, it is just a matter of being careful with your variables.
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Thanks so much for your time and thanks for watching Educator.com.
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Make it a great day.