Angular Variables

Transcript

0:00Introduction

Hello everyone! Before we start, one small puzzle for you. Look at a ceiling fan. Take one point at the tip of a blade, and another point close to the centre. Both go round once in exactly the same time. So, are they moving equally fast? Hmm, think about it. And one more: when you switch that fan off, how many more rounds does it make before it stops? By the end of this video, you will answer both. And remember, all of circular motion is built on today's angular variables. So, let's begin.

0:36Angle, arc, and how fast the angle changes

So, a particle moves on a circle of radius r. On a straight line, one coordinate told us where it is. On a circle, what is the simplest way to say where the particle is? Yes, by an angle! We call it theta, the Greek letter you see on screen: the angular position. And the arc length s is r into theta. So, s equals r theta. But one condition: theta must be in radians. Not degrees, radians. Clear? Now, how fast does this angle change? That is the angular velocity, omega, this curly letter, okay? Omega equals d theta by d t: the rate of change of theta with time. Really simple. And if omega itself changes? Its rate of change is the angular acceleration, alpha, the Greek letter you see here, okay? Alpha equals d omega by d t. So, three angular variables: theta, omega and alpha. Got it?

1:36Angular to linear: multiply by r

Alright, now let's connect these to the ordinary speed. The speed v of the particle is omega into r. v equals omega r. With me? And the same way, a sub t equals alpha r. The small t below a stands for tangential; we will meet it properly later in this chapter. So, to go from angular to linear, just multiply by r. That's it. Now, back to our fan. The tip and the point near the centre both go round once in the same time. So they turn through the same angle every second: the same omega. Same omega, so same speed, right? No, no, careful! Same omega, yes. But v equals omega r. The tip has a bigger r, so it has a bigger v; the double arrow means so, okay? The tip of the blade moves faster than the point near the centre. First puzzle, solved! Never mix up angular speed and linear speed. That is a very common trap.

2:36Period, frequency and rpm

Now take uniform circular motion, where omega stays constant. How long does one full round take? First tell me: one full round is how many radians? Two pi radians, yes! Pi is the Greek letter you know. So the time for one round, the period, is two pi by omega. T equals two pi by omega, okay? And the frequency f, the number of rounds per second, is one by T, okay? So a big T means a small f: a slow fan has a long period and a low frequency. Clear? Now, many JEE problems give the speed in rpm: revolutions per minute. n rpm means omega equals two pi n by sixty, in rad s to the power minus one, that is, radians per second. Why? Because n rounds take sixty seconds, and every round is two pi radians, right? Write this down. We use it in both examples.

3:36Constant angular acceleration

Alright, now the fan speeds up, or slows down, with a constant alpha. In a straight line, constant acceleration gave us three equations, remember? So what do you think we get here? Exactly the same three equations, yes! Just new letters. First: omega equals omega nought plus alpha t. The small zero below omega means the starting angular velocity; we say omega nought, okay? Second: theta equals omega nought t, plus half alpha t squared. Here theta is the angle turned from the start. And third: omega squared equals omega nought squared, plus two alpha theta. Topper's trick: in the straight-line equations, read v, u, a and s as omega, omega nought, alpha and theta. You already know all three. Nice and simple. Now, a question asks for the number of revolutions. You found theta. So theta is the answer, right? Oh no, careful! Theta is in radians. Every revolution is two pi radians, so the number of revolutions is theta by two pi. Students lose easy marks here. Don't be one of them, okay?

4:50Example 1: 300 rpm in rad/s and speed of a wheel point

Let's start with an easy one. Express three hundred rpm in radians per second. And find the speed of a point on a wheel of radius zero point five metres, turning at one hundred and twenty rpm. Pause the video, and try it first. Done? Okay, let's check. Three hundred rpm means three hundred rounds per minute. Divide by sixty, and you get five. So, five radians per second, right? No, no! Five rounds per second, not five radians. Every round is two pi radians, remember? So, three hundred, into two pi by sixty: ten pi radians per second. The cross means into, okay? Now the wheel. At one hundred and twenty rpm, omega equals one hundred and twenty, into two pi by sixty. That is four pi radians per second. Clear? And the speed: v equals omega r. Four pi into zero point five: two pi, in m s to the power minus one, that is, metres per second. See, we converted rpm to omega, and then r turned omega into speed. The same v equals omega r that answered our fan puzzle.

6:07Example 2: fan stops in 10 s, retardation and revolutions

Now a harder one, and it is our fan from the start! Here, a fan turning at six hundred rpm stops uniformly in ten seconds after being switched off. Find the angular retardation, and the number of revolutions it makes while stopping. Pause and try. Hint: stops uniformly means a constant alpha. Done? Okay, let's check. First, convert. Omega nought equals six hundred, into two pi by sixty: twenty pi radians per second. And it stops, so the final omega is zero. Got it? Now the first equation, with the numbers in. Zero equals twenty pi, plus alpha into ten. So alpha equals minus two pi, in rad s to the power minus two, that is, radians per second squared. The minus sign just says the fan is slowing down. Retardation means slowing, so we give its size: two pi radians per second squared. Now the angle, from the second equation. Theta equals twenty pi into ten, minus half, into two pi, into ten squared. Two hundred pi minus one hundred pi: one hundred pi radians. So the fan makes one hundred pi revolutions, right? No, no, wait! Remember the trap! One hundred pi is in radians. Divide by two pi: one hundred pi by two pi is fifty. Fifty revolutions. So that is how many more rounds our fan makes after you switch it off. Second puzzle, solved! And a quick check, the way toppers do it. The fan slows uniformly from six hundred rpm to zero, so its average is three hundred rpm. That is five rev s to the power minus one: five revolutions per second. Five into ten seconds: fifty. Same answer, a perfect match!

8:05Quick revision: v = ωr, rpm to rad/s, revolutions

So, quick revision. To go from angular to linear, multiply by r: v equals omega r, and a sub t equals alpha r, with theta always in radians. For n rpm, omega equals two pi n by sixty, and the period T equals two pi by omega. With constant alpha, use the straight-line equations, and the number of revolutions is theta by two pi. You did really well today. Now go and try the practice problems on this page. All the best!