Another angular rotation question, I seem to suck at these

[emjaya]

A discus thrower accelerates a discus from rest to a speed of 25 m/s by whirling it through 1.25 revolutions. Assume the discus moves on the arc of a circle 1.00m in radius. Calculate the final angular speed of the discus. Determine the magnitude of the angular acceleration of the discus assuming it to be constant. Calculate the time over which the acceleration occurs.

First of all - when they said 1.25 revolutions, thats when the discus thrower is just about to throw it right? Then when it finally leaves the discus thrower's hands the velocity gained is 25 m/s...

so then... I could use V = radius * angular velocity
to find the angular velocity, but do I need to change that from rad/s by multiplying it by 1.25 rev/2pi rad?

If what I just said is what Im supposed to do... how do I get the rest of it? >_<"

[mr_homm]

Your first idea about velocity is correct. You do not use the 1.25 revolutions for this part of the question. What that tells you is how long it took to get the discus up to the final speed, but fore the first question, you don't care how it GOT TO that speed, only what the final speed IS. So you just use velocity = angular velocity * radius and solve for angular velocity. You get angular velocity = 25m/s / 1m = 25 rad/sec.

For the other parts of the question, you just have to remember that rotational kinematics works EXACTLY like linear kinematics. All the equations are the same, the variables just have different names (and different physical meanings, of course, but that doesn't affect the math at all).

Here's a little history lesson to explain why that is:

Suppose it's 300 BC and you are an ancient Greek geometer. You are trying to find a way to measure angles (for the first time in history -- suppose no one has ever really thought hard about measuring angles before). So far, all you know is how to measure distances. What do you do?

First, you have to work with the tools you have, namely distance measurements. So suppose you have two lines that meet at a point and you want to talk about how big the angle is. You've got to use distance somehow, so you try thinking about using the angle to cut a circle. Place a circle centered at the meeting point of the two lines, and look at the arc that the lines cut out of it. (Actually, they cut it into 4 pieces, but if you know the size of one piece that tells you all about the others, so focus on just the smallest circular arc, which you can call S.) If the arc is longer, the angle is wider, so you can use that arc to measure the angle.

But there's a problem here. If you use a bigger circle, you'll get a longer arc for the same two lines. The size of the angle obviously hasn't changed, but the arc did, so it is not a reliable way to measure the angle after all! Still, you can easily fix this up: just define the angle to be the ratio of arc length to radius, θ = S/R. Now it works nicely, because if you make the circle bigger, then both S and R will increase by the same factor, so you'll get the same value for θ.

(By the way, this shows why the units work out the way they do. In the first paragraph, I said that 25m/s / 1m = 25rad/s. This works because meter/meter = radian, since radians are really ratios of distances. Also by the way, you can see now why 1 rev = 2pi radians. For a whole circle, S is the circumference, which is 2pi*R, so when you compute the angle, you get 2pi*R/R = 2pi. People didn't just decide, "Hey, let's make a circle 6.2832 radians!" That would be a pretty silly choice. But it ISN'T a choice, it's forced on you by the ratio definition of angle, i.e. it comes automatically out of the simple and sensible definition θ = S/R.)

So angle is defined as S/R. Now if you turn this around, you get S = Rθ, i.e. distance along the circle is just radius * angle. Since you are now talking about distance along a circle, you can talk about MOTION along the circle too, just by taking rates of change. There's a very important point to notice here: S = Rθ is the DEFINITION of θ, so it is ALWAYS true. That means that the two sides of the equation are always equal, and that means that they must always change at the same rate (otherwise one would get ahead of the other, and they wouldn't be equal any more). So what do you get if you look at the rate of change?

The speed of a point moving around the circle is v = ▲S / ▲t = ▲(Rθ) / ▲t = R*(▲θ / ▲t) = R*w, where w is the angular velocity, w = ▲θ / ▲t. You can do it again: a = ▲v / ▲t = ▲(Rw) / ▲t = R*(▲w / ▲t) = R*A, where A is the angular acceleration A = ▲w / ▲t. (I can't make the Greek letter alpha easily, so I'm using capital A.) So look what you have:
S = Rθ
v = Rw
a = RA.

The angle quantities are all related to the linear quantities by the same factor of R. Now look what this does to the kinematic equations:

s = s0 + v0*t + 1/2*a*t^2 becomes
Rθ = Rθ0 + Rw0*t + 1/2*RA*t^2, and the R cancels out on both sides to give
θ = θ0 + w0*t + 1/2*A*t^2, which looks exactly like the first equation with the names of the variables changed. You can try this out on the other kinematic equation, too, and it works the same way. For any linear kinematic equation, there is a corresponding rotational kinematic equation that you can get by changing s to θ, v to w, and a to A.

This means that you have:

w = w0 + A*t,
θ = θ0 + w0*t + 1/2*A*t^2,
θ = θ0 + w*t - 1/2*A*t^2,
θ = θ0 +1/2*(w0 + w)*t, and
w^2 = w0^2 + 2*A*(θ - θ0),

which are used exactly like the corresponding equations in linear kinematics. You just identify what you know and what you want, and then select the equation that relates those things to each other WITHOUT dragging in anything irrelevant.

For your problem, you know θ - θ0 = 1.25*2pi, w0 = 0, w = 25 (found in the first question), and you want to know A. Notice that t is not mentioned anywhere, so you should use the equation that DOESN'T have t. This is the last equation in the list. From this you can just plug into the data and get A.

The last question is to find t. OK, NOW you want t, but you've lost interest in A (already found that -- on to the next thing), so you choose the equation that doesn't mention A. That's the 4th equation on the list, so you plug in your data and get t.

Sorry this was so long, but I thought you might like to see where all this stuff comes from, instead of just the answer.

Hope this helps!

--Stuart Anderson

[emjaya]

Thank you! I'll have to make sure I save this somehow. This might help in the future x_x knowing me, I'll be needing all the help I can to pass this course. Thanks again!