Computing Radians Per Second

[Greg H.]

Ok, thinking in radians just makes my head hurt, but I need to get a grasp on this for something I am working on.

 

So, let us assume we have a stationary object, object A. For the purposes of this exercise, we can consider it to be a point.

 

Orbiting this object we have another point object, object B. Object B orbits at a distance 5,000 meters with a fixed velocity of 100 m/s.

 

So the question is, how many radians per second is object B's orbital velocity.

 

I already know:

 

[math]V_o = 100 m/s[/math]

[math]C_o = 2\pi \times 5000 m[/math]

 

So would it just be a matter of finding the angle subtended by the arc traveled in one second, and converting that to radians?

Edited July 30, 2013 by Greg H.

[swansont]

Yes. Or you could find the time of a complete orbit and divide by 2*pi, since the re are 2*pi radians in a circle. Or use v = wr, which is found from taking the time derivative of the equation for an arc, which is s = r*theta, which means w = v/r and is already in radians per second.

[overtone]

Isn't the radian here 5000 meters? Why not just divide?

[imatfaal]

Isn't the radian here 5000 meters? Why not just divide?

 

to elaborate - the distance along an arc subtending one radian will always be equal to the radius you can just divide.

[swansont]

Isn't the radian here 5000 meters? Why not just divide?

 

That was one of my options.

[HalfWit]

The radius is 5000 meters.

The circumference is 2i*r = 10,000 pi = 2pi radians; so

1 radian = 5000 meters.

We are circling at 100 m/s so we'll travel 5000 meters = 1 radian in 50 seconds.

If we travel 1 radian in 50 seconds then we'll go 1/50 = .02 radians in 1 second.

So the answer is .02 radians per second.

In general, here is how to stop your headaches and learn to know and love radians.. The unit circle has radius 2pi and everything flows from that. 1/4 of the way around the circle is pi/2; halfway round is pi; 3/4 of the way around is 3pi/2; all the way round is 2pi, which is the same as zero, where we started.

Forget you ever heard of degrees. They're a fading memory from your past. From now on, think only in radians. When you get confused remember that the circumference of the unit circle is 2pi radians and all other circles are proportional to that. And if t is a fraction between zero and one, going t of the way around the circle is 2pi*t radians.

If t = 1/2 then you went halfway around the circle, to pi. If t = 1/3 then you went 1/3 of the way around the circle to 2pi/3. After a while you won't even care about how many degrees that is.

That's the mindset you want to strive for. Think only in radians from now on.

Edited July 31, 2013 by HalfWit

[Greg H.]

Thanks everyone. I suppose the problem is I don't use either degrees or radians in my everyday work, so when I'm forced to think about angles and circles I immediately default to degrees because that's what I understand from school.

 

I appreciate the insight.

 

One question. If I understand what's being said, then 1 radian will always be equal to the radius of the circle, regardless of the value of that radius. Is that correct, of have I misunderstood something?

[swansont]

Thanks everyone. I suppose the problem is I don't use either degrees or radians in my everyday work, so when I'm forced to think about angles and circles I immediately default to degrees because that's what I understand from school.

 

I appreciate the insight.

 

One question. If I understand what's being said, then 1 radian will always be equal to the radius of the circle, regardless of the value of that radius. Is that correct, of have I misunderstood something?

 

 

 

[math]s=r\theta[/math]. So yes, one radian is the angle when you travel an arc with a length (s) equal to the radius [r]. That's true in general, as imatfaal has already stated.

[HalfWit]

 

One question. If I understand what's being said, then 1 radian will always be equal to the radius of the circle, regardless of the value of that radius. Is that correct, of have I misunderstood something?

No that is not true. The radius of the unit circle is 1. The angular measure of the circumference is 2pi radians. The circumference is 2pi but the angular measure is 2pi radians.

 

In other words the length of the circumference is 2pi. The measure of the angle swept out by the radius moving around the circle is 2pi radians. It's the angle that's measured in radians.

Edited July 31, 2013 by HalfWit

[Greg H.]

 

 

 

[math]s=r\theta[/math]. So yes, one radian is the angle when you travel an arc with a length (s) equal to the radius [r]. That's true in general, as imatfaal has already stated.

i thought that was the case, based on his response, but I wanted clarification - sometimes I misread things. Thanks again.

[overtone]

You use radians for all such calculations because they are measured in the same units you are using throughout.

 

Using degrees for such calculations is something like using light penetration in lumens for your water depth unit while calculating volume, pressure, or speed of descent.

[Amaton]

i thought that was the case, based on his response, but I wanted clarification - sometimes I misread things. Thanks again.

 

With these problems, you should always default to radians. They may not be familiar, but one realizes that they are more intuitive and helpful than degrees once practiced. Also, if any elementary calculus is involved, then radians will be your best (and only) option.

 

Anyhow, what could be simpler than [math]\frac{S}{r}=\theta[/math]? (where 'S' is the arc length, 'r' is the radius, and 'theta' is the radian measure to that arc)