How to Calculate the String Angle of a Kite vs. a Balloon

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I’m studying Randall Munroe’s e-book How To: Absurd Scientific Advice for Common Real-World Problems. I most likely do not must let you know this, nevertheless it’s superior (as is all the pieces from Randall Munroe, the creator of xkcd comics). The complete thought of the e-book is to go use some loopy concepts to unravel largely widespread issues. One chapter focuses on cross a river. He offers you a number of choices. You might change the course of the river and even evaporate all of the water within the river (each concepts are foolish and enjoyable). Another choice is to make use of a kite to get throughout the river. And right here is the enjoyable half—Munroe states that each a kite AND a balloon might lengthen over a river. As the wind velocity will increase, a kite will get larger within the sky. However a balloon will get decrease because the wind will increase.

So, at some worth of wind velocity the kite and the balloon would have a string on the similar angle. Oh! I need to calculate this. That will probably be enjoyable.

Let’s begin with a balloon. If you will have a helium-filled balloon and there’s no wind, it’s going to float within the sky and the string will probably be utterly vertical. There are simply three forces appearing on the balloon. There is the downward-pulling gravitational drive that relies on each the mass of the thing (m) and the gravitational subject (g = 9.8 N/kg). Since the balloon displaces air, it has a buoyancy drive that is the same as the burden of the air displaced (Archimedes’ precept). If the balloon solely had these two forces, the web drive would most definitely be upward and the balloon would speed up away. Bye-bye balloon.

Of course you would possibly need to maintain that balloon. That’s why you tie a string to it. This string exerts a downward stress drive (T) with a magnitude to make the web drive equal to zero. With a zero internet drive, the balloon is in equilibrium and stays at relaxation in an effort to get pleasure from your gravity-defying balloon. Here is a diagram representing these forces.

Illustration: Rhett Allain

Adding up simply the vertical parts (let’s let the vertical be the y-direction) of those forces, I can write it as the next sum.

Illustration: Rhett Allain

We have already got an expression for the gravitational drive (m*g), and the stress will probably be no matter worth it must be with the intention to make the entire drive zero (it is a drive of constraint). So, if we’ve an expression for the drive from the air (the buoyancy drive), then we are able to get some stuff collectively. Since this buoyancy drive is the burden of the air displaced, I want the amount of the balloon (V) and the density of air (ρ). Assuming the balloon is a sphere with a radius R, then the buoyancy drive could be:

Illustration: Rhett Allain

OK, now let’s add some wind. Suppose the wind is blowing horizontally with some velocity (v). This means there will probably be one other drive on the balloon, an air drag drive. We can mannequin this air drag as a drive in the identical path because the wind with a magnitude that relies on the wind velocity, the cross-sectional space of the balloon (A), the form of the balloon (C), and the density of air (ρ). If you’re the wind (sure, YOU are the wind), the cross part of the balloon appears to be like like a circle with a radius R. That makes the world equal to πR2 (the world of a circle).

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Illustration: Rhett Allain

But now we’ve an issue. Since there’s a horizontal drive from the wind, there needs to be another horizontal drive in order that the web drive in that path is zero. Yes, this further horizontal drive comes from the string because it pulls at an angle. Here is a brand new diagram. It’s a bit of bit extra sophisticated.

Illustration: Rhett Allain

Notice that I added the wind—only for a enjoyable visible impact. I labeled the string angle with the variable θ. If the balloon remains to be in equilibrium, the web drive should be zero within the each the horizontal (x) AND vertical (y) instructions. The stress within the string has a element of drive in each the x and y instructions such that the next two equations could be true.

Illustration: Rhett Allain

Since the stress is a constraint drive, there isn’t any direct solution to calculate it. That’s high-quality. I can simply remedy for T within the y-forces equation and substitute into the x-forces equation. Problem solved. Now I can get an expression for the lean angle of the balloon. Remember that the drag drive relies on each the radius of the balloon and the rate of the wind, however the buoyancy drive additionally relies upon the radius (due to the amount). Putting all of these items in, I get this crazy-looking expression (nevertheless it’s not as unhealthy because it appears to be like).

Illustration: Rhett Allain

Don’t fear, I’m going to plot the leaning angle of a balloon for various wind speeds, however first let’s take a look at kites. A kite is not a balloon—simply to be clear. However, it could nonetheless fly within the air AND it has a string. Just just like the balloon, the kite additionally interacts with the transferring air (additionally known as “wind”). However, for the kite, the air pushes again (drag) and likewise up (raise). One solution to mannequin each the raise and drag drive for a kite is to make use of the lift-to-drag ratio (it is an actual factor).

It’s not mysterious. The lift-to-drag ratio is actually simply the raise drive divided by the drag drive. Every flying object that produces raise additionally produces drag. They are each because of the similar interplay with the air. So for those who fly quicker (or have quicker wind over a stationary kite), each the raise AND the drag will enhance. Yes, this lift-to-drag ratio relies on the form and measurement of the flying object in addition to orientation with respect to the movement of the air (known as the angle of assault). But for this kite, I’m simply going to calculate the drag after which multiply by CL (raise coefficient) to get the raise drive.

I feel we’re prepared for a diagram. Here is my kite with forces.

Illustration: Rhett Allain

What? This appears to be like identical to the forces for the balloon? OK, it does look related—however there’s a huge distinction. For the balloon, there’s that upward-pushing buoyancy drive, and it is only one worth. It would not change when the wind velocity will increase. For the kite, the upward pushing drive is the raise, and it DOES rely on the wind velocity. So it isn’t the identical. Just contemplate the case when there’s zero wind. The drag drive will probably be zero, which suggests the raise is zero. The kite will not fly—it simply falls down and it is unhappy.

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Again, I get two drive equations that I can use to eradicate the unknown worth of T. With that, I get the next expression for the angle of the kite (θokay). Actually, I put a subscript okay on a bunch of stuff so you possibly can see it is totally different than the values for the balloon. Oh, air nonetheless has the identical density for each objects.

Illustration: Rhett Allain

OK, I’m about to make a plot of the flying angle for each the balloon and a kite at totally different wind speeds. But earlier than I try this, let’s take into consideration the minimal velocity to fly this kite. In order to raise off the bottom, the raise drive should be no less than equal to the burden of the kite. I can then remedy this for the wind velocity. Anything decrease than this and you will not have a flying kite.

Illustration: Rhett Allain

Now I can decide some values for all of the parameters for each the kite and the balloon. From that, I’ll calculate the minimal velocity and plot the string angle for each the balloon and kite. Then I simply enhance the rate and take a look at the gorgeous graph. I’m simply going to make some tough guesses for stuff just like the mass of a kite and the lift-to-drag ratio. But don’t be concerned. If you do not like my decisions, you’ll be able to change the values within the code under. Here is what you get.

Yes, that is precise Python code. If you click on the pencil icon, you’ll be able to edit it and run it once more. But you must discover some necessary options for these two curves (the kite and the balloon).

  • As the wind velocity will increase, the kite’s angle will get bigger and the balloon’s will get smaller. That’s what we anticipate.
  • For some worth of wind velocity, the kite and the balloon are flying on the similar angle (for my values, it is at about 2.19 m/s).
  • This kite won’t ever be straight overhead (angle of 90 levels). Instead, it will get to a most angle of about 61 levels.

If you alter all of the values (mass and drag coefficients for the balloon and kite), you’ll get a distinct wind velocity at which they’ve the identical angle. Oh, and one final thing. It’s true there was fairly a little bit of math on this put up. But it might have been a lot worse. In all of those calculations, I assumed the strings had no mass. Just think about how enjoyable this downside could be with extra real looking strings. I’ll go away that for you as a homework project.


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