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November 26, 2013
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How Heavy is Applejack? by DarkFlame75 How Heavy is Applejack? by DarkFlame75
=Foxy-Noxy did a post about  Applejack breaking that vine, and, although interesting, his math had some errors. Here's my attempt at it. In order for AJ to be able to break the branch without SLAMMING the window, she has to be very heavy. Most arbitrary numbers are from Foxy-Noxy's own calculations, including the tensile strength of the vine. 

Yes, I'm bored. Yes, I probably also made glaring mistakes.
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:iconsmilingshrike:
SmilingShrike Featured By Owner Feb 5, 2014
Cool math bro! But you really don't need lb to convert kilogramme into tons, 1 t = 1000 kg by design, so 17400 kg is 17,4 tons exactly. 
And now we can imagine lift potential of Rainbow Dash and Twilight's baloon.
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:iconrex42:
Rex42 Featured By Owner Nov 27, 2013  Hobbyist Digital Artist
wow.....19-50 tons O_O
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:iconsternguard:
Sternguard Featured By Owner Nov 26, 2013
Easier explanation... earth pony magic!
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:iconhaloprime:
HaloPrime Featured By Owner Nov 26, 2013  Hobbyist Digital Artist
Less easier explanation... Applejack is composed entirely of dark matter.
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:iconfoxy-noxy:
Foxy-Noxy Featured By Owner Nov 26, 2013  Student Digital Artist
Haha, awesome. I didn't actual calculate what I wrote, however. I took it from tumblr because I thought it was worth sharing. (Math was my worst subject in school :c)
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:iconlupus-ignis:
Lupus-Ignis Featured By Owner Nov 26, 2013  Hobbyist Artist
Consider that there is no deformation of the Window frame. That means that the frame itself was not subjected to a pressure larger than its yield strength, so the vine must be of a softer material than speculated. Since Applejack only has one hoof on the frame, the area that supports whatever force she exerts must be the shadow area of her hoof onto the frame. If we consider the frame to be 19mm broad, and for simplicity's sake have half that area on the inside of the glass, that allows only a segment of her hoof to actually transfer force. A hoof is roughly circular, and if we estimate Applejack's shoe size to be radius r=50mm (no insult intended, miss!), the area of the circular segment that can touch the frame is:

A = (r^2)/2 * (θ-sin(θ)).

θ is equal to 2 * acos((r-h)/r), where h is the length from the glass to the edge of the frame (19mm).
θ = 1,253 rad
A = ((50mm^2)/2) * ((1,253rad)-sin(1,253rad)) = 379mm^2

The maximum force transferred is therefore the yield strength of the frame times the area under the hoof. Since it is the "Golden Oak" library, let us use that material for the frame. Oak has a maximum yield strength of σ = 60 MPa (N/mm^2).
Maximum force is therefore:

F = σ * A = 60 N/mm^2 * 379mm^2 = 22 740 N.

Observation shows that the mare's weight is not enough to cut the vine. Only when she uses her famous earth pony strength does the window slam shut. If we assume that her legs are 0,5m long, each of her forelegs must muster a moment of:

M= F*L = 22740N * 0,5m = 11370 N*m

For comparison, that is about 15 horsepower seconds. Applejack cuts the vine in roughly one second with her front legs, using her hind legs for bracing, meaning that he utilises her entire body. My conclusion is therefore that Applejack is approximately 15 times as strong as a standard horse.

Now this engineer should really get some sleep. Any blatant mistakes in the above will be blamed the curse that is midnight mathematics.
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:iconlupus-ignis:
Lupus-Ignis Featured By Owner Nov 26, 2013  Hobbyist Artist
...or rather: The force exerted is at maximum 22,7 kN. We have no way of knowing how strong the vine actually is and thus how strong Applejck really is. The vine must have an ultimate strength (across the grain direction) of maximum A{hoof}/A{vine} * σ{oak, max}. (Bending can be ignored since the glass does not shatter)
σ{vine, max} = (379mmˆ2)/(3800mmˆ2) * 60 MPa = 6MPa, or about a third of that of human skin. Since Applejack has trouble tearing it apart by pulling, we can assume that the strength along the grain direction is considerably higher, as is true for all fibre materials.
While we cannot say with certainty how strong the vine is, I doubt that it is much weaker than 6 MPa, or we would see it tear more from the forces the ponies apply to it, so assuming that Applejack is 5 to 15 times as strong as a normal horse would not be unrealistic.
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:icondarkflame75:
DarkFlame75 Featured By Owner Nov 26, 2013  Student Digital Artist
Midnight mathematics... Exactly how this started...
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:iconlupus-ignis:
Lupus-Ignis Featured By Owner Nov 26, 2013  Hobbyist Artist
Yup, that's one error right there. It is of course not each of her forelegs, but her forelegs combined that exerts 11kNm, as the conclusion states. For her forelegs to be able to create that torque, her hind legs must work equally hard in the oposite direction and her abdominal muscles must counteract both directional forces.
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:iconabyssalemissary:
AbyssalEmissary Featured By Owner Nov 26, 2013  Hobbyist Traditional Artist

There are different problems I see.


What is your source for the tensile strength of the vine? As long as we don't know what material they are it is pretty difficult to assume any value here. So this is maybe the biggest uncertainty here.


What is your assumption for the 0.2 meters width of the contact area? An estimation from viewing the scene I guess?


But most important, we can't actually calculate AJ's weight with this scene because she doesn't use her entire mass to shut the window. She stands on her hind legs and only applies force with her forelegs. So we could only calculate this force she is able to apply this way. To calculate her mass she would have to stand completely on the vine (or rather stick to the window).


So the idea is nice, but it won't work. :)

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