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Blimey, this one grew a bit. I've been working on it for three days, I feel like Isaac Newton.
Okay, I want to state right at the outset that I'm not trying to debunk anything here. I don't know enough about lunar cycles or the mechanism by which a plant grows to have a sensible argument around that whole thing.
I was just intrigued by something PC said (yes, I know, but it's true). He said this in Tony's thread:
This is pure maths and physics gibberish, and some of it might be wrong. Don't read it if you don't like maths and physics :lol:
Here we go.
The equation I'm using is one I dragged out of the back of my brain, which proves that my 26 year old Physics degree wasn't a waste of grant money:
F = GmM/r^2
where:
F is the size of the gravitational force between two bodies in Newtons
G is the Gravitational Constant (6.67 x 10^-11)
m is the mass of one body in kg
M is the mass of the other body in kg
r is the distance between the two bodies in metres (r^2 = r squared, the circumflex being used to denote "raised to the power of" - you'll see it more below)
Right. How heavy is a plant? Well, I've got a bag of parsley downstairs, it weighs 25g, let's use that, and assume it's all water (it mostly is anyway).
How heavy is the Moon? It's 7.3477 x 10^22 kg.
How far apart are the Earth and the Moon? 3.84 x 10^8 metres.
Plugging these numbers in, the gravitational force exerted on a 25g plant's water by the Moon is 8.31 x 10^-7 Newtons.
As an aside, I wanted to work out what this actually meant. How much force is 8.31 x 10^-7 Newtons? We can get an idea by turning to another equation, F=ma. This says that the force applied to an object is equal to its mass multiplied by its acceleration. Flipping it around, we get a=F/m, or the acceleration of an object is equal to the the force applied to it divided by its mass. So we can work out that the acceleration experienced by our 25g plant is:
(8.31 x 10^-7)/0.025 = 0.00003324 m/s/s
So every second, if the force of the Moon's gravity is the only thing acting on it, our 25g plant will speed up by 0.00003324 metres per second.
Now, this is the good bit.
Let's assume that the appearance of the Moon in the sky has an on/off effect, and that the pull is always directly "upwards". Not true by any means, but it's close enough for rock and roll. Let's further assume that the Moon is in the sky for 10 hours at a time. We can work out how far the plant will move in 10 hours.
10 hours = 36000 seconds
To work out how far the plant moves in that time we need:
x = vt + (a * t^2)/2
where:
x is the distance travelled
v is the starting velocity (which is zero in this case)
t is the time in seconds for which the acceleration is applied
a is the acceleration
Plugging the numbers in again, the distance which our plant (or rather, its water) will travel upwards in 10 hours due to the gravitational pull of the Moon is:
21539.52 metres
What? That's outrageous! That's saying that in 10 hours, from a standing start, even from a distance of a quarter of a million miles, the Moon's gravitational pull would move the plant over 13 miles! Someone check my maths!
The problem is, I'm completely ignoring the gravitational pull of the Earth in the opposite direction which, being much closer (right underneath the plant, in fact), has a much greater effect.
I'm going to do a physics thing here, and treat the Earth as a point object located at what is actually the Earth's centre. Why? Because if I don't, the distance between the Earth and the plant is 0 metres, and the F = GmM/r^2 gives us a "divide by zero" error.
Plugging the numbers in again, this time for the Earth:
How heavy is the plant? Still 0.025 kg
How heavy is the Earth? 5.9742 × 10^24 kg
How far away from the centre of the Earth is the plant (i.e. the Earth's radius)? 6.378 x 10^6 m
Therefore the gravitational force exerted on the plant by the Earth is 0.245 Newtons.
(Simple check on this - a=F/m = 0.245/0.025 = 9.8 m/s/s, which is indeed the approximate acceleration due to gravity on the surface of the Earth; my gosh, it's almost like I know what I'm doing.)
So the gravitational force exerted by the Moon = 8.31 x 10^-7 N
and the gravitational force exerted by the Earth = 0.245 N
The effect of the Earth's gravitational force on the plant is a million times that of the Moon's.
Let's turn our attention to a tree!
Oak weighs in at about a tonne per cubic meter according to a random page I found on the interweb, which means a tree is easily going to weigh in the region of 10-20 tonnes, maybe even more. Is that right? We can do a simple check. Imagine the tree is mostly water again, like we did the plant. Now imagine the tree as a cylinder 1m (a yard) in diameter and 30m (~100 feet) high. That sounds reasonable for a tree. The volume of this cylinder is calculated by:
volume = pi * r^2 * h
where:
pi = 3.142
r = radius
h = height
This gives us a volume of approx 24 cubic metres. 1 cubic metre of water weighs in at 1 tonne, so this quick check shows that we're on the right lines, as our cylinder tree weighs 24 tonnes.
We don't need to be too accurate. It's order of magnitude we're after here. You can see, then, that the gravitational force exerted on a tree is going to be 6 orders of magnitude more than that exerted on a plant (if we assume a 25 tonne tree against a 0.025g plant). But the thing is, it's 6 orders of magnitude on both sides, the Earth's and the Moon's. So it cancels out.
Oh, and bringing the Sun into the equation, I found this:
The strength of the sun's gravity is 179 times that of the moon's but the moon is responsible for 56% of the earth's tidal energy while the sun claims responsibility for a mere 44% (due to the moon's proximity but the sun's much larger size).
My conclusion, then, is that whatever causes the sap to rise in a plant, or a tree, it's not gravity.
I'm knackered now.
Okay, I want to state right at the outset that I'm not trying to debunk anything here. I don't know enough about lunar cycles or the mechanism by which a plant grows to have a sensible argument around that whole thing.
I was just intrigued by something PC said (yes, I know, but it's true). He said this in Tony's thread:
I wondered about the physics of this, so I've done some number doodling. Technically it's probably in the wrong thread, but I thought I'd leave it here to run alongside Tony's one, to stop Tony's getting hijacked by... people like me.Pig Cat said:
This is pure maths and physics gibberish, and some of it might be wrong. Don't read it if you don't like maths and physics :lol:
Here we go.
The equation I'm using is one I dragged out of the back of my brain, which proves that my 26 year old Physics degree wasn't a waste of grant money:
F = GmM/r^2
where:
F is the size of the gravitational force between two bodies in Newtons
G is the Gravitational Constant (6.67 x 10^-11)
m is the mass of one body in kg
M is the mass of the other body in kg
r is the distance between the two bodies in metres (r^2 = r squared, the circumflex being used to denote "raised to the power of" - you'll see it more below)
Right. How heavy is a plant? Well, I've got a bag of parsley downstairs, it weighs 25g, let's use that, and assume it's all water (it mostly is anyway).
How heavy is the Moon? It's 7.3477 x 10^22 kg.
How far apart are the Earth and the Moon? 3.84 x 10^8 metres.
Plugging these numbers in, the gravitational force exerted on a 25g plant's water by the Moon is 8.31 x 10^-7 Newtons.
As an aside, I wanted to work out what this actually meant. How much force is 8.31 x 10^-7 Newtons? We can get an idea by turning to another equation, F=ma. This says that the force applied to an object is equal to its mass multiplied by its acceleration. Flipping it around, we get a=F/m, or the acceleration of an object is equal to the the force applied to it divided by its mass. So we can work out that the acceleration experienced by our 25g plant is:
(8.31 x 10^-7)/0.025 = 0.00003324 m/s/s
So every second, if the force of the Moon's gravity is the only thing acting on it, our 25g plant will speed up by 0.00003324 metres per second.
Now, this is the good bit.
Let's assume that the appearance of the Moon in the sky has an on/off effect, and that the pull is always directly "upwards". Not true by any means, but it's close enough for rock and roll. Let's further assume that the Moon is in the sky for 10 hours at a time. We can work out how far the plant will move in 10 hours.
10 hours = 36000 seconds
To work out how far the plant moves in that time we need:
x = vt + (a * t^2)/2
where:
x is the distance travelled
v is the starting velocity (which is zero in this case)
t is the time in seconds for which the acceleration is applied
a is the acceleration
Plugging the numbers in again, the distance which our plant (or rather, its water) will travel upwards in 10 hours due to the gravitational pull of the Moon is:
21539.52 metres
What? That's outrageous! That's saying that in 10 hours, from a standing start, even from a distance of a quarter of a million miles, the Moon's gravitational pull would move the plant over 13 miles! Someone check my maths!
The problem is, I'm completely ignoring the gravitational pull of the Earth in the opposite direction which, being much closer (right underneath the plant, in fact), has a much greater effect.
I'm going to do a physics thing here, and treat the Earth as a point object located at what is actually the Earth's centre. Why? Because if I don't, the distance between the Earth and the plant is 0 metres, and the F = GmM/r^2 gives us a "divide by zero" error.
Plugging the numbers in again, this time for the Earth:
How heavy is the plant? Still 0.025 kg
How heavy is the Earth? 5.9742 × 10^24 kg
How far away from the centre of the Earth is the plant (i.e. the Earth's radius)? 6.378 x 10^6 m
Therefore the gravitational force exerted on the plant by the Earth is 0.245 Newtons.
(Simple check on this - a=F/m = 0.245/0.025 = 9.8 m/s/s, which is indeed the approximate acceleration due to gravity on the surface of the Earth; my gosh, it's almost like I know what I'm doing.)
So the gravitational force exerted by the Moon = 8.31 x 10^-7 N
and the gravitational force exerted by the Earth = 0.245 N
The effect of the Earth's gravitational force on the plant is a million times that of the Moon's.
Let's turn our attention to a tree!
Oak weighs in at about a tonne per cubic meter according to a random page I found on the interweb, which means a tree is easily going to weigh in the region of 10-20 tonnes, maybe even more. Is that right? We can do a simple check. Imagine the tree is mostly water again, like we did the plant. Now imagine the tree as a cylinder 1m (a yard) in diameter and 30m (~100 feet) high. That sounds reasonable for a tree. The volume of this cylinder is calculated by:
volume = pi * r^2 * h
where:
pi = 3.142
r = radius
h = height
This gives us a volume of approx 24 cubic metres. 1 cubic metre of water weighs in at 1 tonne, so this quick check shows that we're on the right lines, as our cylinder tree weighs 24 tonnes.
We don't need to be too accurate. It's order of magnitude we're after here. You can see, then, that the gravitational force exerted on a tree is going to be 6 orders of magnitude more than that exerted on a plant (if we assume a 25 tonne tree against a 0.025g plant). But the thing is, it's 6 orders of magnitude on both sides, the Earth's and the Moon's. So it cancels out.
Oh, and bringing the Sun into the equation, I found this:
The strength of the sun's gravity is 179 times that of the moon's but the moon is responsible for 56% of the earth's tidal energy while the sun claims responsibility for a mere 44% (due to the moon's proximity but the sun's much larger size).
My conclusion, then, is that whatever causes the sap to rise in a plant, or a tree, it's not gravity.
I'm knackered now.
