I was reading the old mailbag Arceive and came across the typical question of feather and hammer. If they are dropped togather which will hit the earth before? http://www.straightdope.com/mailbag/mgravity.html
SDSTAFF member Ian promised to set someone straight but now I would like to set him straight
He gave this function which gives the force on the hammer and the feather.
F = G m1 m2
-------
r^2
Which is quite correct.
He then canceled out all the factors and found the quite correct fact that the acceleration of the hammer and the feather are exacly the same. To do this he used this equation. a = F/m
Now for the correction. Ian forgot that the equation also tells us the force that acts upon the earth do the the feather or the hammer. The force that acts on the earth form the Hammer is surly larger than the force from the feather so the earth will accelerate faster toward the hammer than the feather and there for the hammer should hit first.
Don’t try to tell me that because the feather and the hammer are droppet togather that the earth accelertion should be canceled out since the feather and the hammer are not at the same place and therefor the earth accelertion matters.
By the way. The earth acceleration is small, tiny, smaller then tiny, but it is there!!!
Strictly speaking, Nebula, you are correct, but Ian was very well-justified in his response. Let’s work through the math.
Suppose we have a 2.00kg hammer. Near the earth’s surface (say h=1.00m), the hammer’s acceleration is about a=9.81m/s^2. The force of the earth on the hammer is F=19.6N (F=ma). It will take the hammer t=0.452s to fall the distance h=1.00m (t^2=2h/a).
From Newton’s Third Law, we know the force of the hammer on the earth is also 19.6N (but in the opposite direction). The mass of the earth is about 5.98e24kg, giving an acceleration of a=3.28e-24m/s^2. During that same time t=0.452s, the earth will move 3.34e-25m (2x=at^2).
This is very small. Atomic sizes are on the order of 1e-10m. Nuclear sizes are on the order of 1e-15m. I’m not certain how it would be even theoretically possible to measure a distance as small as 1e-24m. For any intent or purpose, the earth does not move.
Well, yeah the earth will be traveling at exactly the same speed as the earth if the hammer and feather are dropped simultaneously at the same location. But how about if they are dropped simultaneously at opposite sides of the earth? Then you’d have the earth shifting a quark or two towards the hammer… but then the gravitational force of the hammer will be pulling the feather faster than if they were side by side… and, hey, what are you doing with that hammer? Get away from me you freak. Stop it! HELP!! HELP!! OWW!! ugh…
F=Fg b/c the only force on a falling object is gravity
thus M1a = M1M2*G / R^2
Which simplifies to a=M2*G / R^2
In other words the mass of the falling object is irrelevant to it’s acceleration from gravity. The falling objects gravitaional attraction to the planet is irrelevant.
Fg=Force of gravity on an object
F=Force acting on an object
M=Mass ;M1=object M2=planet
a=acceleration
G=Gravitational constant
R=Radius
Hammers fall faster on earth because feathers have more drag in an atmosphere. On the moon though or in an evacuated container you can see them fall at the same rate.
HeadlessCow, you missed the point. The OP had nothing to do with air resistance; it’s just a misapplication of the law of gravitation.
Nebula wrote:
This is not quite exactly true. The acceleration of the hammer and the earth toward each other, is the same as that of the feather and the earth toward each other. The gravitational force always acts equally on Earth and the dropped object, so they move together and meet in the middle someplace.
We usually simplify this, and assume that Earth remains stationary while the dropped object accelerates at 9.8/m/s/s. But in reality, the dropped object accelerates toward Earth at (9.8 minus epsilon) m/s/s, and Earth accelerates toward the object at (epsilon) m/s/s, where epsilon is a teeny tiny puny little number. Epsilon is a little bit bigger in the case of the hammer, but the total acceleration is the same.
So, Nebula, you are incorrect. If you drop the feather and the hammer from the same height at the same time, they both hit the ground at precisely the same instant.
Of course I don’t fit in; I’m part of a better puzzle.
I don’t think Headless Cow missed the point at all. I think Cow’s formulas indicate that the acceleration of the earth towards the object is irrelevant, because the gravitational constant is… well… constant.
Cow’s comment about air resistance is the prelude to the comment about on the moon (replace “mass of earth” in the problem with “mass of moon”).
This is true but the gravitational force is not constant.
Lets make a closer look on how this will affect the accelerations.
F = G m_earth m_object / r^2
We can easily see that the force is in direct relation to the mass of the falling object. So the heavier the object the more force will act upon both the earth and the object. Now since
a_object = F/m_object
a_object = G m_earth / r^2
We can see that the acceleration on the object is constant and in no relation to it’s mass. ( it is in direct relation to the earths mass )
How ever the acceleration of earth will be
a_earth = G m_object / r^2
What we can learn from this.
The acceleration of the object is constant since it is depended on the earths mass
The acceleration of the earth is not constant since it is depended on the objects mass.
Therefore the total acceleration of the earth and the object toward each other is not constant
As Pleonast noted the acceleration of the earth is very small do to our normal live objects and could therefore be ignored since it is within atomic scales.
But hey this is physics and we can make use of the perfect world where everything is possible so lets say that the hammers mass is the same as the earths mass. Now both the earth and the hammer will have an acceleration of around 9.81 toward each other and meet much quicker than if we dropped a 2Kg hammer.
I wouldn’t want to live on the earth if this happened so please don’t try this at home.
By the way IF YOU DROP 2 OBJECTS THEY CAN’T BE AT THE SAME LOCATION.
Gravity is proportional to mass. Naturally the force acting on the hammer is greater because of the greater mass, but there’s more mass to move so it cancels out. That’s why the acceleration is the same, in vacuum that is. Of course, if you do that experiment in your living room, air resistance will interfere significantly with the fall of the feather.
That reminds me of this trick question: what’s heavier, a ton of bricks or a ton of feathers?
“The hammer and feather can’t be in the same place at the same time…”
Blah, blah, blah.
Hey, if you’re going to parse the two inches between the hammer and feather, what are you going to do with the distance between the mass at the top of the hammer with the mass at the bottom of the hammer, which are surely farther than two inches apart?
What if all the atomic bonds between the molecules in the hammer suddenly came apart in midfall? Would that change the acceleration of the ‘hammer’ over all? Would each seperate molecule have a unique acceleration?
Whether the parts are linked together or not, they’re all not in the same place at the same time, so, if you’re going to split hairs in this experiment, you might as well split quarks.
Nebula, scroll back up a bit, and this time read my post before trying to talk about it.
Assuming you can measure accurately enough, the acceleration of any dropped object is slightly less than 9.8 m/s/s. The Earth also accelerates slightly “upward”, toward the object. Together, these two acclerations will always add up to 9.8 m/s/s.
I’ll use your own example. If you drop a hammer with the same mass as the Earth, it will accelerate toward the planet at 4.9 m/s/s. The Earth will accelerate toward the hammer at 4.9 m/s/s. The total acceleration is 9.8 m/s/s.
To reiterate: the acceleration due to gravity depends solely on the mass of the Earth. It doesn’t matter whether you drop a hammer, a feather, a marble or the Moon; they will all experience the exact same acceleration.
Of course I don’t fit in; I’m part of a better puzzle.
You know, if y’all were a little more open to listening, a little less open to being pissed off at people who don’t accept what you say as definitive, and a lot more willing to be specific and comprehensive in your posts, this place might actually solve some questions
I propose for consideration the following problem which should solve the issue. In one case, we have two point masses of equal mass at rest a distance x apart. In the other case, we have two point masses of equal mass at rest the same x apart, but in this case, the masses are 100 times as heavy as the first case.
Now, if I read what Nebula is saying correctly, he feels the smaller masses will take less time to meet as a result of the attraction of gravity than the larger masses. Nebula then carries this over to the feather and the hammer, noting that the combined mass of the Earth-hammer system is greater than the combined mass of the Earth-feather system, if only infinitismally so. More mass = faster acceleration.
Now Aura is saying that the rate of accelleration is not affected by the difference in mass between the hammer and the feather, being totally dependent on the mass of the Earth. In practicality, this is correct. But for Nebula to be ‘wrong,’ then the rate of accelleration for the Earth-hammer system has to be the same as for the Earth-feather system. If this is true, then the time for the two masses to meet in each case in my problem would be the same.
NOW, please, someone solve the problem posed and relate it back to the original post with something OTHER than shrill rhetoric?
This is not true and If you carry out calculation you will find that out.
Now think about this. What if the hammer was the mass of the sun? will the total acceleration still be 9.8 m/s/s even though the gravity of the sun is much more? Is the gravity of the earth some universal constant in your physics? Why don’t 2 soap bars attract each other at the same rate?
Thanks DSYoungEsq for your posting. The two 100times heavier objects will meet faster and I can do calculations for you if you want but I think you already know.
The Universal Law of Gravitation doesn’t tell you the attraction of the earth on the hammer or feather…It tells the force of the attraction between the two objects. It already takes into account the attraction of the earth on the hammer and the attraction of the hammer on the earth. It doesn’t matter how large or small the object being attracted is there is ALWAYS the same acceleration in a perfect case. As the mass increases the force of gravitation increases so does the inertia of the object and they increase at the same rate. This is why the acceleration is constant.
The falling mass is the one that is canceled out and since it is the smaller mass that is percieved as falling in the example of the earth and the sun the earth would be the falling object and the acceleration would then depend on the mass of the sun.
