Physics Lecture: Acceleration of a Free-Falling Body


Uploaded by cafedurhamcollege on 10.08.2012

Transcript:
Ok. We saw how the picket fence dropped and we
got some data. And it looked like there was a constant acceleration
just under 10 metres per second squared. So let’s talk about where that acceleration
comes from. Now there’s two things I want to take a
look at to explain a little bit about gravity. The first is, Newton’s Second Law.
Now Newton’s Second Law stated that, if you have an object and an unbalanced force
acts on it, like a push, then it will accelerate in the direction of that unbalanced force.
So we’ll have an acceleration. Now the amount of acceleration is directly
proportional to the amount of force. Um...push harder, you get more acceleration.
So the greater the force, the greater the acceleration.
But, also the acceleration is inversely proportional to the mass.
That is the larger the object, the smaller the acceleration you’ll get for the same
amount of push. So if you got a car stuck in the snow you
give it a push, you get it moving a little bit with an acceleration.
You try the same push to a massive truck, you’ll get much less acceleration from the
larger mass. So the relationship between acceleration,
force, and mass, is that acceleration is directly proportional to force, and inversely proportional
to the mass. Now we can go from this relationship to an
equation, acceleration equals force over mass, or if we rearrange it a little bit, force
equals mass times acceleration. And that’s what we usually see as Newton’s
Second Law. Now often, I’ll just mention, um...we won’t
always get such an easy conversion from a relationship to a formula.
The reason this converts so easily is that the units of force are defined in terms of
mass and acceleration. So in the SI system, the unit of force is
the Newton and 1 Newton is defined as the force needed to give a 1 kilogram mass an
acceleration of 1 metre per second squared. So 1 Newton equals a kilogram metre per second
squared. So we get that kind of combinational unit
which we prefer combined units, or in this case we use Newtons.
So 1 Newton is a kilogram metre per second squared and we got our idea of force, mass,
and acceleration. Now let’s see what happened to the picket
fence. Now of course, in the picket fence we were
dealing with gravity. Now gravity, we have a form for gravity, the
force due to gravity is G, m1, m2 over r squared. So let’s take a look at this.
If we have two masses separated by a distance. So we’ve got mass 1, mass 2, and separated
by a distance r. The equation for the force is mass 1 times
mass 2 over r squared. So again we’ve got the um...square of the
distances we feel, as we find with most field forces.
Now in this equation we also have this G. Now this G, you can think of it as a conversion
factor. It’s just a number, for the SI system the number is 6.67 times 10 to the negative
11. And what this is a conversion factor.
It allows us to use uh...kilograms for the masses, and metres for the distance, and when
we do we’ll get an answer in Newtons. So it’s just a conversion factor that allows
us to use the SI units. If we were using the CGS system, or if we’re
using foots, feet and pounds, uh...we’d use a different value of G.
It you look up, there’s different values of G depending upon which system you’re
using. But for the SI system, G is approximately
6.67 times 10 to the negative 11. It does have a unit theoretically, uh...not
a single unit, but it’s described as Newton kilograms, sorry, Newton metres squared over
kilograms squared. And what happens is when G is multiplied,
the metres squared cancels the metres squared in the denominator.
The kilograms squared cancels the 2 kilograms multipled to each other, leaving behind just
Newtons, which is what we are trying to calculate. So it theoretically has a complex unit that
allows for the cancellation. And of course, that’s why it’s there,
so that the units work out. So that’s our equation, force, so let me
just clean it up a bit and rewrite it, force equals G mass 1 mass 2 over distance squared.
Now let’s take a look at what happened with our experiment.
When we involve the Earth in the gravity, Earth acts as one of the masses.
So I’ve got the mass of the Earth which is 5.97 times 10 to the 24 kilograms.
And I’ve got the picket fence that we dropped for our experiment.
So we had the piece of plastic, the picket of fence, as one of the masses.
So we’ve got our two masses and their, because they have mass, they will have a gravitational
traction towards each other; of course, the Earth being much, much larger.
Now the distance between the two objects, we could say is basically the radius of the
Earth. Now true, theoretically the um...we’re above
Earth when we did the experiment, but by a couple dozen metres.
And, of course, the Earth isn’t a perfect sphere, so the radius is an approximation.
The radius of the Earth is, excuse me, the radius of the Earth is approximately 6.38
million metres. So about 6.38 million metres from the centre
of the Earth to where the picket fence was when we dropped it.
So we’re going to approximate that the distance between the two objects is the same as the
radius of the Earth. And this is basically true for any object
near or about the surface of the Earth. So using these, our equation for the force
of gravity between the picket fence and the Earth becomes G times the mass of the picket
fence, I’ll use lower case m for that, times the mass of the Earth divided by the radius
of the Earth squared. So in our case in dropping the picket fence,
that becomes equation for the force. Now I can rearrange this a little bit and
see um...sort of an unique situation. If I rearrange this, I get force equals lower
case m, the mass of the picket fence, times G times ME.
So up here in the denominator I haven’t really changed things, I’ve changed the
order, but in multiplication order doesn’t matter.
So G times m times ME is the same as m time G times ME.
Then I still have rE squared. You notice I don’t have the divisional line
going all the way through, but that doesn’t matter.
M times G ME over rE squared is the same as my original equation.
So I’ve changed the format, but I haven’t changed the value.
So notice what happens when we do this. We’ve now got this equation in this form,
force equals mass of the picket fence times G ME over rE squared.
Now this looks just like Newton’s Second Law.
Force equals mass times acceleration. Only for acceleration, I’ve got G ME over
rE squared. And those are all constants.
I know the mass of the Earth, I know the radius of the Earth, and I know G.
So I can actually figure out what acceleration that picket fence was feeling.
If I do the calculation, G times the mass of the Earth over the radius of the Earth
squared, I get 6.67 times 10 to the negative 11 times the mass of the Earth about 5.97
times 10 to the 24 kilograms divided by the radius of the Earth 6.38 million metres squared.
And when I put that through my calculator, I get about 9.81.
Now that is in the same mathematical position as acceleration was in the original formula.
And this is where algebra is used a lot in physics.
If we can rearrange an equation so it’s similar to another equation we can gain some
insights into the relationship. So it holds the same place as acceleration,
but is it the same? Uh...let’s do a unit analysis.
Uh...I’ve got G. Now we mentioned that the units for G are
Newton, Newtons times metres squared over kilograms squared.
So there’s G. Then I had the mass of the Earth.
The mass of the Earth is in kilograms. And then I had the radius of the Earth which
is in metres, and because I am squaring it, I am going to have metres squared.
So if I do a dimensional analysis, and do a little bit of cancelling, I’ve got the
metres squared, cancels out or divides into the metres squared in the numerator.
I’ve got the kilograms cancelling out one of the two kilograms, and I am left with Newtons
over kilograms. Now that’s not yet acceleration.
So I’ve got to break it down a little bit more.
Newtons, we defined Newtons, we go way back up here, and we said that 1 Newton is when
a kilogram, 1 kilogram mass is accelerated a metre per second squared.
So if I take that, 1 Newton is a kilogram times metre per second squared, the kilograms
cancel out and when I do my dimentional analysis I end up with metres per second squared.
So when I do this calculation, it comes down to a number which is in metres per second
squared which is the correct units for acceleration. And what this tells us is that anything on
the surface of the Earth, no matter what it’s mass is, remember we had the formula force
equals mass times acceleration, were this acceleration was G ME over rE squared.
The mass of the object that’s being dropped, in our case the picket fence, has no influence
on the acceleration. The acceleration is controlled by G ME over
rE squared. So the mass of the object being dropped does
not influence the acceleration. It certainly influences the force.
You drop something heavy, it’s going to hit hard.
You drop something light, it won’t hit with as much force.
But, the acceleration is the same. The planet doesn’t change if we change objects.
If I drop a ping pong ball and then I drop a bowling ball,
two completely different weights, in between the two drops the mass of the planet doesn’t
change, and the radius of the planet doesn’t change, so the acceleration due to our planet
will not change. So for any object, basically on the surface
of the Earth, or close to the surface of the Earth, we will see a constant acceleration
of approximately 9.81 metres per second squared. That will vary slightly.
You are on top of a mountain, you’re going to experience a slightly different gravity
than at the bottom. We can measure it.
It is a measurable slight difference. Uh...if you are on a boat over a deep ocean
compared to a standing on land over say a granite deposit you’re actually going to
experience different, slightly different amount of gravity.
And we actually use that in, um...in seismic surveys and oil and gas research.
So that’s our acceleration. We have found then that whenever we drop an
object it will accelerate at a constant 9.8 metres per second squared.
No matter what it’s mass is. Go to a different planet.
Go to the Moon, go to Mars, it’ll be different because we changed the planetary radius and
the planetary mass. But, for Earth our acceleration is 9.8 metres
per second squared. (Music)