Introduction to Newton's Law of Gravitation A little bit on gravity
Introduction to Newton's Law of Gravitation
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- We're now going to learn a little bit about gravity.
- And just so you know, gravity is something that, especially
- in introductory physics or even reasonably advanced
- physics, we can learn how to calculate it, we can learn how
- to realize what are the important variables in it, but
- it's something that's really not well understood.
- Even once you learn general relativity, if you do get
- there, I have to say, you can kind of say, oh, well, it's
- the warping of space time and all of this, but it's hard to
- get an intuition of why two objects, just because they
- have this thing called mass, they are
- attracted to each other.
- It's really, at least to me, a little bit mystical.
- But with that said, let's learn to deal with gravity.
- And we'll do that learning Newton's Law of Gravity, and
- this works for most purposes.
- So Newton's Law of Gravity says that the force between
- two masses, and that's the gravitational force, is equal
- to the gravitational constant G times the mass of the first
- object times the mass of the second object divided by the
- distance between the two objects squared.
- So that's simple enough.
- So let's play around with this, and see if we can get
- some results that look reasonably familiar to us.
- So let's use this formula to figure out what the
- acceleration, the gravitational acceleration, is
- at the surface of the Earth.
- So let's draw the Earth, just so we know what
- we're talking about.
- So that's my Earth.
- And let's say we want to figure out the gravitational
- acceleration on Sal.
- That's me.
- And so how do we apply this equation to figure out how
- much I'm accelerating down towards the center of Earth or
- the Earth's center of mass?
- The force is equal to-- so what's this big G thing?
- The G is the universal gravitational constant.
- Although, as far as I know, and I'm not an expert on this,
- I actually think its measurement can change.
- It's not truly, truly a constant, or I guess when on
- different scales, it can be a little bit different.
- But for our purposes, it is a constant, and the constant in
- most physics classes, is this: 6.67 times 10 to the negative
- 11th meters cubed per kilogram seconds squared.
- I know these units are crazy, but all you have to realize is
- these are just the units needed, that when you multiply
- it times a mass and a mass divided by a distance squared,
- you get Newtons, or kilogram meters per second squared.
- So we won't worry so much about the units right now.
- Just realize that you're going to have to work with meters in
- kilograms seconds.
- So let's just write that number down.
- I'll change colors to keep it interesting.
- 6.67 times 10 to the negative 11th, and we want to know the
- acceleration on Sal, so m1 is the mass of Sal.
- And I don't feel like revealing my mass in this
- video, so I'll just leave it as a variable.
- And then what's the mass 2?
- It's the mass of Earth.
- And I wrote that here.
- I looked it up on Wikipedia.
- This is the mass of Earth.
- So I multiply it times the mass of Earth, times 5.97
- times 10 to the 24th kilograms-- weighs a little
- bit, not weighs, is a little bit more massive than Sal--
- divided by the distance squared.
- Now, you might say, well, what's the distance between
- someone standing on the Earth and the Earth?
- Well, it's zero because they're touching the Earth.
- But it's important to realize that the distance between the
- two objects, especially when we're talking about the
- universal law of gravitation, is the distance between their
- center of masses.
- For all general purposes, my center of mass, maybe it's
- like three feet above the ground, because
- I'm not that tall.
- It's probably a little bit lower than that, actually.
- Anyway, my center of mass might be three feet above the
- ground, and where's Earth's center of mass?
- Well, it's at the center of Earth, so we have to know the
- radius of Earth, right?
- So the radius of Earth is-- I also looked it up on
- Wikipedia-- 6,371 kilometers.
- How many meters is that?
- It's 6 million meters, right?
- And then, you know, the extra meter to get to my center of
- mass, we can ignore for now, because it would be .001, so
- we'll ignore that for now.
- So it's 6-- and soon.
- I'll write it in scientific notation since everything else
- is in scientific notation-- 6.371 times 10 to the sixth
- meters, right?
- 6,000 kilometers is 6 million meters.
- So let's write that down.
- So the distance is going to be 6.37 times 10
- to the sixth meters.
- We have to square that.
- Remember, it's distance squared.
- So let's see if we can simplify this a little bit.
- Let's just multiply those top numbers first. Force is equal
- to-- let's bring the variable out.
- Mass of Sal times-- let's do this top part.
- So we have 6.67 times 5.97 is equal to 39.82.
- And I just multiplied this times this, so now I have to
- multiply the 10's.
- So 10 to the negative 11th times 10 to the negative 24th.
- We can just add the exponents.
- They have the same base.
- So what's 24 minus 11?
- It's 10 to the 13th, right?
- And then what does the denominator look like?
- It's going to be the 6.37 squared times 10
- to the sixth squared.
- So it's going to be-- whatever this is is going to be like 37
- or something-- times-- what's 10 to the sixth squared?
- It's 10 to the 12th, right?
- 10 to the 12th.
- So let's figure out what 6.37 squared is.
- This little calculator I have doesn't have squared, so I
- have to-- so it's 40.58.
- And so simplifying it, the force is equal to the mass of
- Sal times-- let's divide, 39.82 divided by 40.58 is
- equal to 9.81.
- That's just this divided by this.
- And then 10 to the 13th divided by 10 to the 12th.
- Actually no, this isn't 9.81.
- Sorry, it's 0.981.
- 0.981, and then 10 to the 13th divided by 10 to the 12th is
- just 10, right?
- 10 to the first, times 10, so what's 0.981 times 10?
- Well, the force is equal to 9.81 times the mass of Sal.
- And where does this get us?
- How can we figure out the acceleration right now?
- Well, force is just mass times acceleration, right?
- So that's also going to just be equal to the acceleration
- of gravity-- that's supposed to be a small g there-- times
- the mass of Sal, right?
- So we know the gravitational force is 9.81 times the mass
- of Sal, and we also know that that's the same thing as the
- acceleration of gravity times the mass of Sal.
- We can divide both sides by the mass of Sal, and we have
- the acceleration of gravity.
- And if we had used the units the whole way, you would have
- seen that it is kilograms meters per second squared.
- And we have just shown that, at least based on the numbers
- that they've given in Wikipedia, the acceleration of
- gravity on the surface of the Earth is almost exactly what
- we've been using in all the projectile motion problems.
- It's 9.8 meters per second squared.
- That's exciting.
- So let's do another quick problem with gravity, because
- I've got two minutes.
- Let's say there's another planet called the
- planet Small Earth.
- And let's say the radius of Small Earth is equal to 1/2
- the radius of Earth and the mass of Small Earth is equal
- to 1/2 the mass of Earth.
- So what's the pull of gravity on any object, say same
- object, on this?
- How much smaller would it be on this planet?
- Well, actually let me save that to the next video,
- because I hate being rushed.
- So I'll see you
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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