Uploaded by JUSTANEMONERD on 08.12.2012

Transcript:

Today's screencast is on some basics you'll need for 3.091.

And the first basic has to do with Newton's first law.

Newton recognized that in order to get a body to change

its momentum, you have to apply a force. So mathematically, we write that this way.

So the little arrow above each-- the force and the momentum--

means those are vector quantities. They have a direction to them.

And the derivative means the change in time. So on the right hand side, you see that the

momentum is changing in time.

And on the left hand side, you see it's equal to the force.

So Newton said, the force required to change your

momentum a given amount in a given time is given by this equation.

Now, momentum is a vector quantity, and it's really just

the product of the mass of an object times its velocity.

And velocity, of course, is a vectoral quantity. A body is moving in a particular direction

with a particular speed.

And you can see that the momentum has exactly the same

direction as the velocity. So you see, substitute that into this expression,

you have that the force equals the mass times how fast

the velocity is changing with time.

And we say that the acceleration is equal to the

velocity's change with respect to time. And so now you get the familiar equation--

sorry, that should be vectoral quantity, also-- m times a.

Now, lots of times, the body's motion is confined to one

particular dimension. So oftentimes, you'll see this written as

F equals ma, recognizing that if the body's confined to

motion in one direction, these don't need to be vectors

anymore. They're just scalar quantities.

Now, the units are very important here. So we always want you to keep track of them.

If force equals ma, then the units should match on each side.

So the mass, of course, has units of kilograms. And acceleration has units of--

velocity is meters per second, divided by time.

So then the units of acceleration had better be

meters per second per second. So you put the square there.

So that means the units of force had better be kilograms

meters per second squared. We call that funny combination of units--

in MKS-- we call that the Newton, in honor of Sir

Isaac Newton, of course. The situation that we'll run into when forces

and momentum come up has to do with bodies circulating

or orbiting around another.

So let's say we have the nucleus of an atom. As you know, one old model of the situation

in an atoms is that we have an electron in an orbit around

the nucleus. And that electron--

in an orbit, of course-- has a velocity. Let's call it V1.

In the next instant, the electron will have moved in

its orbit, and now it'll be here. We'll call that V2.

Now, the magnitude-- if it's in the stable orbit, of course--

doesn't change. The velocity is staying constant as far as

its magnitude goes.

But it has to, obviously, change direction. Now, remember, in a vectoral sense, the force

and acceleration are both vectoral quantities.

Acceleration is the change in a velocity versus time.

That means any change, not just in magnitude, but also in direction.

So this orbiting body, even though the velocity magnitude

is not changing, its velocity direction is. So there has to be a force being placed on

the electron as it orbits.

We call that the centripetal force. And its magnitude has to be the mass--

of the electron, in this case, or any orbiting body--

times the acceleration that occurs due to this orbital motion.

And, without doing all the mathematics, it turns out that

the centripetal acceleration, as we call it, depends on the

magnitude of the velocity. So V here, without the vector sign, just means

the magnitude of the velocity divided by the radius of the

orbit. So that means the centripetal force to keep

it in a stable orbit is the mass times the acceleration.

So there you have it. The key equations here, of course, are this

and this.

And the first basic has to do with Newton's first law.

Newton recognized that in order to get a body to change

its momentum, you have to apply a force. So mathematically, we write that this way.

So the little arrow above each-- the force and the momentum--

means those are vector quantities. They have a direction to them.

And the derivative means the change in time. So on the right hand side, you see that the

momentum is changing in time.

And on the left hand side, you see it's equal to the force.

So Newton said, the force required to change your

momentum a given amount in a given time is given by this equation.

Now, momentum is a vector quantity, and it's really just

the product of the mass of an object times its velocity.

And velocity, of course, is a vectoral quantity. A body is moving in a particular direction

with a particular speed.

And you can see that the momentum has exactly the same

direction as the velocity. So you see, substitute that into this expression,

you have that the force equals the mass times how fast

the velocity is changing with time.

And we say that the acceleration is equal to the

velocity's change with respect to time. And so now you get the familiar equation--

sorry, that should be vectoral quantity, also-- m times a.

Now, lots of times, the body's motion is confined to one

particular dimension. So oftentimes, you'll see this written as

F equals ma, recognizing that if the body's confined to

motion in one direction, these don't need to be vectors

anymore. They're just scalar quantities.

Now, the units are very important here. So we always want you to keep track of them.

If force equals ma, then the units should match on each side.

So the mass, of course, has units of kilograms. And acceleration has units of--

velocity is meters per second, divided by time.

So then the units of acceleration had better be

meters per second per second. So you put the square there.

So that means the units of force had better be kilograms

meters per second squared. We call that funny combination of units--

in MKS-- we call that the Newton, in honor of Sir

Isaac Newton, of course. The situation that we'll run into when forces

and momentum come up has to do with bodies circulating

or orbiting around another.

So let's say we have the nucleus of an atom. As you know, one old model of the situation

in an atoms is that we have an electron in an orbit around

the nucleus. And that electron--

in an orbit, of course-- has a velocity. Let's call it V1.

In the next instant, the electron will have moved in

its orbit, and now it'll be here. We'll call that V2.

Now, the magnitude-- if it's in the stable orbit, of course--

doesn't change. The velocity is staying constant as far as

its magnitude goes.

But it has to, obviously, change direction. Now, remember, in a vectoral sense, the force

and acceleration are both vectoral quantities.

Acceleration is the change in a velocity versus time.

That means any change, not just in magnitude, but also in direction.

So this orbiting body, even though the velocity magnitude

is not changing, its velocity direction is. So there has to be a force being placed on

the electron as it orbits.

We call that the centripetal force. And its magnitude has to be the mass--

of the electron, in this case, or any orbiting body--

times the acceleration that occurs due to this orbital motion.

And, without doing all the mathematics, it turns out that

the centripetal acceleration, as we call it, depends on the

magnitude of the velocity. So V here, without the vector sign, just means

the magnitude of the velocity divided by the radius of the

orbit. So that means the centripetal force to keep

it in a stable orbit is the mass times the acceleration.

So there you have it. The key equations here, of course, are this

and this.