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Let's consider another example of continuous mass transfer. Suppose we have a truck, and that truck has some type of plow. And it's plowing snow. And there's some type of external force acting on this truck, friction, pushing the truck forward, so let's just assume we have some type of force, F, on the truck. And this is our snow. And what's happening in this problem is that the truck connects-- picks up the snow. And then, which is at rest initially, gets the snow up to the speed of the truck. And then the snow falls off the plow. So how do we model this problem? Well, let's look at our situation at time t. And what we're going to do is, we're going to consider a certain mass of snow, delta ms, that's at rest. And our truck, it's a fixed mass truck, is moving with a velocity vt at time t, the truck. So now, what happens at time t plus delta t? Well, the truck has picked up the mass of the snow. And the truck has now changed its spe...

Let's consider what we call the window washer problem. What we have is suspended from some ceiling. We have a pulley. And the pulley is suspended by a rope, which we're going to call this string 3. And we have a rope that is wrapping around this pulley. And then it wraps around another pulley. So this rope is going around another pulley. And it's fixed to the ceiling. And this is what we're going to call string 1. And then this string, there's another string that comes down to a platform. And this one we're going to call string 2. And sitting on the platform is a person. So we have a person sitting on the platform. And that person is pulling the rope down. Now this is a very complicated problem. And it's a classic example of how do we choose systems so that we can apply Newton's second law. Now, one of the important things we're going to do is learn to see when we choose a system what forces are internal and external. And that will enab...

PROFESSOR: Let us consider the anharmonic oscillator, which means that you're taking the unperturbed Hamiltonian to be the harmonic oscillator. And now, you want to add an extra term that will make this anharmonic. Anharmonic reflects the fact that the perturbations are oscillations of the system are not exactly harmonic. And in the harmonic oscillator, the energy difference between levels is always the same. That's a beautiful property of the harmonic oscillator. That stops happening in an anharmonic oscillator. The energy differences can vary. So the things, if you have a transition from one level, first level to the ground state, or second level to the ground state, one is not the harmonic of the other because they're not exactly twice as big as each other. So let's try to add an x to the 4th perturbation, which is intuitively very clear. You have a potential. And you're adding now an extra piece that behaves like x to the 4th. And it's going to...

We will now consider a very practical application of the weak law of large numbers, and the calculations associated with it. The application has to do with polling. There's a certain referendum that's going to take place. We're close enough to the day of the referendum so that voters have made up their minds, and there is a fraction p of the population that represents the voters that are going to vote yes. But the referendum has not yet taken place, and you want to find out, to predict or estimate what p actually is. What you do is that you go ahead, and you select at random a number of people out of the population. And for each person, you record their answer, whether they intend to vote yes, or whether they intend to vote no. When we say that the people are randomly selected, what we mean is that we choose them uniformly from the population. And since there's a fraction p that will vote yes, this means that this random variable will be 1 with probability...

We will now consider an example that illustrates the difference between the notion of independence of a collection of events and the notion of pairwise independence within that collection. The model is simple. We have a fair coin which we flip twice. So at each flip, there is probability 1/2 of obtaining heads. Furthermore, we assume that the two flips are independent of each other. Let H1 be the event that the first coin toss resulted in heads, which corresponds to this event in this diagram. Let H2 be the event that the second toss resulted in heads, which is this event in the diagram-- the two ways that we can have the second toss being heads. Now, we're assuming that the tosses are independent. So the event heads-heads has a probability which is equal to the probability that the first toss resulted in heads-- that's 1/2-- times the probability that the second toss resulted in heads, which is 1/2. So the product is 1/4. We have probability 1/4 for this outcome....

We can consider a given coordinate system as a reference frame within which we can describe the kinematics of an object. By "the kinematics," I mean the position, the velocity, and the acceleration as a function of time, basically a geometric description of the motion. Some aspects of these kinematics will look different in different reference frames and I'd like to examine that now. First, I want to define what I mean by an "inertial reference frame." An inertial reference frame is one in which an isolated body, one with no net force acting on it, moves at constant velocity, where that constant velocity might be zero. Another way of saying this is that an inertial reference frame is one in which Newton's laws of motion apply. Recall that Newton's first law of motion states that an isolated object with no forces acting on it moves at constant velocity. So let's begin by considering an observer in a particular reference frame. We'll ...

Let's consider a very famous problem the, Atwood machine. We have a pulley, A, suspended from a ceiling. And a rope is wrapped around the pulley. And on each side of the rope, there's different masses. So here is block 1, and block 2, and we can say here-- it doesn't matter-- but we'll say that M2 is bigger than M1. And that gives us some intuition that we expect block 2 to go down and block 1 to go up. Now in this problem there is friction between the rope and the pulley, so the rope is not sliding. And what that means is that the pulley will rotate. And also the mass of the pulley is not 0. So these were all assumptions we made way back when were analyzing Newton's second law, but now we have to take into effect that there some rotational inertia to make the pulley start to have angular acceleration. So what we'd like to do is to identify our three objects-- mass 1, the pulley, and mass 2. And for mass 1 and mass 2, use Newton's second law, f...

Consider a rod of mass m, and suppose I apply a force to this rod. Let's say a force like that. So we know that as a result of that applied force, the center of mass of the rod, which we can imagine is right at the center the rod, will translate with some acceleration, such that the vector f is equal to the mass of the rod times the acceleration of the center of mass. Now recall that for a rigid body, this equation will be true regardless of where on the rod I apply the force. So for example, if I draw the rod again over here, and I apply the same force, vector f, but I apply it, let's say, on the right hand side, I'll still have the same f equals ma. All I specify with f is its magnitude and direction. But for a rigid body, no matter where on the body I apply that force, the acceleration of the center of mass will be the same. Now, we know from experience, however, that the motion of the rod is different if I push it at the center at one end or at the other e...

Let's consider the motion of a car on a circular track, and the track is frictionless. And it's also banked. So this is the overhead view of our circular track. It has radius, r. And here's our car moving at a constant velocity. Now, from the side view when we want to look at that bank turn-- let's draw a side view. So here's our side view, and the car is moving with a velocity into the plane of the figure. Now this surface here is frictionless. And what we'd like to do is find out what speed the car can move such that it doesn't slide up or down the inclined plane. So how should we analyze that? Well our approach will be to apply Newton's second laws. Now what's very important to realize is this is circular motion. And for circular motion we know that the car is accelerating towards the center of the circle. Now from the side view, towards the center of the circle is in this direction. So the car is accelerating radially inward. And th...