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Calculus - Example 2

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The notion of a function is one of the most fundamental concepts in mathematics. An example is the function that relates each real number x to its square. The output of a function f corresponding to an input x is denoted by f(x), which is read as "f of x" or "f at x", or simply "f of x", when the context makes it clear. Functions of various kinds appear in many areas of mathematics, and their study is one of the central topics of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, or it could be described explicitly, as a formula or as a graph. The input and output of a function could be real numbers, the integers, a subset of the rational numbers, a set of real numbers, or more general objects such as vectors. The set used to define a function is called the domain of the function. The set of permissible outputs

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Eric M.

When an object is rolling without slipping, the rolling friction force is much less than the friction force when the object is sliding; a silver dollar will roll on its edge much farther than it will slide on its flat side (see Section 5.3). When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that $a_x$ and $a_z$ are approximately zero and $v_x$ and $\omega_z$ are approximately constant. Rolling without slipping means $v_x = r\omega_z$ and $a_x = r\alpha_z$ . If an object is set in motion on a surface $without$ these equalities, sliding (kinetic) friction will act on the object as it slips until rolling without slipping is established. A solid cylinder with mass $M$ and radius $R$, rotating with angular speed $\omega_0$ about an axis through its center, is set on a horizontal surface for which the kinetic friction coefficient is $\mu_k$. (a) Draw a free-body diagram for the cylinder on the surface. Think carefully about the direction of the kinetic friction force on the cylinder. Calculate the accelerations $a_x$ of the center of mass and $a_z$ of rotation about the center of mass. (b) The cylinder is initially slipping completely, so initially $\omega_z = \omega_0$ but $v_x =$ 0. Rolling without slipping sets in when $v_x = r\omega_z$ . Calculate the $distance$ the cylinder rolls before slipping stops. (c) Calculate the work done by the friction force on the cylinder as it moves from where it was set down to where it begins to roll without slipping.

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Physics 101 Mechanics

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Video Transcript

Welcome to our math review video where we're going to talk about how the plots of derivatives and integral czar important in physics. Um, in particular. This is going to be, uh this is gonna be an instructive, uh, instructive video for those of you who are just starting out doing the first unit in physics because this almost always will show up. So what I'm talking about here is if I have a function f of X versus X, and I say, Okay, my function looks like this. And I say, Well, if I took the derivative of that, I would just get zero. If I took the integral of it, I would get some number. But what does it mean? Okay, in physics were really worried about What does it mean? Why would we apply this? Well, the reason we would apply it might be to say, if a knob checked is moving in this case, we might say, What if I have an object for which its position looks something like that? So we're plotting against t here. Eso its position looks something like this. And I say, Well, what if I take the derivative of that I've already told you that the derivative of position with respect to time turns about to be speed. Okay, well, the speed then we see we have a constant slope, which means we're gonna have a constant speed. Okay, so that's that's very interesting. Were able to go from the picture that we have for position down to the picture that we have for speed. Similarly, if I were to say, I have speed with respect to time and it looks something like this and say I want to take the integral from here Thio here. Well, what I'm going to do is when I take the integral from here to here, I'm going to get that much. When I take it from here to here, I'm going to get a slightly more amount. When I take it from here to here, I'm going to get more. And so it's increasing at a constant rate, which means that if it's increasing a constant rate than my position, which remember, position is equal to the integral of speed with over speed, with respect to time, then my position is going to be changing at a constant rate because we were as we took the integral. We got a standard increase. Okay, so this is this is kind of an interesting relationship here, and we can export more, and we will explore a lot more when we come to cinematics. So, for example, what if we were to have some chicken a metric functions instead? Stay. We have sign of X versus X, and I want to take the derivative of this well, so we remember it looks like this. And if I want to take the derivative of it, then I can see well, here, it's gonna be a maximum in positive. Here it will be. Zero here will be a maximum and negative. Here it will be zero, and I start to come up with the cosine function, finding that the derivative of sign will be cosign again, you might say, Well, where does this apply? Well, when we do Oslo Torrey emotion, we'll find that the position is described by a sign or a cosine function. And in order to find its speed with respect to time, will take the derivative and come up with either a cosine or negative sign. So, uh, the plots can tell us a lot and being able to look at a plot and figure out the derivative or integral of that plot just by staring at it can give you a lot of insight into what the answer to your question will be. I really recommend that you practice this with a couple of well known functions, things like Pollen Oh, Meals during the metric functions and exponential.

RC
University of North Carolina at Chapel Hill
Top Physics 101 Mechanics Educators
Andy C.

University of Michigan - Ann Arbor

Zachary M.

Hope College

Aspen F.

University of Sheffield

Meghan M.

McMaster University