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A constant force $\mathbf{P}$ is applied to a piston and rod of total mass $m$ to make them move in a cylinder filled with oil. As the piston moves, the oil is forced through orifices in the piston and exerts on the piston a force of magnitude $k v$ in a direction opposite to the motion of the piston. Knowing that the piston starts from rest at $t=0$ and $x=0,$ show that the equation relating $x, v,$ and $t$ where $x$ is the distance traveled by the piston and $v$ is the speed of the piston, is linear in each of these variables.

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05:29

Eric Mockensturm

Physics 101 Mechanics

Chapter 12

Kinetics of Particles: Newton’s Second Law

Newton's Laws of Motion

Cornell University

Simon Fraser University

University of Winnipeg

McMaster University

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

09:37

Isaac Newton (4 January 1643 – 31 March 1727) was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

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ported to find the relation between X we anti we apply Newton's law in horizontal direction. There, the summer forces in FX direction is equal. Teoh, i m a mass times acceleration with, um we can write to some of these forces in X axis. Direction will be be minus ah que times we is equal to Emmy. Then from here we will solve for explosion. So exploration will be then simply b minus care We divided by, um Moss. Then we know from the definition off exploration that is exploration is a call to rate of change or for lost e d video entity. Um then we will replace the equation that brought here the let's call this question one. And this is questionable too. We subsitute two into one. Then we get P minus K. We divided by Momus. Is it cool to Devi, divided by D. T from here, Then we solve for a GT. So Aditi will be then Moss divided by minus P minus K b times a story This is K times Ah Devi. So let's call this equation number three in next step we integrate DT from zero to t um so integrating DT from 0 to 18 0 to t d t is 1/4 we integrate right inside from zero to be moss divided by minus B minus k. We do you reading this integral? We get t year left inside on. Right inside we get minus in Divided by K. Ellen off our P minus Khairi divided by P. Then we solve for we so we will be equal. Teoh be divide by K and minus P divided by K exponential So exponential minus Katie divided by him Let's call this equation number four Then we know from the definition off off the last e which is the rate of change of displacement. So we'll use that definition and right in we So we we is equal to the X divided by DJ. Next step all let's call this cuisine more. Five. The next step we substitute five into four. So the relation that we get here is DX divided by D t physical to pee or K minus P or K exponential in two K minus Kenny Times t divided by him then, uh, or next task is to integrate the d X problem zero to x and find X So, um, integrating from zero to x d. X integrate right inside from zero to t soapy or K minus P or K Exponential ar minus Katie defined by AM Oh, this whole times DT year. Then you're reading this integral we get X to be ah pt divided by K plus p divided by em into one minus G exponential power. Um, Katie divided by him. Let's call this equation number six. Then we substituted questionable foreign to six and finally find the relation for X B. Ah, tnk So that is X is equal. Teoh pt divided by K minus. Okay, we divided by him. So this is the relation between X, we and T, which is a linear equation.

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