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As an object moves along the $x$ axis from $x = 0.0 \mathrm { m }$ to $x = 20.0 \mathrm { m }$ it is acted upon by a force given by $F = \left( 100 - ( x - 10 ) ^ { 2 } \right) \mathrm { N }$ . Determine the work done by the force on the object: $( a )$ by first sketching the $F$ vs. $x$ graph and estimating the area under this curve; $( b )$ by evaluating the integral $\int _ { x = 0.0 \mathrm { m } } ^ { x = 20 \mathrm { m } } F d x$ .

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a.$1000 \mathrm{J}$b.$1330 \mathrm{J}$

Physics 101 Mechanics

Chapter 7

Work and Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Rutgers, The State University of New Jersey

Simon Fraser University

Hope College

Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

03:47

In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The kinetic energy of a rotating object is the sum of the kinetic energies of the object's parts.

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A force $\vec{F}$ in the p…

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force shown in the figure …

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(II) The force on a partic…

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The force acting on a part…

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so this would be the diagram. This would be the graph, rather. And for part A. They want us to find the work by approximating the area. So here the work is simply going to be the area under a force versus displacement graph. So here it would simply be this area and the magnitude of this area of other the ah, the quantity. Oh, are like the Ah, my apologies. The area here under the Graff would be equal to the amount of work done. So if we could approximate this is a triangle. We could say that the work done is going to equal half times base times, height. And here we're going to say that work is going to equal 1/2 times base here, the base again. We're going to call it 20 meters, and then the height would be, ah, 100 Newtons. So the work here is going to be equal to a approximately 1000 jewels. If we approximate it as a triangle Now, Teo, get the actual work. We can integrate the force function. So the work is going to be equal to ex initial the integral from ex initial tow X final of act of f of x times DX. So we can say that the work is going to be equal to the square to the interval of 0 to 20 zero meters, 2 20 meters of 100 minus acts of minus 10 squared D X. So the work is going to be equal to 0 to 20 the integral from 0 to 20 of 20 x minus X squared time's DX and so we can solve and integrate here and say that we're going to add one to the power divide and say that the work is going to be equal to 10 x squared minus X cubed over three, evaluated at zero and at 20 and this will equal 10 times 20 squared, minus 20 cube over three. So the work done here is going to be 1300 and 33 jewels. So the approximation that we made 1000 Jules is a bit low and ah, it makes sense because a triangle would actually hey essentially be shaped like this and here would be missing all of this area here if we did approximated as a triangle. But by integrating, we can get the true value of the total work done. And again, that answers 1330 each. Three jewels. That is the end of the solution. Thank you for watching.

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