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The force required to compress an imperfect horizontal spring an amount $x$ is given by $F = 150 x + 12 x ^ { 3 } ,$ where $x$ is in meters and $F$ in newtons. If the spring is compressed $2.0 \mathrm { m } ,$ what speed will it give to a 3.0 -kg ball held against it and then released?

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$$=[15 \mathrm{m} / \mathrm{s}]$$

Physics 101 Mechanics

Chapter 7

Work and Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

University of Michigan - Ann Arbor

University of Washington

University of Winnipeg

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.

03:19

The force required to comp…

03:07

The magnitude of the f…

02:32

A spring of spring constan…

00:49

The force, $F$, required t…

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A $1.50-\mathrm{kg}$ objec…

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A vertical spring with a f…

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A 1.50 -kg object is held …

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In a physics lab experimen…

So we want to find the final, the work done and the final velocity. Rather, we need to find a final velocity. And we can do this by finding the amount of work done. So we know that the work is going to be equal to the change of kinetic energy. And you know what? The change of kinetic energy with equal 1/2 em times velocity final, squared, minus velocity initial squared. We know that philosophy National squared is going to be zero. So this will simply equal 1/2 and the final squared. In order to find the work we need to integrate from 0 to 2 of 150 x plus 12 x cube of thie x at other times the ex. So this legal 75 x squared plus three x to the fourth and again we're evaluating from 3 0 to 2 meters, and at this point, we can solve. So the work done is going to be 75 times two squared plus three times two to the fourth power and this is giving us 348 Jules. So at this point, let's keep note of this, and we know that the force final is going to be equal to two times the work divided by mass to the 1/2 power. Ah, this is simply alter break manipulating this top equation here. So and at this point, we can solve. So there will be two times 348 Jules, divided by the mass of 3.0 kilograms and then all to the 1/2 power. So the final philosophy is going to be equal to 15.2 meters per seconds. That'll be our final answer. That is the end of the solution. Thank you for watching.

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