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A particle of mass $m$ moves under the influence of apotential energy$$U(x)=\frac{a}{x}+b x$$where $a$ and $b$ are positive constants and the particle isrestricted to the region $x>0 .$ Find a point of equilibriumfor the particle and demonstrate that it is stable.

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$\sqrt{a / b}$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

University of Washington

Hope College

University of Sheffield

McMaster University

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.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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so there is a particle off mass M and it moves under the influence of a potential energy function which is given by you. Effects is a by X Blaise BX. You can be a positive and it is moving a region which is X is greater than zero. Find the point of equilibrium for the particle and we need to demonstrate that at this table. So the a point off stable equilibrium. So we'll have. Do you buy D X should be close to zero and then we should have d'you or were the second directive should be greater than zero, indicating a minimum in the potential energy function. So this is we're playing the minimal rule in this location here. So if you effects is equals to a over X less B X, then the derivative of that which is d'you buy DX should be close to minus a by X squared. Let's be, we said this equals to zero, which implies that X squared is equals to a over B or ex is equals to bless minus square root off a bi bi No, the problem restricts us, So the problem restricts X to be greater than zero. Therefore, we take the positive early effects which is square root off if I be so now we find the second derivative off d two x by d. X two at X ISS equals to a square root off a bi bi so second derivative we'll give you to a buy x cubed. If he saw the signature and this is zero well, the dirty would be the second door over there was to it by X cubed and we'll find the value at excess. He called the square root off a bi bi That should be do a by a bee three by two. Now, if you can see that I'm just plugging in the value off ex witches A upon be half to the square to the part of half and multiplied by three here, as you can see So this gives you three by two and this is greater than zero, as you can see. So therefore the point and X is equals two square root off a bi bi gives a minimum in the potential energy function and that is her answer for this

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