A conservative force F⃗=(6.0 x-12) î N, where x is in meters, acts on a particle moving along an

A conservative force F⃗=(6.0 x-12) î N, where x is in...

A conservative force $\vec{F}=(6.0 x-12) \hat{\mathrm{i}} \mathrm{N}$, where $x$ is in meters, acts on a particle moving along an $x$ axis. The potential energy $U$ associated with this force is assigned a value of $27 \mathrm{~J}$ at $x=0$. (a) Write an expression for $U$ as a function of $x$. with $U$ in joules and $x$ in meters. (b) What is the maximum positive potential energy? At what (c) negative value and $(\mathrm{d})$ positive value of $x$ is the potential energy equal to zero?

A conservative force $\vec{F}=(6.0 x-12) \hat{\mathrm{i}} \mathrm{N}$, where $x$ is in meters, acts on a particle moving along an $x$ axis. The potential energy $U$ associated with this force is assigned a value of $27 \mathrm{~J}$ at $x=0$.
(a) Write an expression for $U$ as a function of $x$. with $U$ in joules and $x$ in meters. (b) What is the maximum positive potential energy? At what (c) negative value and $(\mathrm{d})$ positive value of $x$ is the potential energy equal to zero?

Key Concepts

Finding Extrema and Zeros of the Potential Energy Function

Analyzing the potential energy function may involve identifying its maximum or minimum values and determining the points where it becomes zero. This is done by finding the critical points, which are locations where the derivative of the potential energy (the negative of the force) is zero, or by directly solving the potential energy equation for zero. Such analysis is important for understanding stability and the energy landscape in a mechanical system.

Boundary Conditions and Determination of the Integration Constant

The integration of the force function to find potential energy results in an arbitrary constant, which represents the zero of potential energy. To uniquely determine this constant, a specific boundary condition or reference point is provided (such as a given value of U at a specific position). Utilizing this additional information ensures that the potential energy function accurately reflects the physical situation.

Integration with Respect to x

When deriving the potential energy from a given force function, the process involves integrating the force with respect to the position variable. Since the force equals the negative derivative of the potential energy, integrating the negative of the force yields the potential energy function. This mathematical operation is essential in transitioning from force analysis to energy analysis.

Relation Between Force and Potential Energy

The potential energy function U is related to the conservative force F through the equation F = –dU/dx in one-dimensional motion. This relationship means that by integrating the force with respect to position, one can obtain the potential energy function (up to an arbitrary constant). This procedure is crucial in converting force-based descriptions of physical systems into energy-based descriptions.

Conservative Force

A conservative force is one for which the work done in moving an object between two points is independent of the path taken. This property allows the existence of a scalar potential energy function U, where the force can be expressed as the negative gradient of U. This concept is fundamental in mechanics as it underpins energy conservation and simplifies the analysis of dynamical systems.

Transcript

00:01 Okay, everyone, so today we'll be looking at question 26 from chapter 8.

00:07 And honestly, this is more of a math question than anything.

00:10 It says you have a conservative force.

00:13 Okay, we have a conservative force, and that is equal to 6 .0x minus 12.

00:20 All right, newton's.

00:24 Well, it wants to know, the conservative force where x is in meters, x on a particle moving along the x -axis.

00:32 So the potential energy, u is associated with this force, is assigned a value of 27 joules when x is equal to zero.

00:39 And a wants to know if we can write an expression for you as a function of x, with u in joules and x in meters.

00:48 Write an expression for you.

00:51 Okay.

00:53 Well, this is very simple.

00:55 All we have to do is integrate this.

00:57 So, all right.

01:00 So if we integrate 6 .6x minus 12, if we integrate it, what do we get? well, you do 6x, you add.

01:13 So it will be to the second degree, and you divide by 2 minus 12, you add an x.

01:20 And we know that since it's equal to 27 when x is equal to 0, all you have to do is plus 27.

01:32 And this is it.

01:33 So you is equal to 3x squared minus 12x plus 27.

01:40 That is the answer for a.

01:43 Very simple.

01:44 All you have to do is integrate the conservative force given to you.

01:48 And now b wants to know what is the maximum positive potential energy? well, all we do is set.

01:58 Your conservative force equal to zero...

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