Respuesta :
Answer:
[tex]F(x)=-3 N - (3N)x^2[/tex]
Explanation:
The force is defined as the negative of the derivative of the potential energy:
[tex]F=-\frac{dU}{dx}[/tex]
If we use the potential energy function given in this problem:
[tex]U(x)=3.00 x + 1.00 x^3[/tex]
and we calculate the force, we get:
[tex]F(x)=-\frac{d}{dx}(3x+x^3)=-3-3x^2[/tex]
So, the force is
[tex]F(x)=-3 N - (3N)x^2[/tex]
The function of the force associated with the given energy function is [tex]\boxed{ - 3 - 3{x^2}}[/tex] .
Further Explanation:
Given:
The potential energy function for the system is [tex]U=\left( {3.00\,{\text{J}}} \right)x + \left( {1.00\,{{\text{J}} \mathord{\left/{\vphantom {{\text{J}} {{{\text{m}}^{\text{2}}}}}}\right.\kern-\nulldelimiterspace} {{{\text{m}}^{\text{2}}}}}} \right){x^3}[/tex].
Concept:
From the work-energy theorem, the force function for a given potential function can be expressed as the first derivative of the potential energy function with respect to the position of the body.
The force function is represented mathematically as:
[tex]\boxed{F= -\dfrac{{dU}}{{dx}}}[/tex] ......(1)
The potential energy function for the system is given as:
[tex]U = 3x + 1{x^3}[/tex]
Substitute [tex]3x + 1{x^3}[/tex] for [tex]U[/tex] in equation (1).
[tex]\begin{aligned}F&=-\frac{{d\left({3x + 1{x^3}}\right)}}{{dx}}\\&=-3- 3{x^2}\\\end{aligned}[/tex]
Thus, the function of the force associated with the given energy function is [tex]\boxed{ - 3 - 3{x^2}}[/tex].
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Answer Details:
Grade: College
Subject: Physics
Chapter: Work and Force
Keywords: Potential energy function, force function, associated with, [tex]u(x)=(3.00j)x+(1.00j/m^2)x^3, F=-dU/dx[/tex], derivative of potential function.
