Chapter 1: Q23DQ (page 228)

The potential energy function for a force $\vec{F}$ is $U = α x^{3}$ , where $α$ is a positive constant. What is the direction of $\vec{F}$ ?

Short Answer

Expert verified

$\vec{F}$ is directed to the negative direction of the x-axis.

Step by step solution

Significance of the force

Force can be described as the negative gradient of the potential function for any object.

Determination of the direction of the force

The equation of the potential energy function is expressed as:

$U = α x^{3}$

Here, U is the potential energy, $α$ is the constant and x is the distance moved by the object.

The formula for the magnitude of force is

$F = - \frac{\partial U}{\partial x}$

Here $\vec{F}$ is the force exerted.

Substitute $α x^{3}$ for U in the above equation.

$F = - \frac{\partial (α x^{3})}{\partial x} = - 3 α x$

The negative sign indicates that the force is directed to the negative direction of the x-axis.

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The potential energy function for a force $\vec{F}$ is $U = α x^{3}$ , where $α$ is a positive constant. What is the direction of $\vec{F}$ ?

Short Answer

Step by step solution

Significance of the force

Determination of the direction of the force

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The potential energy function for a force $\vec{F}$ is $U = α x^{3}$ , where $α$ is a positive constant. What is the direction of $\vec{F}$ ?