Question

A particle moving along the x-axis is in a system with potential energy $U=10/xJ$, where x is in m. What is the x-component of the force on the particle at$x=2m,5m,and8m$?

Short answer

x-component of force at $x=2m$$=2.5N$.

x-component of force at $x=5m$localid="1649681135207" $=0.4N$.

x-component of force at $x=8m$localid="1649681142148" $=0.1562N$.

Step-by-step solution

Given information

The potential energy equation is given as

$U=\frac{10}{x}J$where x is in 'm'.

To find the x-component of the force on a particle at$x=2m,5m$and$8m$.

The force exerted on a mass item that causes it to alter velocity.

Explanation : 

Potential energy of a motion equation along x-axis,

$U=\frac{10}{x}J$

On the potential energy diagram, the equation gives the x-component of force,

localid="1649681265497" $F_x=-\frac{dU}{dx}J/m$

The slope of the potential energy diagram is the x-component of force.

On substituting potential energy equation in the above force equation, we get the Particle slope,

localid="1649681337433" role="math" $$\begin{gathered} F_x=-\frac{d}{dx}\left(\frac{10}{x}\right) \\ =-\left(-\frac{10}{x^2}\right)......\left(1\right) \\ =\frac{10}{x^2}N \end{gathered}$$

At localid="1649681348187" $\left(x=2m\right)$, the x-component of force is,

localid="1649681356801" $$\begin{gathered} F_{x=2m}=\frac{10}{{\left(2\right)}^2} \\ =2.5N \end{gathered}$$

At localid="1649681364777" $\left(x=5m\right)$, the x-component of force is,

localid="1649681375104" $$\begin{gathered} F_{x=5m}=\frac{10}{{\left(5\right)}^2} \\ =0.4N \end{gathered}$$

At localid="1649681384241" $\left(x=8m\right)$, the x-component of force is,

localid="1649681392464" $$\begin{gathered} F_{x=8m}=\frac{10}{{\left(8\right)}^2} \\ =0.1562N \end{gathered}$$

Hence,

x-component of force at $x=2m$ localid="1649681402563" $=2.5N$.

x-component of force at $x=5m$ localid="1649681410158" $=0.4N$.

x-component of force at $x=8m$localid="1649681419101" $=0.1562N$.