40P - Fundamentals Of Physics

Question

A can of sardines is made to move along an x axis from $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{25}\mathbf{ }\mathbf{m}$ to $\mathbf{x}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{25}\mathbf{ }\mathbf{m}$ by a force with a magnitude given by $\mathbf{F}\mathbf{=}\mathbf{exp}\left(-4x^2\right)$ with x in meters and F in newtons. (Here exp is the exponential function.) How much work is done on the can by the force?

Step-by-Step Solution

Answer

 The work done on the can by the force is, 0.21 J.

Step 1: Given

1. The motion of can starts from ${\mathrm{x}}_{\mathrm{i}}=0.25 \mathrm{m}$
2. The final position of the can is, ${\mathrm{x}}_{\mathrm{f}}=1.25 \mathrm{m}$ 
3.  The function of force is, $\mathrm{F}={\mathrm{e}}^{-4{\mathrm{x}}^2}$ 
4. The scale for vertical axis is, ${\mathrm{F}}_{\mathrm{s}}=10.0 \mathrm{N}$ 

Step 2: Concept of calculating force done by a variable force.

The problem deals with the concept of work done. It includes both forces exerted on the body and the total displacement of the body. We can use the equation of the work done by a variable force.

 

Formula:

$\mathbf{W}\mathbf{=}{\mathbf{\int }}_{{\mathbf{x}}_{\mathbf{i}}}^{{\mathbf{x}}_{\mathbf{f}}}\mathbf{F}\left(x\right)\mathbf{d}\mathbf{x}$ 

Step 3: Calculate the work done

The equation for work done is,

 $\mathrm{W}={\int }_{{\mathrm{x}}_{\mathrm{i}}}^{{\mathrm{x}}_{\mathrm{f}}}\mathrm{F}\left(\mathrm{x}\right)d\mathrm{x}$

 

Substitute the value of F(x) in the above equation.

$\mathrm{W}={\int }_{0.25}^{1.25}{\mathrm{e}}^{-4{\mathrm{x}}^2}d\mathrm{x}$ 

 

By solving this integral, we get

 $\mathrm{W}=0.21 \mathrm{J}$

 

Therefore, work done by the force on the can is 0.21 J.