Find work on a straight line path

In summary, the force exerted on a proton in an experiment is F=<-ax2,0>, where a =12N/m2. The work done by this force is calculated as the proton moves along a straight-line path from the point ri=<0.10m,0> to the point rf=<0.30m, 0.40 m>.
  • #1
steph3824
4
0

Homework Statement



In an experiment, one of the forces exerted on a proton is F=<-ax2,0> where a =12N/m2. Calculate the work done by this force as the proton moves along a straight-line path from the point ri=<0.10m,0> to the point rf=<0.30m, 0.40 m>

Homework Equations


I'm not sure if I would use W=F*change in x (don't know how to type in the delta symbol) since it says it's a straight line path, or if I would use W=F*change in r since the problem contains r's in it


The Attempt at a Solution


I'm not too sure what to do here. I see that I'm given a Fx and Fybut I don't know exactly what to do with the multiple r's. Thanks for any help!
 
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  • #2
steph3824 said:

Homework Statement



In an experiment, one of the forces exerted on a proton is F=<-ax2,0> where a =12N/m2. Calculate the work done by this force as the proton moves along a straight-line path from the point ri=<0.10m,0> to the point rf=<0.30m, 0.40 m>

Homework Equations


I'm not sure if I would use W=F*change in x (don't know how to type in the delta symbol) since it says it's a straight line path, or if I would use W=F*change in r since the problem contains r's in it


The Attempt at a Solution


I'm not too sure what to do here. I see that I'm given a Fx and Fybut I don't know exactly what to do with the multiple r's. Thanks for any help!
Where the force is not constant as is the case here, you must use the general defintion of the work done be a force. That is

[tex]W = \int_\gamma \bold{F}\left(\bold{r}\right)\bold{\cdot}d\bold{\gamma}[/tex]

Where [itex]\bold{\gamma}[/itex] is the path.
 
  • #3
Oh ok I didn't realize that the force wasn't constant so I understand why that equation would be used. I still need help though..I'm not sure where to go with that equation
 
  • #4
steph3824 said:
Oh ok I didn't realize that the force wasn't constant so I understand why that equation would be used. I still need help though..I'm not sure where to go with that equation
Okay. Generally with these types of questions, it is best to determine the path first. You have already figure out that the path is a straight line, so the next thing to do is to parametrise it. Can you do that?
 
  • #5
No, I'm not sure what parametrise means.
 
  • #6
steph3824 said:
No, I'm not sure what parametrise means.
Okay, it simply means write the equation of the straight line in terms of a parameter. You know that the general form of a straight line is y=mx+c, but writing the path in this form doesn't really make it easy to evaluate the path integral. Instead we want to write the x and y coordinates separately in terms of a parameter, say t.

In this case we know that the path is a straight line and can be written in the form

[tex]y=mx+c = \frac{0.4}{0.2}x - 0.2 = 2x - \frac{1}{5}[/tex]

Now, we need to parametrise this line. Let's start by letting

[tex]x = t \hspace{1cm}\text{where }\frac{1}{10}\leq t \leq \frac{3}{10}[/tex]

This will give us the change in x as we follow the path. Now we need to obtain an equation for y. All we need to do now is substitute x=t into the equation for our straight line to obtain

[tex]y = 2t - \frac{1}{5}[/tex]

Hence we have parameterise our path

[tex]\gamma = \left\{\begin{array}{l} x = t \\ y = 2t - \frac{1}{5}\end{array}\right.\hspace{1cm}\frac{1}{10}\leq t \leq \frac{3}{10}[/tex]

Do you follow?
 

1. How do you find the work on a straight line path?

To find the work on a straight line path, you must first determine the force applied along the path and the distance traveled. Then, you can use the formula W = Fd, where W is the work, F is the force, and d is the distance.

2. Can the work on a straight line path be negative?

Yes, the work on a straight line path can be negative if the force and displacement are in opposite directions. This indicates that energy is being taken away from the system, rather than added.

3. Why is the work on a straight line path important?

The work on a straight line path is important because it is a measurement of the energy transferred to or from a system. This can help in understanding the behavior and performance of systems, such as machines or vehicles.

4. Is the work on a straight line path affected by the mass of an object?

No, the work on a straight line path is not affected by the mass of an object. Instead, it is determined by the force applied and the distance traveled. However, an object's mass can affect the force required to move it, which in turn can impact the work done.

5. How is the work on a straight line path different from the work on a curved path?

The work on a straight line path is calculated using the formula W = Fd, while the work on a curved path requires the use of calculus. Additionally, on a curved path, the force and displacement may not be in the same direction, resulting in a different calculation for work.

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