Work done by a force along the path

[MyoPhilosopher]

Homework Statement

Work done by a force along the path

Relevant Equations

\vec F(\vec r)=α|\ vec r 2 \vecr.
r(s)=sˆx+sˆy,

If the first equation for force is not formatted please refer to the pic.
I did not post my full attempt as the problematic equation leads to a complex integral.
Should I be converting r vector into its univector. The two outlined equations refer to the force. I am making an error transforming the force into x_hat and y_hat format. Any advice is appreciated I am quite stuck

 

[vela]

To calculate the work, you're trying to evaluate
$$\int_C \vec F \cdot d\vec r = \int_0^1 \vec F \cdot \frac{d\vec r}{ds}\,ds,$$ right? You have expressions for $\vec F$, $\vec r$, and $d\vec r/ds$. Just substitute those into the integral and then evaluate the dot product. What do you get?

 

[MyoPhilosopher]

To calculate the work, you're trying to evaluate
$$\int_C \vec F \cdot d\vec r = \int_0^1 \vec F \cdot \frac{d\vec r}{ds}\,ds,$$ right? You have expressions for $\vec F$, $\vec r$, and $d\vec r/ds$. Just substitute those into the integral and then evaluate the dot product. What do you get?

How do I do the dot product if F is in terms of r(vector) and dr is in the form ˆx+ˆy. How do I convert r_vector in its x_hat and y_hat components

 

[vela]

You have an expression for $\vec r$ in terms of $\hat x$ and $\hat y$, don't you? Substitute that into the expression for $\vec F$ where appropriate.

 

[MyoPhilosopher]

You have an expression for $\vec r$ in terms of $\hat x$ and $\hat y$, don't you? Substitute that into the expression for $\vec F$ where appropriate.

No that's the issue. My $\vec r$ in terms of $\hat x$ and $\hat y$ is not correct.
Is the $\vec r$ in my force equation maybe related to $\vec r$(s) and therefore just the sine and cosine components of a straigth line graph with gradient 1 between the specified points?

 

[haruspex]

No that's the issue. My $\vec r$ in terms of $\hat x$ and $\hat y$ is not correct.

Draw a vector from the origin representing $\vec r$. What are the coordinates of its endpoint?

 

[MyoPhilosopher]

Draw a vector from the origin representing $\vec r$. What are the coordinates of its endpoint?

0 and 1, so I am assuming it is as easy as $\vec r$. 1cos(45)$\hat x$ and 1sin(45)$\hat y$

 

[haruspex]

0 and 1

Eh? I have no idea how you get that.
Let me put it another way: In Cartesian coordinates, what does the vector $\vec r$ represent?

 

[MyoPhilosopher]

Eh? I have no idea how you get that.
Let me put it another way: In Cartesian coordinates, what does the vector $\vec r$ represent?

Sorry, we integrate from 0->1 So cartesian be 1,1 endpoint.

This seems way too complex an integral and I can't see where I am going wrong

 

[haruspex]

Sorry, we integrate from 0->1 So cartesian be 1,1 endpoint.

You do not seem to understand my question. It is not specific to the problem in this thread, but quite general.
In Cartesian coordinates, what is the usual representation of the vector $\vec r$? There are no numbers in the answer.

 

[MyoPhilosopher]

You do not seem to understand my question. It is not specific to the problem in this thread, but quite general.
In Cartesian coordinates, what is the usual representation of the vector $\vec r$? There are no numbers in the answer.

I understand your questions as though you are referring to the origin? In which case the $\vec r$ can be seen as a straight line in the form y = x? I am just trying to understand what is specifically incorrect with the picture above as I can't see a mathematical error.

 

[haruspex]

I understand your questions as though you are referring to the origin? In which case the $\vec r$ can be seen as a straight line in the form y = x? I am just trying to understand what is specifically incorrect with the picture above as I can't see a mathematical error.

No, $\vec r$ represents a point, not a line.
Yes, you can draw a line from the origin to that point, but why should it be the line y=x? It is not necessarily at 45 degrees.

 

[MyoPhilosopher]

No, $\vec r$ represents a point, not a line.
Yes, you can draw a line from the origin to that point, but why should it be the line y=x? It is not necessarily at 45 degrees.

,

The given equation is r(s)=sˆx+sˆy

 

[Lnewqban]

The given equation is r(s)=sˆx+sˆy

Do you have a graph to show us?

 

[MyoPhilosopher]

Do you have a graph to show us?

Domain for s is including and from 0 to including 1

 

[Lnewqban]

Is that the path?
How is the force acting along it?

 

[haruspex]

The given equation is r(s)=sˆx+sˆy

Ok, I see. You listed it as a "relevant equation", which is supposedly to be where you list standard equations you suspect may be relevant. The specification of the path belongs in the problem statement.

So, you wrote $r=\hat x+\hat y$. I presume you meant $\hat r=\hat x+\hat y$, but that is incorrect. What is the magnitude of $\hat x+\hat y$?

 

[MyoPhilosopher]

Ok, I see. You listed it as a "relevant equation", which is supposedly to be where you list standard equations you suspect may be relevant. The specification of the path belongs in the problem statement.

So, you wrote $r=\hat x+\hat y$. I presume you meant $\hat r=\hat x+\hat y$, but that is incorrect. What is the magnitude of $\hat x+\hat y$?

Yep sorry I corrected it to its components in above pic, so I made $\hat r=1cos(45)\hat x+1sin(45)\hat y$,

 

[haruspex]

Sorry, we integrate from 0->1 So cartesian be 1,1 endpoint.

View attachment 257519
This seems way too complex an integral and I can't see where I am going wrong

I don't understand the final step in the above. You seem to have substituted s²+x² for r² in one place and s²+y² for r² in another. Neither is correct.

Btw, it would have been easier without switching to x and y coordinates. Just write the integral in terms of $\vec r$.

 

[MyoPhilosopher]

I don't understand the final step in the above. You seem to have substituted s²+x² for r² in one place and s²+y² for r² in another. Neither is correct.

Btw, it would have been easier without switching to x and y coordinates. Just write the integral in terms of $\vec r$.

Alright so instead of integral ds just use integral dr. Out of interest what would be the correct expression of r² in terms of s,x and y?

 

[hutchphd]

The given equation is r(s)=sˆx+sˆy

So r² ≡ r⋅r = ?

 

[haruspex]

Alright so instead of integral ds just use integral dr.

No, integral $.\vec {dr}$.

Out of interest what would be the correct expression of r² in terms of s,x and y?

$\vec r=s\hat r$, so what is r² in terms of s?
$\vec r=x\hat x+y\hat y$, so what is r² in terms of x and y?