Mixing
How to find the work integral?
$begingroup$
The question is, Let $P_{1}$ and $P_{2}$ be the points at distances $s_{1}$ and $s_{2}$ from origin. Show that the work done by gravitational force $$vec{F}=-frac{GMm (xhat{i}+yhat{j}+zhat{k})}{(x^2+y^2+z^2)^frac{3}{2}}$$ in moving a particle from $P_{1}$ to $P_{2}$ is $$W=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg).$$
The attempt what I have made is. I know that the gravitational field of force is conservative, so we can find the potential function from these three equations
- $frac{partial f}{partial x}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} x$
- $frac{partial f}{partial y}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} y$
- $frac{partial f}{partial z}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} z$
I find the potential function, i.e, $f(x,y, z)=frac{GMm}{sqrt{x^2+y^2+z^2}}+C,$ where $C$ is constant of integration.
Now I assume the two points $P_{1}=x_{1}hat{i}+y_{1}hat{j}+z_{1}hat{k}$ and $P_{2}=x_{2}hat{i}+y_{2}hat{j}+z_{2}hat{k}$ with given assumptions that
$|OP_{1}|=sqrt{x_{1}^2+y_{1}^2+z_{1}^2}=s_{1}$ and $|OP_{2}|=sqrt{x_{2}^2+y_{2}^2+z_{2}^2}=s_{2}$
Now the work done by conservative field of force is equal
$$W=int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}vec{nabla}f bullet dvec{r}= int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}df=f(P_{2})-f(P_{1})$$
$$ = bigg(frac{GMm}{sqrt{x_{2}^2+y_{2}^2+z_{2}^2}}+Cbigg)-bigg(frac{GMm}{sqrt{x_{1}^2+y_{1}^2+z_{1}^2}}+Cbigg)$$
$$=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg)$$
I think what I have done is correct, but I need just confirmation!
Thanks in advance!
integration multivariable-calculus definite-integrals multiple-integral
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
$endgroup$
add a comment |
$begingroup$
The question is, Let $P_{1}$ and $P_{2}$ be the points at distances $s_{1}$ and $s_{2}$ from origin. Show that the work done by gravitational force $$vec{F}=-frac{GMm (xhat{i}+yhat{j}+zhat{k})}{(x^2+y^2+z^2)^frac{3}{2}}$$ in moving a particle from $P_{1}$ to $P_{2}$ is $$W=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg).$$
The attempt what I have made is. I know that the gravitational field of force is conservative, so we can find the potential function from these three equations
- $frac{partial f}{partial x}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} x$
- $frac{partial f}{partial y}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} y$
- $frac{partial f}{partial z}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} z$
I find the potential function, i.e, $f(x,y, z)=frac{GMm}{sqrt{x^2+y^2+z^2}}+C,$ where $C$ is constant of integration.
Now I assume the two points $P_{1}=x_{1}hat{i}+y_{1}hat{j}+z_{1}hat{k}$ and $P_{2}=x_{2}hat{i}+y_{2}hat{j}+z_{2}hat{k}$ with given assumptions that
$|OP_{1}|=sqrt{x_{1}^2+y_{1}^2+z_{1}^2}=s_{1}$ and $|OP_{2}|=sqrt{x_{2}^2+y_{2}^2+z_{2}^2}=s_{2}$
Now the work done by conservative field of force is equal
$$W=int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}vec{nabla}f bullet dvec{r}= int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}df=f(P_{2})-f(P_{1})$$
$$ = bigg(frac{GMm}{sqrt{x_{2}^2+y_{2}^2+z_{2}^2}}+Cbigg)-bigg(frac{GMm}{sqrt{x_{1}^2+y_{1}^2+z_{1}^2}}+Cbigg)$$
$$=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg)$$
I think what I have done is correct, but I need just confirmation!
Thanks in advance!
integration multivariable-calculus definite-integrals multiple-integral
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
$endgroup$
$begingroup$
It seems correct to me.
$endgroup$
– Rafa Budría
Jan 27 at 19:26
add a comment |
$begingroup$
The question is, Let $P_{1}$ and $P_{2}$ be the points at distances $s_{1}$ and $s_{2}$ from origin. Show that the work done by gravitational force $$vec{F}=-frac{GMm (xhat{i}+yhat{j}+zhat{k})}{(x^2+y^2+z^2)^frac{3}{2}}$$ in moving a particle from $P_{1}$ to $P_{2}$ is $$W=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg).$$
The attempt what I have made is. I know that the gravitational field of force is conservative, so we can find the potential function from these three equations
- $frac{partial f}{partial x}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} x$
- $frac{partial f}{partial y}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} y$
- $frac{partial f}{partial z}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} z$
I find the potential function, i.e, $f(x,y, z)=frac{GMm}{sqrt{x^2+y^2+z^2}}+C,$ where $C$ is constant of integration.
Now I assume the two points $P_{1}=x_{1}hat{i}+y_{1}hat{j}+z_{1}hat{k}$ and $P_{2}=x_{2}hat{i}+y_{2}hat{j}+z_{2}hat{k}$ with given assumptions that
$|OP_{1}|=sqrt{x_{1}^2+y_{1}^2+z_{1}^2}=s_{1}$ and $|OP_{2}|=sqrt{x_{2}^2+y_{2}^2+z_{2}^2}=s_{2}$
Now the work done by conservative field of force is equal
$$W=int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}vec{nabla}f bullet dvec{r}= int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}df=f(P_{2})-f(P_{1})$$
$$ = bigg(frac{GMm}{sqrt{x_{2}^2+y_{2}^2+z_{2}^2}}+Cbigg)-bigg(frac{GMm}{sqrt{x_{1}^2+y_{1}^2+z_{1}^2}}+Cbigg)$$
$$=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg)$$
I think what I have done is correct, but I need just confirmation!
Thanks in advance!
integration multivariable-calculus definite-integrals multiple-integral
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
$endgroup$
The question is, Let $P_{1}$ and $P_{2}$ be the points at distances $s_{1}$ and $s_{2}$ from origin. Show that the work done by gravitational force $$vec{F}=-frac{GMm (xhat{i}+yhat{j}+zhat{k})}{(x^2+y^2+z^2)^frac{3}{2}}$$ in moving a particle from $P_{1}$ to $P_{2}$ is $$W=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg).$$
The attempt what I have made is. I know that the gravitational field of force is conservative, so we can find the potential function from these three equations
- $frac{partial f}{partial x}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} x$
- $frac{partial f}{partial y}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} y$
- $frac{partial f}{partial z}=-frac{GMm }{(x^2+y^2+z^2)^frac{3}{2}} z$
I find the potential function, i.e, $f(x,y, z)=frac{GMm}{sqrt{x^2+y^2+z^2}}+C,$ where $C$ is constant of integration.
Now I assume the two points $P_{1}=x_{1}hat{i}+y_{1}hat{j}+z_{1}hat{k}$ and $P_{2}=x_{2}hat{i}+y_{2}hat{j}+z_{2}hat{k}$ with given assumptions that
$|OP_{1}|=sqrt{x_{1}^2+y_{1}^2+z_{1}^2}=s_{1}$ and $|OP_{2}|=sqrt{x_{2}^2+y_{2}^2+z_{2}^2}=s_{2}$
Now the work done by conservative field of force is equal
$$W=int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}vec{nabla}f bullet dvec{r}= int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})}df=f(P_{2})-f(P_{1})$$
$$ = bigg(frac{GMm}{sqrt{x_{2}^2+y_{2}^2+z_{2}^2}}+Cbigg)-bigg(frac{GMm}{sqrt{x_{1}^2+y_{1}^2+z_{1}^2}}+Cbigg)$$
$$=GMmbigg(frac{1}{s_{2}}-frac{1}{s_{1}}bigg)$$
I think what I have done is correct, but I need just confirmation!
Thanks in advance!
integration multivariable-calculus definite-integrals multiple-integral
integration multivariable-calculus definite-integrals multiple-integral
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
asked Jan 27 at 15:08
Noor AslamNoor Aslam
16012
16012
$begingroup$
It seems correct to me.
$endgroup$
– Rafa Budría
Jan 27 at 19:26
add a comment |
$begingroup$
It seems correct to me.
$endgroup$
– Rafa Budría
Jan 27 at 19:26
$begingroup$
It seems correct to me.
$endgroup$
– Rafa Budría
Jan 27 at 19:26
$begingroup$
It seems correct to me.
$endgroup$
– Rafa Budría
Jan 27 at 19:26
add a comment |
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$begingroup$
It seems correct to me.
$endgroup$
– Rafa Budría
Jan 27 at 19:26