United States Physics 1 With Calculus - Gravitation

[GreenPrint]

Homework Statement

See Attachment

Calculate the force of gravity on the point mass due to the line mass in terms of the gravitational constant G, m, M, D, and L. The line mass has a uniform density.

Homework Equations

The Attempt at a Solution

Ok so apparently I'm suppose to consider the point at were the point mass is to be the origin of the Cartesian coordinate system. To the right of this point is positive. Up from this point is positive. Below this point is negative. If the dotted line is extended through the line mass than the y components of the gravity contributed by the infinitismal masses were cancel each other out directly.

See Attachment

I get (G m M L)/(D sqrt(D^2 + L^2/4) )
in my paper I accidentally put

(G m M L)/(D^2 sqrt(D^2 + L^2/4) )
by accident

for some reason I feel as if this answer is wrong and was wondering if anyone could help me with this problem. I question if I'm really suppose to multiply it by two. It thought that I was suppose to because the horizontal components add together directly.

 

[Delphi51]

I used λ = M/L for the linear mass density and ended up with a slightly different integral.
If you look carefully at the step where you first use the integral sign, the dimensions are not correct. The dL cancels one of the L's on the bottom, so you have dimensions of GMm/L instead of GMm/L². If you write dF = GM*dm/R² where dm = λ*dL, it is easier to see (I left out the cosine). There is a similar mistake in the very last step, replacing a D with a D² or you would have seen that the dimensions were wrong in the end.

You were wise to integrate 0 to L/2 and multiply by 2. I did -L/2 to L/2 and it was a little more awkward to work out.

 

[GreenPrint]

Is

(2GmM)/(D sqrt(D^2 + L^2/4) )

what you got for the final answer?

 

[Delphi51]

Without the 2. I never had a 2. Your 2 disappeared when you evaluated at L/2.

 

[GreenPrint]

Right, thanks for the help.

 

[Delphi51]

Most welcome.