- #1
joex444
- 44
- 0
Homework Statement
What is the gravitational force caused by a thin uniform rod of length L on a point mass located perpendicular to the rod at it's center? Assume the point mass is a distance R perpendicular to the rod.
Homework Equations
[tex]g = - \frac{GM}{r^2}\hat{r} = - G\rho \int \frac{dV}{r^2}\hat{r}[/tex]
The Attempt at a Solution
[tex]g = -G\rho \int \frac{Rdx}{\sqrt{x^2+R^2}^3} [/tex]
[tex]x = Rtan(\phi) [/tex]
[tex]g = -\frac{G\rho}{R} * ( \frac{x}{\sqrt{x^2+R^2}} ) [/tex]
Using limits of integration from -L/2 to +L/2 to yield:
[tex]g=-\frac{2G\rho}{R}*(L/\sqrt{L^2+4d^2})[/tex] after removing a factor of 4 from the (L/2)^2 from the square root.
Since the y components cancel, I took theta to be the angle nearest the point mass, and called rhat cos(theta) which I evaluated to R/sqrt(x^2+R^2) which could combine with the r^2 in the denominator from the gravity equation.
If this is wrong, I'll be glad to provide more steps to see where I went wrong. I ended up with a 1/(x^2+R^2)^(3/2) in the integral, so I did a trig substition to integrate it, and came up with the formula wikipedia has in its irrational integrals table, so I'm pretty sure that's alright, but wasn't sure if I set it up right.
Last edited: