Problem: Two identical particles, each of mass m, are located on the x axis at x = + x0 and x = - x0.A. Determine a formula for the gravitational field due to these two particles for points on the y axis; that is, write as a function of y, m, x0, and so on.B. At what point (or points) on the y axis is the magnitude of a maximum value, and what is its value there? [Hint: Take the derivative dg/dy.]C. What is the maximum value of the magnitude of g?

FREE Expert Solution

We're told that the mass of each particle is m.

The gravitational field is expressed as:

$\overline{){\mathbf{g}}{\mathbf{=}}\frac{\mathbf{G}\mathbf{m}}{{\mathbf{r}}^{\mathbf{2}}}}$

A.

The magnitude of gravitational field for each particle on the y-axis is given by:

$\begin{array}{rcl}{\mathbf{g}}_{\mathbf{1}}& \mathbf{=}& {\mathbf{g}}_{\mathbf{2}}\\ & \mathbf{=}& \frac{\mathbf{G}\mathbf{m}}{{\mathbf{r}}^{\mathbf{2}}}\\ & \mathbf{=}& \frac{\mathbf{G}\mathbf{m}}{\mathbf{\left(}{{\mathbf{x}}_{\mathbf{0}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{y}}_{\mathbf{0}}}^{\mathbf{2}}\mathbf{\right)}}\end{array}$

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Problem Details

Two identical particles, each of mass m, are located on the x axis at x = + x0 and x = - x0.

A. Determine a formula for the gravitational field due to these two particles for points on the y axis; that is, write as a function of y, m, x0, and so on.

B. At what point (or points) on the y axis is the magnitude of a maximum value, and what is its value there? [Hint: Take the derivative dg/dy.]

C. What is the maximum value of the magnitude of g?

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