Question

2. Consider the system of two identical masses on the y axis that we worked with in class: one mass M at position (r, y)-(0, a) and another identical mass M at position (x,y)- (0, -a). (a) Draw a well labeled diagram like we did in class showing an arbitrary point (x, y) and write out the full expression for the total gravitational field (in component form) at that point. Also draw a set of gravitational field lines throughout the region showing the magnitude, direction, and symmetry of the field (b) Use your expression from part (a) to write an expression for the total gravitational field (in component form) at an arbitrary position on the y axis. Do the components of your result (for points above, below, and between the masses) make sense given the symmetry of the system? Explain. (c) Describe in words and/or equations) what you think the magnitude and direction of the gravitational field should be at a point on the y axis very far above the origin (i.e. where x 0 and y>>a). Show that your expression from (b) demonstrates this behavior if you take the appropriate limit

Answer 1

Part (a) in red color rightarrow Gravitational field lines symmetry rightarrow Across x-z plane& y-z plane Field at any arbitrary point (x, y) = GM/(squareroot x^2 + (y - a)^2)^2[- x I + (a - y) j/ squareroot x^2 + (y - a)^2) + GM/(squareroot (x^2 + (y - a)^2))^2 [- x I + (- a - y) j/ squareroot x^2 + (y - a)^2)] = -GM [x I + (y - a) j/[x^2 + (y - a)^2]^3/2 + x I + (y - a) j/[x^2 + (y + a)^2]^3/2 part (B): - on y axis x = 0 pit in eq field = -GM [(y - a) j/(y - a)^3 + (y + a) j/(y + a)^3] case I for point above the masses y > a Field = -G M [(y - a)/(y - a)^3 + (y + a)/(y + a)^3] j field is in - j direction i. e circleminus y_0 direction