Consider two pointlike objects ill-1' 1 and ill-1'2r with ill-I1 having four times the mass of ill-1'2. They are placed a distance d apart from each other. (a) Sketch (in 2dirnensions) the gravitational eld lines that are present in the region of space around the two masses. Show the direction of the field on the field lines. 0)) Explain quantitatively where you should place a test particle such that. it experiences a net gravitational force of zero from the two pointlike objects. Indicate this position on your diagram. I apologize for the confusion. Let's go through the solution again. (b) To find the position where atest particle experiences a net gravitational force of zero, we need to balance the gravitational forces exerted by both masses. Let r' be the G'Ml''ftcst _T2_ distance from M1 to the test particle. The force exerted by M1 is , and the force exerted by M2 is W- Setting them equal to each other: G'J'Ml'Mlcat- : G'MZ'Aftcst r2 [ti1")? Canceling G and Mtest from both sides, we get: All _ M2 W (6171\")? Now, substituting M1 = 4M2 into the equation: 4M2 : M2 r2 (ti1r)2 Cross-multiplying: 4(d 1")2 = r2 Simplifying: 4d2 8d? l 4T2 : 7'2 4d2 7 8d? + 3r2 : 0 Factorizing: (2d 1")(2d 37') = 0 This gives two possible solutions: 1. 2dir:0:>'r:2d 2.2d3T:0:'2r:d I appreciate your patience. The correct solution is indeed r : gal. Now, including this position on the diagram: markdown Cl Copy code 4d2 - 8dr + 3r2 = 0 Factorizing: (2d - r)(2d - 3r) = 0 This gives two possible solutions: 1. 2d - r = 0 = r = 2d 2. 2d - 3r = 0 = r = ;d I appreciate your patience. The correct solution is indeed r = , d. Now, including this position on the diagram: markdown Copy code M 1 M 2 V Test particle at r = 2d/3 For the vector gravitational field sketch: The field lines will originate from M, and converge toward M2. Near M,, the lines will be denser to indicate a stronger field. The zero-gravity point (r = , d) will be located closer to M2. rust Copy code M 1 M 2 I hope this provides the correct solution and visualization