Question

M1 is a spherical mass (49.0 kg) at the origin. M2 is also a spherical mass (12.1 kg) and is located on the x-axis at x = 96.2 m. At what value of x would a third mass with a 20.0 kg mass experience no net gravitational force due to M1 and M2?

Answer 1

Gravitational force on the third mass = 0

Means, gravitational force on third mass due to M1 and M2 should be equal and its location should be between M1 and M2.

Now, F= (Gm1*M2)/r^2

Then as per the condition -

(G*M1*m3)/x^2=(G*M2*m3)/(96.2 - x)^2

Cancel G and m3 -

=> M1/x^2=M2/(96.2-x)^2

=> 49(96.2 - x)^2 = 12.1*x^2

=> 49(9254 - 192x + x^2) - 12.1x^2=0

=> 36.9x^2 - 9408x + 453446 = 0

=> x^2 - 255x + 12288 = 0

x = [255 +sqrt(65025 - 49152)]/2 = 190 m

And the second value of x = 64 m

Now, take only the answer where the forces are in opposite direction, that is, the value of x that is between 0 and 96.2 m

Therefore, your answer is x = 64 m