Two particles are located on the x axis. particle 1 has a mass m and is at the origin. particle 2 has a mass 2m and is at x = + l. a third particle is placed between particles 1 and 2. where on the x axis should the third particle be located so that the magnitude of the gravitational force on both particle 1 and particle 2 doubles? express your answer in terms of l.

Question

Rogelio Vega

Physics

9 June, 12:32

Two particles are located on the x axis. particle 1 has a mass m and is at the origin. particle 2 has a mass 2m and is at x = + l. a third particle is placed between particles 1 and 2. where on the x axis should the third particle be located so that the magnitude of the gravitational force on both particle 1 and particle 2 doubles? express your answer in terms of l.

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Answers (1)

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Kayleigh Vance · Answer 1

The solution would be like this for this specific problem:

The force on m is:

GMm / x^2 + Gm (2m) / L^2 = 2[Gm (2m) / L^2] - > 1

The force on 2m is:

GM (2m) / (L - x) ^2 + Gm (2m) / L^2 = 2[Gm (2m) / L^2] - > 2

From (1), you’ll get M = 2mx^2 / L^2 and from (2) you get M = m (L - x) ^2 / L^2

Since the Ms are the same, then

2mx^2 / L^2 = m (L - x) ^2 / L^2

2x^2 = (L - x) ^2

xsqrt2 = L - x

x (1 + sqrt2) = L

x = L / (sqrt2 + 1) From here, we rationalize.

x = L (sqrt2 - 1) / (sqrt2 + 1) (sqrt2 - 1)

x = L (sqrt2 - 1) / (2 - 1)

x = L (sqrt2 - 1)

= 0.414L

Therefore, the third particle should be located the 0.414L x axis so that the magnitude of the gravitational force on both particle 1 and particle 2 doubles.