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A particle of mass 3m is located 1.00 m from a particle of

Chapter 13, Problem 9E

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QUESTION:

A particle of mass 3m is located 1.00 m from a particle of mass m.

(a) Where should you put a third mass M so that the net gravitational force on M due to the two masses is exactly zero?

(b) Is the equilibrium of M at this point stable or unstable (i) for points along the line connecting m and 3 m, and (ii) for points along the line passing through M and perpendicular to the line connecting m and 3 m?

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QUESTION:

A particle of mass 3m is located 1.00 m from a particle of mass m.

(a) Where should you put a third mass M so that the net gravitational force on M due to the two masses is exactly zero?

(b) Is the equilibrium of M at this point stable or unstable (i) for points along the line connecting m and 3 m, and (ii) for points along the line passing through M and perpendicular to the line connecting m and 3 m?

ANSWER:

Solution 9E Step 1 a) Suppose, the distance between the mass m and M as “x” metres. Then, the gravitational force of attraction between these two masses would be, F = GMm 1 x2 The distance between the masses M and 3m is, (1-x) metres Therefore, the gravitational force acting between these two forces would be, 3GMm F =2 2 (1x) At equilibrium point, both these forces will be equal in magnitude and opposite in direction and we can find out that point. F1= F2 GMm = 3GMm x2 (1x) That is, 12 = 3 2 x (1x) 2 2 That is,(1 x) = 3x 1+x - 2x = 3x 2 That is, 2x + 2x - 1 = 0 Solve for x and we will get the point. 2± 4+8 x = 4 2±2 3 x = 4 x = 1± 3 2 We can take only positive values. x = 1+1.732 = 0.732/2 = 0.366 m 2 The equilibrium point for M is 0.366 m from the mass “m”.

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