Find the magnitude of the net gravitational force exerted by these objects on a

Question

Find the magnitude of the net gravitational force exerted by these objects on a 44.0-kg object placed midway between them. 2.87e-10 Draw a picture of the three objects, including the forces exerted on the third. N At what position (other than an infinitely remote one) can the 44.0-kg object be placed so as to experience a net force of zero from the other two objects? 1.27 This is the correct distance to the smaller mass. The problem asks for the distance to the larger mass. Do you expect the position will be closer to the larger or smaller mass? m from the 445 kg mass toward the 145 kg mass

Explanation / Answer

Let the 2 original masses be m1, m2 separated by distance d, and the third mass be m3 at distance x from m2, and G be the gravitational constant.

The resultant force F towards m2 is:
F = G m2 m3 / x^2 - G m1 m3 / (d - x)^2
G m3 [ m2 / x^2 - m1 / (d - x)^2 ] ...(1)

(a)
Putting x = d / 2 in (1):
F = 4G m3 (m2 - m1) / d^2

= 4 * 6.673 × 10^(- 11) * 44.0(445 - 145) / 3.50^2
= 2.876 * 10^(- 7) N.

(b)
Equating F to 0 in (1):
m2 / x^2 = m1 / (d - x)^2
m1 x^2 = m2(d^2 + x^2 - 2dx)
(m2 - m1)x^2 - 2d m2 x + m2 d^2 = 0
x = (2d m2 +/- sqrt[ 4d^2 m2^2 - 4m2 d^2 (m2 - m1) ]) / [ 2(m2 - m1)
= d( m2 +/- sqrt[ m1 m2 ]) / (m2 - m1)

The negative sign gives the root between 0 and 3.50:
x = 3.50[ 445 - sqrt(145 * 445) ] / (445 - 145)
= 2.228 m.

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