Objects with masses of 110 kg and a 410 kg are separated by 0.470 m. (a) Find th
ID: 1776566 • Letter: O
Question
Objects with masses of 110 kg and a 410 kg are separated by 0.470 m.
(a) Find the net gravitational force exerted by these objects on a 46.0-kg object placed midway between them.
toward the 110-kg mass? -or- toward the 410-kg mass?-or- The magnitude is zero.
(b) At what position (other than infinitely remote ones) can the 46.0-kg object be placed so as to experience a net force of zero?
m from the 410-kg mass
toward the 110-kg mass? -or- toward the 410-kg mass?-or- The magnitude is zero.
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) * 46.0(410 - 110) / 0.470^2
= 1.67 * 10^(- 5) N.
The direction of net force is towards 410 kg mass
(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 0.470:
x = 0.470[ 410 - sqrt(110 * 410) ] / (410 - 110)
= 0.309 m.
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