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Answer :
(a) The magnitude of the net gravitational force exerted by the 115 kg and 415 kg objects on the 38.0 kg object placed midway between them is 3.33 × [tex]10^(^-^9^)[/tex] N.
(b) The 38.0 kg object can be placed at a distance of 3.2 m from the 415 kg mass toward the 115 kg mass in order to experience a net force of zero from the other two objects.
To find the magnitude of the net gravitational force, we can use the formula for gravitational force:
F = G * ((m1 * m2) /[tex]r^2[/tex])
Where:
F is the magnitude of the gravitational force,
G is the gravitational constant (6.67430 × [tex]10^(^-^1^1^)[/tex]N[tex]m^2[/tex]/[tex]kg^2[/tex]),
m1 and m2 are the masses of the objects, and
r is the distance between the centers of the objects.
For the 115 kg object and the 415 kg object, the distance between them is 4.80 m. Since the 38.0 kg object is placed midway between them, it is equidistant from both objects, which is half of the given separation distance, 2.40 m.
Substituting the values into the formula, we have:
F = (6.67430 × [tex]10^(^-^1^1^)[/tex] N[tex]m^2[/tex]/[tex]kg^2[/tex]) * ((115 kg * 415 kg) / [tex](2.40 m)^2[/tex])
F ≈ 3.33 ×[tex]10^(^-^9^)[/tex] N
Therefore, the magnitude of the net gravitational force exerted by the 115 kg and 415 kg objects on the 38.0 kg object is approximately 3.33 × [tex]10^(^-^9^)[/tex]N.
To find the position where the 38.0 kg object experiences a net force of zero from the other two objects, we need to consider the balance of gravitational forces. Since the 115 kg object is lighter than the 415 kg object, the 38.0 kg object needs to be closer to the heavier mass.
Let's assume the distance from the 415 kg mass toward the 115 kg mass as 'x' meters. The distance from the 115 kg mass toward the 415 kg mass would then be 4.80 m - x meters.
By setting up an equation for the gravitational forces on the 38.0 kg object, we can find the position where the net force is zero. The gravitational force exerted by the 415 kg object is equal to the gravitational force exerted by the 115 kg object, but in the opposite direction. Mathematically, this can be represented as:
G * ((415 kg * 38.0 kg) / (x m)²) = G * ((115 kg * 38.0 kg) / ((4.80 m - x m)²))
By solving this equation, we find that the position where the net force on the 38.0 kg object is zero is at approximately x = 3.2 m.
Therefore, the 38.0 kg object can be placed at a distance of 3.2 m from the 415 kg mass toward the 115 kg mass in order to experience a net force of zero from the other two objects.
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