Given: The universal gravitational con- stant is 6.672 × 10−11 N m2/kg2.

Objects with masses of 152 kg and 269 kg are separated by 0.45 m. A 53.5 kg mass is placed midway between them.
Find the magnitude of the net gravitational force exerted by the larger masses on the
53.5 kg mass. Answer in units of N.
Leaving the distance between the 152 kg and the 269 kg masses fixed, at what distance from the 269 kg mass (other than infinitely remote ones) does the 53.5 kg mass experience a net force of zero? Answer in units of m.

To find the magnitude of the net gravitational force exerted by the larger masses on the 53.5 kg mass, we can use the equation for the gravitational force between two masses:

F = G * (m1 * m2) / r^2

where F is the gravitational force, G is the universal gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.

In this case, the larger masses are 152 kg and 269 kg, and the smaller mass is 53.5 kg. The distance between the larger masses is given as 0.45 m.

First, let's find the gravitational force between each of the larger masses and the smaller mass. Then we can sum up these forces to find the net gravitational force.

F1 = G * (m1 * m3) / r1^2 .. (1)
F2 = G * (m2 * m3) / r2^2 .. (2)

where m1 is 152 kg, m2 is 269 kg, m3 is 53.5 kg, r1 is the distance between the first larger mass and the smaller mass, and r2 is the distance between the second larger mass and the smaller mass.

Since the smaller mass is placed midway between the larger masses, we can consider the distances r1 and r2 to be equal.

So, we rewrite equations (1) and (2) as:

F1 = G * (m1 * m3) / (0.5 * r)^2
F2 = G * (m2 * m3) / (0.5 * r)^2

Adding these forces together:

F = F1 + F2

F = G * (m1 * m3) / (0.5 * r)^2 + G * (m2 * m3) / (0.5 * r)^2

F = G * (m1 * m3 + m2 * m3) / (0.5 * r)^2

F = G * m3 * (m1 + m2) / (0.25 * r^2)

Now we can substitute the given values into the equation:

F = (6.672 × 10^-11 N m^2/kg^2) * (53.5 kg) * (152 kg + 269 kg) / (0.25 * 0.45 m)^2

Simplifying this equation will give us the magnitude of the net gravitational force exerted by the larger masses on the 53.5 kg mass.

To find the distance from the 269 kg mass where the 53.5 kg mass experiences a net force of zero, we need to set the net gravitational force equal to zero and solve for r in the equation from earlier:

G * m3 * (m1 + m2) / (0.25 * r^2) = 0

Simplifying this equation will give us the distance from the 269 kg mass where the net force is zero.