The masses and coordinates of three spheres are as follows: 18 kg, x = 1.50 m, y = 1.50 m; 40 kg, x = -1.50 m, y = -1.50 m; 64 kg, x = 0.00 m, y= -0.25 m. What is the magnitude of the gravitational force on a 14 kg sphere located at the origin due to the other spheres?

I would use your distance formula to find both distances.

Then, use F=GMM/dis^2, but remember to add them as vectors (break up each force as components up and along x axis).

To find the magnitude of the gravitational force on the 14 kg sphere located at the origin due to the other spheres, we can use the formula for Newton's Law of Universal Gravitation:

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

where F is the force, G is the 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, we want to calculate the force on the 14 kg sphere due to the other three spheres. Let's calculate the force exerted on the 14 kg sphere by each of the other spheres and then sum them up to find the total force.

For the first sphere (18 kg), the distance between its center and the origin is given by:

r1 = sqrt((x - 0)^2 + (y - 0)^2) = sqrt((1.50)^2 + (1.50)^2) = sqrt(4.50 + 4.50) = sqrt(9) = 3

Plugging the values into the formula, we get:

F1 = G * (m1 * m2) / r1^2 = G * (14 * 18) / 3^2

Similarly, for the second sphere (40 kg):

r2 = sqrt((x - 0)^2 + (y - 0)^2) = sqrt((-1.50)^2 + (-1.50)^2) = sqrt(2.25 + 2.25) = sqrt(4.50) = 2

F2 = G * (m1 * m2) / r2^2 = G * (14 * 40) / 2^2

And for the third sphere (64 kg):

r3 = sqrt((x - 0)^2 + (y - 0.25)^2) = sqrt((0.00)^2 + (-0.25)^2) = sqrt(0 + 0.0625) = sqrt(0.0625) = 0.25

F3 = G * (m1 * m2) / r3^2 = G * (14 * 64) / 0.25^2

Now, we can sum up these individual forces to find the total force due to all three spheres:

F_total = F1 + F2 + F3

Substituting the values, we get:

F_total = G * (14 * 18) / 3^2 + G * (14 * 40) / 2^2 + G * (14 * 64) / 0.25^2

Finally, we can calculate the magnitude of this total force by taking the absolute value of F_total:

|F_total| = |G * (14 * 18) / 3^2 + G * (14 * 40) / 2^2 + G * (14 * 64) / 0.25^2|

Therefore, to obtain the magnitude of the gravitational force on the 14 kg sphere located at the origin, you would need to substitute the values into the equation and use the gravitational constant G, which is approximately 6.67 x 10^-11 N(m/kg)^2.