Sphere A, with a mass of 67 kg, is located at the origin of an xy-coordinate system. Sphere B, with a mass of 54 kg, is located at coordinates (0.22 m, 0). Sphere C, with a mass of 0.55 kg, is located at coordinates (0.16 m, 0.11 m).

In unit-vector notation, what is the gravitational force on Sphere C due to Spheres A and B?

The gravitational force on sphere C due to spheres A and B can be calculated by finding the individual forces between C and A, and C and B, and then adding them together using vector addition.

Explanation:

To find the gravitational force on sphere C due to spheres A and B, we need to calculate the individual forces between C and A, and C and B, and then add them together using vector addition.

First, let's calculate the force between C and A. The formula to calculate the gravitational force is F = G * (m1 * m2) / r^2, where G is the gravitational constant (approximately 6.67 x 10^-11 N*m^2/kg^2), m1 is the mass of sphere C (0.55 kg), m2 is the mass of sphere A (67 kg), and r is the distance between them.

The distance between C and A can be calculated using the coordinates. Since sphere A is located at the origin, the distance between C and A is the magnitude of the position vector of C, which is given by sqrt((0.16 m)^2 + (0.11 m)^2).

Now, let's calculate the force between C and B. The formula is the same, but this time m2 is the mass of sphere B (54 kg) and r is the distance between C and B, which can be calculated using the coordinates of C and B.

Once we have calculated the individual forces, we can add them together using vector addition. In unit-vector notation, the gravitational force on C due to A and B can be represented as F = F_CA + F_CB, where F_CA is the force between C and A and F_CB is the force between C and B.

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