a square of edge length 15.0 cm is formed by four spheres of masses m1 = 8.00 g, m2 = 3.50 g, m3 = 1.50 g, and m4 = 8.00 g. In unit-vector notation, what is the net gravitational force from them on a central sphere with mass m5 = 2.20 g?
To simplify this, look at diagonols. The net force on the center from two opposite corners is G(2.2)/d^2 (M1-Mopposite)
do that twice, for opposite corners. then do the vector addition of two vectors at 90 degrees, the result is easy to compute using pyth theorm.
To find the net gravitational force on the central sphere, we need to calculate the gravitational force between the central sphere and each of the other four spheres, and then sum them up.
Assuming all the spheres are point masses, we can use the formula for gravitational force:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.
We can calculate the gravitational force between the central sphere and each of the other four spheres using the given information:
For sphere 1:
Mass of sphere 1 (m1) = 8.00 g
Distance between sphere 1 and the central sphere (r1) = 15.0 cm
Gravitational force from sphere 1 on the central sphere (F1) = G * (m1 * m5) / r1^2
Similarly, we can calculate the forces from spheres 2, 3, and 4:
For sphere 2:
Mass of sphere 2 (m2) = 3.50 g
Distance between sphere 2 and the central sphere (r2) = 15.0 cm
Gravitational force from sphere 2 on the central sphere (F2) = G * (m2 * m5) / r2^2
For sphere 3:
Mass of sphere 3 (m3) = 1.50 g
Distance between sphere 3 and the central sphere (r3) = 15.0 cm
Gravitational force from sphere 3 on the central sphere (F3) = G * (m3 * m5) / r3^2
For sphere 4:
Mass of sphere 4 (m4) = 8.00 g
Distance between sphere 4 and the central sphere (r4) = 15.0 cm
Gravitational force from sphere 4 on the central sphere (F4) = G * (m4 * m5) / r4^2
Once we have calculated all these individual forces, we can find the net gravitational force by summing them up using vector notation:
Net gravitational force (F_net) = F1 + F2 + F3 + F4
So, in unit-vector notation, the net gravitational force from the four spheres on the central sphere can be expressed as:
F_net = (Fx, Fy, Fz)
Where Fx, Fy, and Fz are the x, y, and z components of the force vector respectively.