Two 2.160 kg crystal balls are 0.870 m apart in the vertical direction. Calculate the magnitude of the gravitational force they exert on a 8.80 g marble located 0.260 m horizontally out from the center of a line that intersects the centers of the balls. (For your own knowledge, determine the direction of the force.)

I feel like I'm close to understanding this, but I still don't quite get it. Do I have to calculate the mag. of grav. force the crystal balls exert on each other first?

<<Do I have to calculate the mag. of grav. force the crystal balls exert on each other first? >>

No, just calculate the forces between the 0.0088 kg ball and either of the 2.160 kg balls, using the universal law of gravity. The value of R, from the small marble to either ball, is
sqrt(0.435^2 + 0.260^2)= 0.5068 m
The force vectors are the same in magnitude but act in different directions. The vertical components of the two forces (with the upper and the lower ball cancel. The gravity force vectors make an angle of arctan 0.435/0.260 = 59.1 degrees with the horizontal axis of symmetry. Only the horizontal components of the gravitaional force are additive.

(6.67E-11)(2.16)(0.0088) / (0.5068)^2 = 4.94E-12 N

Is that correct?

That is the force between the mabloe and one ball. For the net force with two balls, multiply by twice the cosine of the angle I calculated

Oh you're right, I forgot that step.

So 5.07E-12 N is the net force with both balls.

Place a small wooden stick over the edge of a desk and Hit the end of the stick overhanging the table so that it flies away. How is the flight distance related to the relevant parameters? What is the condition for a maximum horizontal distance?

if you scuff electrons from your hair onto comb, are you positively or negatively charged? how bout the comb?

No, you don't need to calculate the magnitude of the gravitational force between the crystal balls to find the magnitude of the gravitational force they exert on the marble.

To calculate the magnitude of the gravitational force between two objects, you can use the equation:

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

where F is the magnitude of the gravitational force, G is the gravitational constant (approximately 6.674 x 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.

Given information:
Mass of each crystal ball (m1 and m2) = 2.160 kg
Distance between the two crystal balls (r) = 0.870 m
Mass of the marble (m3) = 8.80 g = 0.00880 kg
Horizontal distance from the center of the line that intersects the centers of the crystal balls to the marble (d) = 0.260 m

To find the magnitude of the gravitational force between the crystal balls and the marble, you can first calculate the gravitational force F1 between each crystal ball and the marble using the equation above. Then, you can find the vector sum of these forces to determine the magnitude and direction of the resultant force.

Let's calculate the magnitude of the gravitational force exerted by each crystal ball on the marble:

F1 = (G * m1 * m3) / (r1^2)
= (6.674 x 10^-11 N m^2/kg^2) * (2.160 kg) * (0.00880 kg) / (0.260 m)^2

Use this equation to calculate the magnitude of the gravitational force exerted by the other crystal ball on the marble:

F2 = (G * m2 * m3) / (r2^2)

To find the resultant force, you can add the magnitudes of F1 and F2 using the Pythagorean theorem:

resultant force = sqrt(F1^2 + F2^2)

The direction of the force will depend on the vector sum of the individual forces. If the forces have the same direction, the resultant force will be in that direction. If they have opposite directions, the resultant force will be the difference between their magnitudes, with the direction of the larger force.