four uniform spheres, with masses MA= 100kg, MB= 650 kg, MC= 1800 kg, and MD = 100 kg, have (x,y) coordinates of (0, 50 cm), (O,O), (-80 cm, 0), and (40,0) respectively. What is the net gravitational force F2 on spere B due to the other spheres?
F = G M1 m/r^2 direction attractive
let M1 = 650. the mass of B
then G M1 = 6.67*10^-11 *650
= 4.34* 10^-8
so I will call
k = 4.34*10^-8 and do the arithmetic later
now each force = k m/r^2
now let's do forces in the X direction
no x force from A on B
x force C on B = k(1800/80^2) = -2812 k
x force D on B = k (100/.4^2) = +625 k
total x force Fx = - 2187 k
The only y force is from A in + y direction
Fy = k (100/.5^2) = k 100/.25 = 400 k
so
Fx = -2187(4.34*10^-8)
Fy = + 400 (4.34*10^-8)
You can multiply that out
if magnitude and direction are asked then
F = sqrt(Fx^2+Fy^2)
tan angle to - x axis in quadrant 2 = Fy/|Fx|
To calculate the net gravitational force on sphere B due to the other spheres, we need to find the individual gravitational forces between sphere B and each of the other spheres, and then add them up.
The gravitational force between two spheres can be calculated using Newton's Law of Universal Gravitation:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), m1 and m2 are the masses of the two spheres, and r is the distance between their centers.
Let's calculate the individual gravitational forces first:
1. Gravitational force between sphere B and sphere A:
- Mass of sphere B (m1) = 650 kg
- Mass of sphere A (m2) = 100 kg
- Distance between their centers (r) = square root of ((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of their centers.
- x1 = 0 cm, y1 = 0 cm (center of sphere B)
- x2 = 0 cm, y2 = 50 cm (center of sphere A)
- Plugging in the values, we get:
r = square root of ((0 cm - 0 cm)^2 + (50 cm - 0 cm)^2) = square root of (0 cm^2 + 2500 cm^2) = square root of (2500 cm^2)
- Calculating the gravitational force:
F1 = (G * m1 * m2) / r^2
2. Gravitational force between sphere B and sphere C:
- Mass of sphere C (m2) = 1800 kg
- Distance between their centers (r) = square root of ((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of their centers.
- x1 = 0 cm, y1 = 0 cm (center of sphere B)
- x2 = -80 cm, y2 = 0 cm (center of sphere C)
- Plugging in the values, we get:
r = square root of ((-80 cm - 0 cm)^2 + (0 cm - 0 cm)^2) = square root of (6400 cm^2 + 0 cm^2) = square root of (6400 cm^2)
- Calculating the gravitational force:
F2 = (G * m1 * m2) / r^2
3. Gravitational force between sphere B and sphere D:
- Mass of sphere D (m2) = 100 kg
- Distance between their centers (r) = square root of ((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of their centers.
- x1 = 0 cm, y1 = 0 cm (center of sphere B)
- x2 = 40 cm, y2 = 0 cm (center of sphere D)
- Plugging in the values, we get:
r = square root of ((40 cm - 0 cm)^2 + (0 cm - 0 cm)^2) = square root of (1600 cm^2 + 0 cm^2) = square root of (1600 cm^2)
- Calculating the gravitational force:
F3 = (G * m1 * m2) / r^2
Finally, to find the net gravitational force F2 on sphere B due to the other spheres, we add up the individual gravitational forces:
F2 = F1 + F2 + F3
Substituting the calculated values into the equation gives you the answer.