Find the change in the gravitational force between two planets if the masses of both planets are doubled but the distance between them stays the same.
Express your answer as an integer.
I have no idea what exactly they are wanting here?? please help!
No problem! I can help you understand and solve this problem step by step.
First, let's understand the basics. The gravitational force between two planets is given by Newton's Law of Universal Gravitation:
F = (G * m1 * m2) / r^2
Where:
F: Gravitational force between the two planets
G: Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1, m2: Masses of the two planets
r: Distance between the centers of the two planets
Now, the problem states that both masses are doubled while the distance remains the same. So, let's denote the initial values with a subscript "i" and the final values with a subscript "f":
mi1: Initial mass of planet 1
mi2: Initial mass of planet 2
mf1: Final mass of planet 1 (doubled)
mf2: Final mass of planet 2 (doubled)
ri: Initial distance between the planets
rf: Final distance between the planets (same as ri)
To find the change in gravitational force (ΔF), we need to calculate the initial force (Fi) and the final force (Ff), and then take the difference:
ΔF = Ff - Fi
Let's calculate Fi using the initial values:
Fi = (G * mi1 * mi2) / ri^2
Now, calculate Ff using the final values:
Ff = (G * mf1 * mf2) / rf^2
Since the distance between the two planets is the same, rf = ri. So, we can simplify the equation:
Ff = (G * mf1 * mf2) / ri^2
Now, substitute the values into the equations, keeping in mind that both masses are doubled:
Fi = (G * 2 * mi1 * 2 * mi2) / ri^2
Ff = (G * 2 * mf1 * 2 * mf2) / ri^2
Simplify the equations:
Fi = (4 * (G * mi1 * mi2)) / ri^2
Ff = (4 * (G * mf1 * mf2)) / ri^2
Finally, substitute the equation for Fi and Ff into the equation for the change in gravitational force:
ΔF = Ff - Fi
= (4 * (G * mf1 * mf2)) / ri^2 - (4 * (G * mi1 * mi2)) / ri^2
= (4 * G * (mf1 * mf2 - mi1 * mi2)) / ri^2
Now, you can calculate the change in gravitational force (ΔF) by substituting the appropriate values for G, mf1, mf2, mi1, mi2, and ri into the equation. The final answer should be expressed as an integer.
I hope this explanation helps! Let me know if you have any further questions.