Hi all,
I have been given component x- and y- for F1=(8.0*10^7N,7.0*10^7N)
F2=(4.0*10^7N,-2.0*10^7N) for an object.
Which following my textbooks have found the following vector components:
F1+F2=(1.2*10^8N,5.0*10^7N)
Now I want to find the distance from earth that this object has the same magnitude force as above.
So I was thinking of using F=Gm1+m2/d^2
but in order to do this I want to rearrange the equation to solve for d^2... however im unsure how to work with component vectors in this form i.e should I add the x and y components together to end up with 1.7*10^8N? and then set that as F?
Any pointers would be great!
Thanks
|F1+F2| = √((1.2*10^8)^2+(5.0*10^7)^2)
= 10^8 √(1.2^2 + 0.5^2)
= 1.3*10^8 N
So, you want d such that
GMm/d^2 = 1.3*10^8
Now just plug in G and the masses and crank out d.
Thanks Steve I assumed as much but just wanted to be sure.
To find the distance at which the object has the same magnitude force as the vector sum F1+F2, we can use the formula you mentioned: F = G * (m1 * m2) / d^2.
First, let's calculate the magnitude of the force vector F1+F2. The magnitude of a vector (F) with x- and y-components (Fx, Fy) can be found using the Pythagorean theorem, given by:
|F| = sqrt(Fx^2 + Fy^2)
For F1+F2, the x-component (Fx) is (1.2 * 10^8 N) and the y-component (Fy) is (5.0 * 10^7 N). Therefore:
|F1+F2| = sqrt((1.2 * 10^8)^2 + (5.0 * 10^7)^2)
Now, to find the distance (d), we can rearrange the formula:
d^2 = G * (m1 * m2) / F
Since G, m1, and m2 are constants (given for the Earth's gravitational force), we can substitute their values into the equation:
d^2 = (6.67430 * 10^-11 m^3 kg^-1 s^-2) * (m1 * m2) / |F1+F2|
Given that m1 and m2 are the masses of the Earth and the object, respectively, and we are assuming the object is much smaller than the Earth, we can consider m2 as negligible compared to m1. Therefore, m1 * m2 can be approximated to just m1.
d^2 ≈ (6.67430 * 10^-11 m^3 kg^-1 s^-2) * m1 / |F1+F2|
Now, we can substitute the values you were given to find the distance squared (d^2):
d^2 ≈ (6.67430 * 10^-11 m^3 kg^-1 s^-2) * (mass of Earth) / |F1+F2|
To find the actual distance (d), take the square root of d^2:
d ≈ sqrt[(6.67430 * 10^-11 m^3 kg^-1 s^-2) * (mass of Earth) / |F1+F2|]
Remember to use the appropriate units when calculating. This will give you the distance from Earth at which the object experiences a force of the same magnitude as F1+F2.