as a rocket carrying a space probe acceleraes away from earth the fuel is being used up and the rocket's mass becomes less when the mass of a rocket(and its fuel) is M and the distance of the rocket from eart's centre is 1.5rE, the force of gravitational attraction between eartrh and rocket is F1, when some fruel is consumed causing the mass to become 0.5M and the distance from Earth's centre is 2.5rE the new gravitational attraction is F2. Determine the ratio of F2 to F1. Tkhe symbalrE is earth's radius.
The earth's gravity reaches out forever but the force of attraction on bodies at great distances would be extremely small depending on the mass of the body. The Law of Universal Gravitation states that each particle of matter attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Expressed mathematically,
F = GM(m)/r^2
where F is the force with which either of the particles attracts the other, M and m are the masses of two particles separated by a distance r, and G is the Universal Gravitational Constant. The product of G and, lets say, the mass of the earth, is sometimes referred to as GM or µ (the greek letter pronounced meuw as opposed to meow), the earth's gravitational constant. Thus the force of attraction exerted by the earth on any particle within, on the surface of, or above it, is F = 1.40766x10^16 ft^3/sec^2(m)/r^2 where m is the mass of the object being attracted = W/g, and r is the distance from the center of the earth to the mass.
F1 = GMm1/r1^2
F2 = GMm2/r2^2
F2/F1 = [GMm2/r2^2]/[GMm1/r1^2]
......= m2(r1^2)/m1(r2^2)
To determine the ratio of F2 to F1, we can use the universal law of gravitation:
F = (G * m1 * m2) / r^2
where F is the force of gravitational attraction, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.
Let's first find the gravitational attraction F1 when the distance from Earth's center is 1.5rE and the mass of the rocket (including fuel) is M.
F1 = (G * M * mE) / (1.5rE)^2
Now, let's find the gravitational attraction F2 when the distance from Earth's center is 2.5rE and the mass of the rocket (including fuel) is 0.5M.
F2 = (G * 0.5M * mE) / (2.5rE)^2
To determine the ratio of F2 to F1, we divide F2 by F1:
Ratio (F2/F1) = (F2 / F1) = [(G * 0.5M * mE) / (2.5rE)^2] / [(G * M * mE) / (1.5rE)^2]
Canceling out the common terms in the numerator and denominator:
Ratio (F2/F1) = [(0.5M) / (2.5rE)^2] / [(M) / (1.5rE)^2]
Simplifying further:
Ratio (F2/F1) = 0.5 * [(1.5rE)^2] / [(2.5rE)^2]
Expanding this equation:
Ratio (F2/F1) = 0.5 * (1.5^2) * (rE^2) / (2.5^2) * (rE^2)
Simplifying:
Ratio (F2/F1) = 0.5 * (2.25) / (6.25)
Ratio (F2/F1) = 0.9
Therefore, the ratio of F2 to F1 is 0.9.
To determine the ratio of F2 to F1, we can make use of Newton's law of universal gravitation, which states that the force of gravitational attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The equation can be written as:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravitational attraction
G is the gravitational constant
m1 and m2 are the masses of the two objects
r is the distance between their centers
In this scenario, we have two situations: F1 when the rocket's mass is M and the distance is 1.5rE, and F2 when the mass is 0.5M and the distance is 2.5rE.
First, let's find the ratio of F2 to F1:
F2 / F1 = (G * (m1 * m2) / (2.5rE)^2) / (G * (m1 * m2) / (1.5rE)^2)
In this case, the mass of the rocket and the fuel is changing, but the mass of the Earth remains constant. Therefore, we can simplify the equation by canceling out the gravitational constant, m2, and rearranging the terms:
F2 / F1 = ((m1 * (0.5M)) / (2.5rE)^2) / ((m1 * M) / (1.5rE)^2)
F2 / F1 = ((0.5 * M) / (2.5)^2) / ((M) / (1.5)^2)
F2 / F1 = (0.5 / 6.25) / (1 / 2.25)
F2 / F1 = 0.08 / 0.4444
Hence, the ratio of F2 to F1 is approximately 0.18 (or 18%).
Please note that the specific values of the constants (G, rE) were not provided, so the final numerical ratio may differ, but the general calculation and process remain the same.