determine the distance from earths centre where the force of gravity acting on a space probe is only 11% of the force acting on the same probe at earths surface. Express your answer in terms of earths radius, rE
Force is inversely prop to distance squared, so...
.11=1/(distance/rE)^2
F = G Me m/R^2
Fe= G Me m/Re^2
F/Fe = 11/100 = 1/R^2 / 1/Re^2 = Re^2/R^2
11/100 = Re^2/R^2
R/Re = sqrt(100/11) = 10/sqrt 11
= 3.01
so
R = 3.01 Re
The force of gravity is proportional to 1/r^2, where r is measured from the center of the earth. Therefore
(r/rE)^-2 = 0.11
r/rE = sqrt (1/0.11) = sqrt 9.09 = 3.015
To determine the distance from Earth's center where the force of gravity acting on a space probe is only 11% of the force acting on the same probe at Earth's surface, we can use the concept of the gravitational force equation.
The gravitational force acting on an object is given by the formula:
F = (G * m1 * m2) / r^2
Where:
F = gravitational force
G = gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2)
m1, m2 = masses of the two objects (in this case, the probe and Earth)
r = distance between the centers of the two objects
Since we are interested in finding the distance where the force is only 11% of the force at the Earth's surface, we can set up the following equation:
(0.11 * F_surface) = (G * m_probe * m_earth) / r^2
Here, F_surface represents the force acting on the probe at Earth's surface and m_probe and m_earth are the masses of the probe and Earth, respectively.
Now, let's express the force at Earth's surface in terms of Earth's radius, rE. The distance from the center of the Earth to its surface is equal to the radius of the Earth.
Since the force of gravity at Earth's surface is given by:
F_surface = (G * m_probe * m_earth) / rE^2
We can substitute this expression into the equation we set up earlier:
(0.11 * [(G * m_probe * m_earth) / rE^2]) = (G * m_probe * m_earth) / r^2
Next, we can cancel out the mass of the probe, m_probe, and the mass of the Earth, m_earth, from both sides of the equation:
0.11 / rE^2 = 1 / r^2
Now, let's solve for r. We can cross-multiply the equation:
(0.11 * r^2) = rE^2
Divide both sides of the equation by 0.11:
r^2 = (rE^2) / 0.11
Take the square root of both sides:
r = √[(rE^2) / 0.11]
Therefore, the distance from Earth's center where the force of gravity acting on a space probe is only 11% of the force acting on the same probe at Earth's surface is given by the expression r = √[(rE^2) / 0.11].