A planet with density equal to Earth is 2R times far away from earth, if we double the distancs from 2R to 4R then "g" becomes:-
please check what you wrote
"Earth is 2R times far away from earth "
makes no sense to me.
g = k/r^2
so, if r doubles, g is cut by a factor of 1/4
thanku Sir
To find out how "g" changes when we double the distance from 2R to 4R, we need to understand the relationship between gravity (g), distance (r), and the mass of the planet (M).
The formula to calculate the gravitational field strength or acceleration due to gravity (g) is:
g = (G * M) / r^2
Where:
- g is the gravitational field strength
- G is the gravitational constant (approximately 6.67 x 10^-11 N m^2/kg^2)
- M is the mass of the planet
- r is the distance from the center of the planet
Given that the density of the planet is equal to Earth, we know that the ratio of the mass (M) to the volume (V) of the planet is the same as that of Earth.
Now, let's solve the problem step by step:
Step 1: Compare the density of the planet to that of Earth.
Since the density is equal to Earth, we can assume that the ratio of mass to volume is the same for both Earth and the planet.
Step 2: Calculate the mass of the planet.
Since we are given that the planet's density is equal to Earth, we can use the formula for density (ρ = M / V) and assume the volume (V) is 4/3 * π * (2R)^3 (the volume of a sphere with radius 2R).
Substituting the values, we get:
ρ = (4/3 * π * (2R)^3 * ρ_earth
Rearranging the formula and substituting ρ_earth (density of Earth) gives:
M = (4/3 * π * (2R)^3 * ρ_earth)
Step 3: Calculate the initial gravitational field strength (g_1).
Using the first given distance (2R), we can substitute the values into the formula:
g_1 = (G * M) / (2R)^2
Step 4: Calculate the final gravitational field strength (g_2).
Using the new distance (4R), we can substitute the values into the formula:
g_2 = (G * M) / (4R)^2
Step 5: Calculate the ratio of the final gravitational field strength to the initial gravitational field strength (g_2 / g_1).
Substituting the formulas from steps 3 and 4 gives:
(g_2 / g_1) = [(G * M) / (4R)^2] / [(G * M) / (2R)^2]
Simplifying gives:
(g_2 / g_1) = (2R)^2 / (4R)^2
Step 6: Simplify the ratio and find the final answer.
(g_2 / g_1) = 1/4
Therefore, when we double the distance from 2R to 4R, the gravitational field strength (g) becomes one-fourth (1/4) of its initial value.