Advertisements
Advertisements
Question
Explain the variation of g with depth from the Earth’s surface.
Advertisements
Solution
Consider a particle of mass m which is in a deep mine on the Earth. (Example: coal mines in Neyveli). Assume the depth of the mine as d. To calculate g’ at a depth d, consider the following points.
Particle in a mine
The part of the Earth which is above the radius (Re – d) does not contribute to the acceleration. The result is proved earlier and is given as
g’ = `"GM’"/(("R"_"e" - "d")^2)`
Here M’ is the mass of the Earth of radius (Re – d)
Assuming the density of Earth ρ to be constant, ρ = `"M"/"V"`
where M is the mass of the Earth and V its volume, thus,
ρ = `"M’"/"V’"`
`"M’"/"V’" = "M"/"V"` and M’ = `"M"/"V""V’"`
M’ = `("M"/(4/3π"R"_"e"^3)) (4/3π("R"_"e" - "d")^3)`
M’ = `"M"/"R"_"e"^3 ("R"_"e" - "d")^3`
g’ = `"G" "M"/"R"_"e"^3 ("R"_"e" - "d")^3 . 1/("R"_"e" - "d")^2`
g’ = `"GM" ("R"_"e" (1 - "d"/"R"_"e"))/("R"_"e"^3)`
g’ = `"GM" ((1 - "d"/"R"_"e"))/"R"_"e"^2`
Thus, g’ = `"g" (1 - "d"/"R"_"e")`
Here also g’ < g. As depth increase, g’ decreases. It is very interesting to know that acceleration due to gravity is maximum on the surface of the Earth but decreases when we go either upward or downward.
APPEARS IN
RELATED QUESTIONS
Is there any meaning of “weight of the earth”?
If heavier bodies are attracted more strongly by the earth, why don't they fall faster than the lighter bodies?
The earth revolves round the sun because the sun attracts the earth. The sun also attracts the moon and this force is about twice as large as the attraction of the earth on the moon. Why does the moon not revolve round the sun? Or does it?
What is the acceleration due to gravity on the top of Mount Everest? Mount Everest is the highest mountain peak of the world at the height of 8848 m. The value at sea level is 9.80 m s−2.
A particle is fired vertically upward with a speed of 15 km s−1. With what speed will it move in interstellar space. Assume only earth's gravitational field.
Explain the variation of g with latitude.
One can easily weigh the earth by calculating the mass of the earth by using the formula:
The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity ______.
A person whose mass is 100 kg travels from Earth to Mars in a spaceship. Neglect all other objects in the sky and take acceleration due to gravity on the surface of the Earth and Mars as 10 m/s2 and 4 m/s2 respectively. Identify from the below figures, the curve that fits best for the weight of the passenger as a function of time.
The percentage decrease in the weight of a rocket, when taken to a height of 32 km above the surface of the earth will, be ______.
(Radius of earth = 6400 km)
