Concept explainers
Suppose you stood atop a ladder that was so tall that you were three Earth radii from Earth’s center. Show that your weight would be one ninth its present value.
To show: Change in the weight of body is one-ninth of its present value.
Explanation of Solution
Given:
Given body is at a distance thrice the radius of earth from the center of the earth.
Formula used:
Weight of a body is nothing but the product of mass of the body and acceleration due to gravity ( g ).
Weight of an object is equal to the gravitational force acting on the object of mass m
(1)
Where,
Gis the Gravitational constant=
M is Earth’s mass
m is object’s mass
R is Earth’s radius
Calculation:
From equation (1), acceleration due to gravity on the surface of the earth is,
So acceleration due to gravity at height H from the surface of the earth is,
Where,
In this case the height above the surface of the Earth is equal to 2R so
The ratio will be
This proves that acceleration due to gravity at a distance three times the radius of the Earth is one-ninth the acceleration due to gravity at a distance R from the center of the earth.
Multiply equation (2) by mass of the body so as to obtain the weight of the body at height H.
Conclusion:
Therefore, weight of the body is decreased by its one-ninth when compared with the weight of the body on the surface of the earth.
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