
Discuss variation of with
(A) Altitude
(B) Depth
Answer
487.5k+ views
Hint : Acceleration due to gravity varies differently with altitude and depth. For altitude, we simply put the height in the formula and get the required result. But for its accurate value, we need to calculate the mass of the earth in terms of its density and then change the radius according to the depth.
Formula used:
Acceleration due to gravity is given as,
where, Universal gravitational constant, is the mass of the earth, is the radius of the earth.
Complete step by step answer
(a)
Let us calculate at a height of .
We know that,
At the height , changes to
Where, is the value of at height .
Dividing equation by , we get
Cancelling out , we get
Using the binomial theorem, we can write is as
This is the required variation of with an altitude .
We can vary the value of height i.e. and we will get a different value of .
(b)
The density of earth is given as,
For earth, can be written as
Using this in equation , we get
We use this value of in equation
Now, the value of at a depth is given as
Dividing equation by , we get
We can write this as
This is the required variation of with depth.
In this case of measuring at a depth , we can vary the value of and we will get different values of .
Note
In the case of measuring at an altitude of , while using binomial theorem, we make an assumption that . Thus this formula holds true only for those cases where the height at which we are measuring is negligible with respect to .
Formula used:
Acceleration due to gravity is given as,
where,
Complete step by step answer
(a)
Let us calculate
We know that,
At the height
Where,
Dividing equation
Cancelling out
Using the binomial theorem, we can write is as
This is the required variation of
We can vary the value of height i.e.
(b)
The density of earth is given as,
For earth,
Using this in equation
We use this value of
Now, the value of
Dividing equation
We can write this as
This is the required variation of
In this case of measuring
Note
In the case of measuring
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