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The height, in terms of R (radius of Earth), at which the acceleration due to gravity becomes g/9 (where g = acceleration due to gravity on the surface of Earth), is nR. What is the value of n?
Correct answer is '2'. Can you explain this answer?
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The height, in terms of R (radius of Earth), at which the acceleratio...
The acceleration due to gravity at a height h from the ground is given as g/9.
At the surface of Earth, the acceleration due to gravity is 
At r = R + h:
Acceleration due to gravity 
∴ r = 3R
The height above the ground = h = r - R = 2R
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The height, in terms of R (radius of Earth), at which the acceleratio...
Understanding the Problem
To find the height at which the acceleration due to gravity is g/9, we can use the formula for gravitational acceleration at a distance from the center of the Earth.
Gravitational Acceleration Formula
The acceleration due to gravity at a height h above the Earth's surface is given by:
- g' = g * (R / (R + h))^2
Where:
- g' = acceleration due to gravity at height h
- g = acceleration due to gravity on Earth's surface
- R = radius of the Earth
- h = height above the Earth's surface
Setting Up the Equation
We want to determine the height h where g' = g/9.
- g/9 = g * (R / (R + h))^2
Dividing both sides by g gives:
- 1/9 = (R / (R + h))^2
Solving for h
Taking the square root of both sides:
- 1/3 = R / (R + h)
Cross-multiplying yields:
- R + h = 3R
Thus:
- h = 3R - R
- h = 2R
Conclusion
The height h at which the acceleration due to gravity is g/9 is 2R. Therefore, the value of n is:
- n = 2
This result shows that at a height of 2 times the Earth's radius, the gravitational force reduces to one-ninth of its surface value.
To find the height at which the acceleration due to gravity is g/9, we can use the formula for gravitational acceleration at a distance from the center of the Earth.
Gravitational Acceleration Formula
The acceleration due to gravity at a height h above the Earth's surface is given by:
- g' = g * (R / (R + h))^2
Where:
- g' = acceleration due to gravity at height h
- g = acceleration due to gravity on Earth's surface
- R = radius of the Earth
- h = height above the Earth's surface
Setting Up the Equation
We want to determine the height h where g' = g/9.
- g/9 = g * (R / (R + h))^2
Dividing both sides by g gives:
- 1/9 = (R / (R + h))^2
Solving for h
Taking the square root of both sides:
- 1/3 = R / (R + h)
Cross-multiplying yields:
- R + h = 3R
Thus:
- h = 3R - R
- h = 2R
Conclusion
The height h at which the acceleration due to gravity is g/9 is 2R. Therefore, the value of n is:
- n = 2
This result shows that at a height of 2 times the Earth's radius, the gravitational force reduces to one-ninth of its surface value.
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The height, in terms of R (radius of Earth), at which the acceleration due to gravity becomes g/9 (where g = acceleration due to gravity on the surface of Earth), is nR. What is the value of n?Correct answer is '2'. Can you explain this answer?
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