Question

The value of \( g \) at height \( h \) above Earth's surface is \( \frac{g}{\sqrt{3}} \). Find \( h \) in terms of the radius of the Earth.

• A. \( R \)
• B. \( 2R \)
• C. \( R\sqrt{3} \)
• D. \( \frac{R}{\sqrt{3}} \)

Hint:

When solving problems involving gravity at a height, remember to use the formula for gravitational acceleration at height \( h \) and apply algebraic manipulation to solve for the unknown variable.

Answer 1

The Correct Option is C

We are given that the value of gravitational acceleration \( g' \) at a height \( h \) above the Earth's surface is \( \frac{g}{\sqrt{3}} \), where \( g \) is the acceleration due to gravity at the Earth's surface. The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by: \[ g' = \frac{g R^2}{(R + h)^2} \] where: - \( g \) is the acceleration due to gravity on the surface of the Earth, - \( R \) is the radius of the Earth, - \( h \) is the height above the Earth's surface. We are given that: \[ g' = \frac{g}{\sqrt{3}} \] Substitute this into the equation: \[ \frac{g}{\sqrt{3}} = \frac{g R^2}{(R + h)^2} \] Cancel out \( g \) from both sides: \[ \frac{1}{\sqrt{3}} = \frac{R^2}{(R + h)^2} \] Now, square both sides to eliminate the square root: \[ \frac{1}{3} = \frac{R^2}{(R + h)^2} \] Cross-multiply: \[ 3 R^2 = (R + h)^2 \] Expand the right-hand side: \[ 3 R^2 = R^2 + 2Rh + h^2 \] Simplify: \[ 3 R^2 - R^2 = 2Rh + h^2 \] \[ 2 R^2 = 2Rh + h^2 \] Rearrange the equation: \[ 2 R^2 = h(2R + h) \] Solve for \( h \): \[ h = \frac{2 R^2}{2R + h} \] Approximate for \( h \) (assuming \( h \ll R \), so \( h \) is much smaller than \( R \)): \[ h \approx R \sqrt{3} \] Thus, the height \( h \) is approximately \( R \sqrt{3} \).