Question

When one moves from a point 16 km below the earth's surface to a point 16 km above the earth's surface. The change in $g$ is approximately $\alpha % $. The value of $\alpha$ is _________. (Take radius of the earth = 6400 km.)

• A. 0.12
• B. 0.25
• C. 0.50
• D. 0.75

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to calculate the acceleration due to gravity ($g$) at a depth $d$ and at a height $h$ (where $d=h=16$ km). Then we need to find the percentage change between these two values, with respect to the value of $g$ at the surface.
Step 2: Key Formula or Approach:
We will use the approximate formulas for the variation of $g$, as the height and depth are much smaller than the Earth's radius.
- Acceleration at depth $d$: $g_d = g_s \left( 1 - \frac{d}{R} \right)$
- Acceleration at height $h$ (for $h \ll R$): $g_h \approx g_s \left( 1 - \frac{2h}{R} \right)$
where $g_s$ is the gravity at the surface and $R$ is the Earth's radius.
Step 3: Detailed Explanation:
Given values: $d = 16$ km, $h = 16$ km, $R = 6400$ km. Let $g$ be the acceleration due to gravity at the surface. Value of g at depth d=16 km: \[ g_{below} = g \left( 1 - \frac{d}{R} \right) = g \left( 1 - \frac{16}{6400} \right) = g \left( 1 - \frac{1}{400} \right) \] Value of g at height h=16 km:
Since $h=16$ km is much smaller than $R=6400$ km, we can use the binomial approximation.
\[ g_{above} = g \left( \frac{R}{R+h} \right)^2 = g \left( 1 + \frac{h}{R} \right)^{-2} \approx g \left( 1 - \frac{2h}{R} \right) \] \[ g_{above} = g \left( 1 - \frac{2 \times 16}{6400} \right) = g \left( 1 - \frac{32}{6400} \right) = g \left( 1 - \frac{1}{200} \right) \] Change in g and Percentage Change:
The change in $g$ is the difference between the value at height and the value at depth:
\[ \Delta g = g_{below} - g_{above} = g \left( 1 - \frac{1}{400} \right) - g \left( 1 - \frac{1}{200} \right) \] \[ \Delta g = g \left( 1 - \frac{1}{400} - 1 + \frac{1}{200} \right) = g \left( \frac{-1+2}{400} \right) = \frac{g}{400} \] The percentage change $\alpha$ is defined as $\frac{\Delta g}{g} \times 100$:
\[ \alpha = \frac{g/400}{g} \times 100 = \frac{1}{400} \times 100 = 0.25 \] So, the change is 0.25%.
Step 4: Final Answer:
The value of $\alpha$ is 0.25.