Understanding Gravitational Acceleration at Depth

The acceleration due to gravity, often denoted by $g$, is not constant throughout the Earth. While we usually use a standard value for calculations near the surface, its value changes with both altitude above the surface and depth below the surface.

The question asks about how gravity changes as we go deeper into the Earth. As we move below the surface, the mass of the Earth pulling us downwards effectively reduces because the shell of mass above us no longer contributes to the net gravitational pull in the same way. This causes the gravitational acceleration to decrease linearly with depth, assuming uniform density (a common approximation for these types of problems).

Formula for Gravity Variation Below Surface

The formula that describes the acceleration due to gravity ($g_d$) at a depth $d$ below the Earth's surface is given by:

$\qquad g_d = g_0 \left(1 - \frac{d}{R}\right)$

Where:

• $g_d$ is the acceleration due to gravity at depth $d$.
• $g_0$ is the acceleration due to gravity on the Earth's surface.
• $d$ is the depth below the surface.
• $R$ is the radius of the Earth.

Solving for Depth Where Gravity is Half

We are given that the value of gravitational acceleration at a certain depth $d$ is half of the value on the Earth's surface. Mathematically, this condition is $g_d = \frac{1}{2} g_0$.

We can substitute this condition into the formula for gravity at depth:

$\qquad \frac{1}{2} g_0 = g_0 \left(1 - \frac{d}{R}\right)$

To find the depth $d$, we can simplify this equation. Assuming $g_0$ is not zero (which is true on the Earth's surface), we can divide both sides of the equation by $g_0$:

$\qquad \frac{1}{2} = 1 - \frac{d}{R}$

Now, we need to isolate the term $\frac{d}{R}$. We can subtract 1 from both sides:

$\qquad \frac{1}{2} - 1 = - \frac{d}{R}$

$\qquad - \frac{1}{2} = - \frac{d}{R}$

Multiplying both sides by -1 gives:

$\qquad \frac{1}{2} = \frac{d}{R}$

Finally, we solve for $d$ by multiplying both sides by $R$:

$\qquad d = \frac{R}{2}$

Calculating the Specific Depth Below Earth

The radius of the Earth ($R$) is given in the question as 6400 km. We can now substitute this value into our equation for $d$:

$\qquad d = \frac{6400 \text{ km}}{2}$

$\qquad d = 3200 \text{ km}$

Conclusion

Therefore, the depth below the Earth's surface at which the gravitational acceleration is half of its value on the surface is 3200 km.