(a) Assuming the earth to be a sphere of uniform density, calculate the value of acceleration due to gravity at a point (i) $1600km$ above the earth, (ii) $1600km$ below the earth, (b) Also find the rate of variation of acceleration due to gravity above and below the earth's surface. Radius of earth $=6400km,g9.8m/s^2$.

Step 1: Find the value of g at depth x $g_x=g\left(\frac{1-x}{R}\right)=g\left(1-\frac{1600km}{6400km}\right)=g\times \frac{3}{4}=7.35m/s^2$ Step 2: Find the formula for g at a distance r from the center of the earth above the earth's surface $g=\frac{GM}{r^2}$ Step 3: Find the derivative of g with respect to r $\frac{dg}{dr}=-\frac{2GM}{r^3}$ Step 4: Use the relationship between r and h to find the derivative of g with respect to h $r=R+h,dr=dh$$\frac{dg}{dh}=\frac{dg}{dr}\times \frac{dr}{dh}=-\frac{2GM}{(R+h{)}^3}$ Step 5: Find the formula for g at a distance x below the earth's surface $g=g\left(\frac{1-x}{R}\right)$ Step 6: Find the derivative of g with respect to x $\frac{dg}{dx}=-\frac{g}{R}=-\frac{GM}{R^2}$ Step 7: Use the relationship between mass and density to find the formula for g in terms of density $\frac{4}{3}\pi R^3\rho =M$$g=\frac{4}{3}\pi G\rho R$