Question

The ratio of the weight of a body at a height of \(\dfrac{R}{10}\) from the surface of the earth to that at a depth of \(\dfrac{R}{10}\) is \((R\) is radius of earth\()\)

• A. \(4:5\)
• B. \(1:1\)
• C. \(9:8\)
• D. \(2:3\)
• E. \(8:9\)

Hint:

Use: \[ g_h \approx g\left(1-\frac{2h}{R}\right), \qquad g_d=g\left(1-\frac{d}{R}\right) \] for small heights and depths.

Answer 1

Answer: (E)

At height \(h\), for small \(h\ll R\), \[ g_h \approx g\left(1-\frac{2h}{R}\right) \] Here, \[ h=\frac{R}{10} \] So, \[ g_h=g\left(1-\frac{2}{10}\right)=\frac{4g}{5} \] At depth \(d\), \[ g_d=g\left(1-\frac{d}{R}\right) \] Here, \[ d=\frac{R}{10} \] So, \[ g_d=g\left(1-\frac{1}{10}\right)=\frac{9g}{10} \] Therefore, \[ g_h:g_d=\frac45:\frac9{10}=8:9 \] Hence, \[ \boxed{(E)\ 8:9} \]