In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth). The value of OA : AB will be x : y. The value of x is ________.

Solution

Physics sir · 4 steps

Find g at C with the inverse-square law, then go inside and find where g falls to the same value.

1. Gravity at C. C is 3200 km = R/2 above the surface, so its distance from the centre is 3R/2. Outside the earth, g falls as 1/r², so g_C = g(R/(3R/2))² = 4g/9.
2. Gravity at A. Inside the earth, g grows linearly with distance from the centre. At OA = r, g_A = gr/R.

   gr/R = 4g/9 ⇒ r = 4R/9

3. Split OB. OA = 4R/9 and AB = R - 4R/9 = 5R/9, that is about 2844 km and 3556 km.
4. So. OA : AB = 4 : 5, so x = 4.

The common slip: 5. Reading the ratio the other way round, as AB : OA, gives 5, not 4. Here x belongs to OA, the distance from the centre.

The idea to keep. Outside the earth g ∝ 1/r², inside it g ∝ r. Use the exact forms whenever the height is comparable to R; the small-height approximation fails here.

Answer from the NTA final answer key. Solution by PrepWiser.