Two bodies of mass 1 kg and 3 kg have position vectors î+2 ĵ+k̂ and -3 î-2 ĵ+k̂ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

Solution

Physics sir · 3 steps

Find the centre of mass vector first, then compare lengths, not directions, with each option.

1. Mass-weighted average. The centre of mass is the average of the position vectors, each weighted by its mass. Total mass is 1 + 3 = 4 kg.

   r⃗_(cm) = (1(î+2ĵ+k̂) + 3(-3î-2ĵ+k̂))/4 = (-8î-4ĵ+4k̂)/4 = -2î-ĵ+k̂

2. Its magnitude. |r⃗_(cm)| = √((-2)² + (-1)² + 1²) = √6. Now find which option has the same length.

   Option	Vector	Magnitude
   (a)	î+2ĵ+k̂	√(1+4+1) = √6
   (b)	-3î-2ĵ+k̂	√(9+4+1) = √14
   (c)	-2î+2k̂	√(4+0+4) = √8
   (d)	2î-ĵ+2k̂	√(4+1+4) = 3

3. So. Only (a) has magnitude √6, the same as |r⃗_(cm)|. The two vectors point in different directions, but the question asks only about magnitude.

The idea to keep. The centre of mass is a mass-weighted average: multiply each position vector by its mass, add, divide by total mass. Then read the question carefully: magnitude or direction?

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