4 Masses 8 kg, 2 kg, 4 kg and 2 kg are placed at the corners A, B, C, D respectively of a square ABCD of diagonal 80 cm. The distance of centre of mass from A will be

Options

A. 20 cm
B. 30 cm
C. 40 cm
D. 60 cm

Answer

B

Explanation

Let corner A of square ABCD is at the origin and the mass 8 kg is placed at this corner (given in problem) Diagonal of square `d = a sqrt(2)=80 cm rArr a = 40sqrt(2)cm`
`m_(1)=8kg, m_(2)=2kg, m_(3)=4 kg, m_(4)=2 kg`
Let `vec(r )_(1), vec(r )_(2), vec(r )_(3), vec(r )_(4)` are the position vectors of respective masses
`vec(r )_(1)=0hat(i)+0hat(j), vec(r )_(2)=a hat(i)+0hat(j), vec(r )_(3)=a hat(i)+a hat(j), vec(r )_(4)=0hat(i)+a hat(j)`
From the formula of centre of mass
`vec(r )=(m_(1)vec(r )_(1)+m_(2)vec(r )_(2)+m_(3)vec(r )_(3)+m_(4)vec(r )_(4))/(m_(1)+m_(2)+m_(3)+m_(4))=15 sqrt(2)i+15 sqrt(2)hat(j)`
`therefore` co-ordinates of centre of mass `= (15sqrt(2), 15sqrt(2))` and co-ordination of the corner = (0, 0)
From the formula of distance between two points `(x_(1), y_(1))` and `(x_(2), y_(2))`
distance `= sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2))=sqrt((15sqrt(2)-0)^(2)+(15sqrt(2)-0)^(2))=sqrt(900)=30 cm`