Question

Four particles of masses m, 2m, 3m and 4m are placed at the vertices of a square of side a. At an
instant when the velocity of the particles are as shown in figure, the velocity of C.M is

Answer 1

Explanation:

Let the square be located at a position so that its one corner is at origin.

Therefore,

Coordinates of other corners :

( a , 0 ) : Another point at x-axis.

( 0 , a ) : Another point at y-axis.

( a , a ) : Remaining corner.

From the properties of centre of mass :

Co-ordinates of centre of mass : ( { mx + m x + ...... m x } / { m + m + .....m } , [ my + m y + ...... m y ] / { m + m + .....m } )

*Where m are representing the mass of the body , located at ( x , y ).

Here,

We have four masses with the coordinates ( 0 , 0 ) , ( 0 , a ) , ( a , a ) and ( a , 0 ) .

Therefore,

= > Coordinates of centre of mass of this square : ( [ { 0 x m } + { 0 x 2m } + { 3m x a } + { 4m x a } ] / [ m + 2m + 3m + 4m ] , [ { 0 x m } + { 2m x a } + { 3m x a } + { 4m x 0 } ] / [ m + 2m + 3m + 4m ] )

= > Coordinates of centre of mass of this square : ( [ 0 + 0 + 3ma + 4ma ] / [ 10 m ] , [ 2ma + 3ma + 0 ] / [ 10 m ] )

= > Coordinates of centre of mass of this square : ( 7ma / 10m , 5ma / 10m ) i.e. ( 0.7a , 0.5a )

Hence the required coordinates of the centre of mass of this square are ( 0.7a , 0.5a ).