Two semicircular rings of linear mass densities λ and 2 λ and of radius ' R ' each are joined to form a complete ring. The distance of the center of the mass of complete ring from its centre is

Question

Two semicircular rings of linear mass densities λ and 2 λ and of radius ' R ' each are joined to form a complete ring. The distance of the center of the mass of complete ring from its centre is (A) (3 R/8 π) (B) (2 R/3 π) (C) (4 R/21 π) (D) None of these

Tardigrade Admin · Accepted Answer

Let the two half rings be placed in left and right of

y

-axis with centre as shown in figure.
Then the coordinate of centre of mass of
left and right half rings are

(- \frac{2 R}{π}, 0)

and

(\frac{2 R}{π}, 0)

.

∴ x

-coordinates of centre of mass of complete ring is

\frac{m ( - \frac{2 R}{π} ) + 2 m ( \frac{2 R}{π} )}{3 m} = \frac{2 R}{3 π}

Q. Two semicircular rings of linear mass densities $λ$ and $2 λ$ and of radius ' $R$ ' each are joined to form a complete ring. The distance of the center of the mass of complete ring from its centre is

$\frac{3 R}{8 π}$

$\frac{2 R}{3 π}$

$\frac{4 R}{21 π}$

None of these

Q. Two semicircular rings of linear mass densities λ and 2λ and of radius ' R ' each are joined to form a complete ring. The distance of the center of the mass of complete ring from its centre is

8π3R​

3π2R​

21π4R​

None of these

Let the two half rings be placed in left and right of y-axis with centre as shown in figure. Then the coordinate of centre of mass of left and right half rings are (−π2R​,0) and (π2R​,0). ∴x-coordinates of centre of mass of complete ring is 3mm(−π2R​)+2m(π2R​)​=3π2R​

Q. Two semicircular rings of linear mass densities $λ$ and $2 λ$ and of radius ' $R$ ' each are joined to form a complete ring. The distance of the center of the mass of complete ring from its centre is

$\frac{3 R}{8 π}$

$\frac{2 R}{3 π}$

$\frac{4 R}{21 π}$