Two solid hemispheres of radii R and (R/2) with centers O and Oâ² respectively as shown in figure. The density of bigger hemisphere is ρ and that of smaller hemisphere is 2ρ . Taking center of bigger hemisphere is at origin and the distance between centers of two hemispheres OOâ² is (R/10) find co-ordinates of center of mass of the system.

Question

Two solid hemispheres of radii R and (R/2) with centers O and Oâ² respectively as shown in figure. The density of bigger hemisphere is ρ and that of smaller hemisphere is 2ρ . Taking center of bigger hemisphere is at origin and the distance between centers of two hemispheres OOâ² is (R/10) find co-ordinates of center of mass of the system. <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-pdm2q1u3xqf3qdvd.jpg" /> (A) (- (R/50) , (21 R/80)) (B) (- (R/30) , (21 R/80)) (C) (- (R/50) , (7 R/16)) (D) (- (R/30) , (7 R/16))

Tardigrade Admin · Accepted Answer

Let, M2=M , M1=4M xc m=(4 M (O) - M ((R/10))/4 M + M)⇒ xcm=-(R/50) yc m=(4 M ((3 R/8)) - M ((R/16))/4 M + M)⇒ ycm=(21 R/80)

$(- \frac{R}{50}, \frac{21 R}{80})$

$(- \frac{R}{30}, \frac{21 R}{80})$

$(- \frac{R}{50}, \frac{7 R}{16})$

$(- \frac{R}{30}, \frac{7 R}{16})$

Let, $M_{2} = M$ , $M_{1} = 4 M$
$x_{c m} = \frac{4 M ( O ) - M ( \frac{R}{10} )}{4 M + M} \Rightarrow x_{c m} = - \frac{R}{50}$
$y_{c m} = \frac{4 M ( \frac{3 R}{8} ) - M ( \frac{R}{16} )}{4 M + M} \Rightarrow y_{c m} = \frac{21 R}{80}$

(−50R​,8021R​)

(−30R​,8021R​)

(−50R​,167R​)

(−30R​,167R​)

Let, M2​=M , M1​=4M xcm​=4M+M4M(O)−M(10R​)​⇒xcm​=−50R​ ycm​=4M+M4M(83R​)−M(16R​)​⇒ycm​=8021R​

$(- \frac{R}{50}, \frac{21 R}{80})$

$(- \frac{R}{30}, \frac{21 R}{80})$

$(- \frac{R}{50}, \frac{7 R}{16})$

$(- \frac{R}{30}, \frac{7 R}{16})$

Let, $M_{2} = M$ , $M_{1} = 4 M$
$x_{c m} = \frac{4 M ( O ) - M ( \frac{R}{10} )}{4 M + M} \Rightarrow x_{c m} = - \frac{R}{50}$
$y_{c m} = \frac{4 M ( \frac{3 R}{8} ) - M ( \frac{R}{16} )}{4 M + M} \Rightarrow y_{c m} = \frac{21 R}{80}$