The coordinates of the center of mass of the following quarter circular arc are

Question

The coordinates of the center of mass of the following quarter circular arc are <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-4ocayyzvkjdq.png" /> (A) ((r/2) , (r/2)) (B) ((2 r/3) , (2 r/3)) (C) ((2 r/π ) , (2 r/π )) (D) ((4 r/π ) , (4 r/π ))

Tardigrade Admin · Accepted Answer

Consider a small element as shown in the diagram

Solution

d m = λ R d θ = \frac{2 m}{π} d θ

x_{c m} = \frac{1}{m} \int_{0}^{\frac{π}{2}} R cos θ \frac{2 m}{π} d θ

x_{c m} = \frac{2 R}{π} \int_{0}^{\frac{π}{2}} cos θ d θ = \frac{2 R}{π} [s in π /2 - s in 0] = \frac{2 R}{π}

Similarly

y_{c m} = \frac{1}{m} \int_{0}^{\frac{π}{2}} R s in θ \frac{2 m}{π} d θ

y_{c m} = \frac{2 R}{π} \int_{0}^{\frac{π}{2}} s in θ d θ = \frac{2 R}{π} [- cos π /2 - (- cos 0)] = \frac{2 R}{π}

Q. The coordinates of the center of mass of the following quarter circular arc are

$(\frac{r}{2}, \frac{r}{2})$

$(\frac{2 r}{3}, \frac{2 r}{3})$

$(\frac{2 r}{π}, \frac{2 r}{π})$

$(\frac{4 r}{π}, \frac{4 r}{π})$

Q. The coordinates of the center of mass of the following quarter circular arc are

(2r​,2r​)

(32r​,32r​)

(π2r​,π2r​)

(π4r​,π4r​)

$(\frac{r}{2}, \frac{r}{2})$

$(\frac{2 r}{3}, \frac{2 r}{3})$

$(\frac{2 r}{π}, \frac{2 r}{π})$

$(\frac{4 r}{π}, \frac{4 r}{π})$