A hemispherical shell of mass 2M and radius 6 | vedclass

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(D) The center of mass of a hemispherical shell of radius $6R$ is at a distance of $3R$ from its center. Let the mass of the shell be $m_1 = 2M$ and t

Vedclass · Accepted Answer

$\omega_2 = \sqrt{\frac{GM}{72R^3}}$

Vedclass · Answer

(D) The center of mass of a hemispherical shell of radius $6R$ is at a distance of $3R$ from its center. Let the mass of the shell be $m_1 = 2M$ and the point mass be $m_2 = M$. The distance between them is $d = 3R$.For a two-body system,the radii of the circular orbits about the center of mass are $r_1 = \frac{m_2 d}{m_1 + m_2} = \frac{M(3R)}{2M + M} = R$ and $r_2 = \frac{m_1 d}{m_1 + m_2} = \frac{2M(3R)}{2M + M} = 2R$.The gravitational force between them is $F = \frac{G m_1 m_2}{d^2} = \frac{G(2M)(M)}{(3R)^2} = \frac{2GM^2}{9R^2}$.For the point mass $M$,the centripetal force is provided by the gravitational force: $M \omega^2 r_2 = F$.$M \omega^2 (2R) = \frac{2GM^2}{9R^2}$.$\omega^2 = \frac{GM}{9R^3}$.Since both bodies rotate with the same angular speed $\omega_1 = \omega_2 = \omega$,we have $\omega = \sqrt{\frac{GM}{9R^3}}$.Looking at the options,there seems to be a discrepancy in the provided options based on the standard calculation. However,if we re-evaluate the force calculation assuming the distance between the center of mass of the shell and the point mass is $3R$,the result is $\omega = \sqrt{\frac{GM}{9R^3}}$. Given the options,$D$ is the closest in form,but based on the physics,the correct angular speed is $\sqrt{\frac{GM}{9R^3}}$.

Similar Questions

Particles of mass $100 \, g$ each are placed at the vertices of an equilateral triangle of side $20 \, cm$. How much work must be done to increase the separation between them to infinity?

Three equal masses of $m \; kg$ each are fixed at the vertices of an equilateral triangle $ABC.$
$(a)$ What is the force acting on a mass $2 \; m$ placed at the centroid $G$ of the triangle?
$(b)$ What is the force if the mass at the vertex $A$ is doubled?
Take $AG = BG = CG = 1 \; m$.

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