A spherical hollow is made in a lead sphere o | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $\rho$ be the density of the lead sphere.The mass of the original sphere is $M = \frac{4}{3} \pi R^3 \rho$.The hollowed part is a sphere of ra

Vedclass · Accepted Answer

$\frac{R}{14}$

Vedclass · Answer

(B) Let $\rho$ be the density of the lead sphere.The mass of the original sphere is $M = \frac{4}{3} \pi R^3 \rho$.The hollowed part is a sphere of radius $r = \frac{R}{2}$. Its mass is $m_1 = \frac{4}{3} \pi (\frac{R}{2})^3 \rho = \frac{1}{8} (\frac{4}{3} \pi R^3 \rho) = \frac{M}{8}$.The remaining mass is $m_2 = M - m_1 = M - \frac{M}{8} = \frac{7M}{8}$.Let the center of the original sphere be the origin $(0,0)$. The center of the removed sphere is at $x_1 = \frac{R}{2}$.Using the center of mass formula for the original sphere: $X_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = 0$.Substituting the values: $0 = \frac{(\frac{M}{8})(\frac{R}{2}) + (\frac{7M}{8})(x_2)}{M}$.Solving for $x_2$: $\frac{MR}{16} + \frac{7M}{8} x_2 = 0 \implies \frac{7}{8} x_2 = -\frac{R}{16} \implies x_2 = -\frac{R}{14}$.The negative sign indicates the center of mass shifts to the left by $\frac{R}{14}$.

$A$ spherical hollow is made in a lead sphere of radius $R$,such that its surface touches the outside surface of the lead sphere and passes through the center. What is the shift in the center of mass of the lead sphere due to the hollowing?

Similar Questions

$Na_2S_2O_3$ reacts with moist $Cl_2$ to form $Na_2SO_4$,$HCl$ and $X$. Which one of the following is $X$?

Ionic radius (in $\mathring{A}$) of $As^{3+}$,$Sb^{3+}$ and $Bi^{3+}$ follow the order

Let $A, B$ and $C$ be three points in a plane. The locus of a point $P$ moving such that $PA^2 + PB^2 = 2PC^2$ is a

$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ horizontal force $F$ is applied at one end of the rod perpendicular to its length. The acceleration of the centre of mass of the rod is:

In a college of $300$ students,every student reads $5$ newspapers and every newspaper is read by $60$ students. The number of newspapers is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module