COM of Other Bodies

Q. Surface mass density of a semicircular disk varies with position as $\sigma ={\sigma }_0\frac{r}{R}$ where ${\sigma }_0$ is constant, $R$ is the radius of the disc and $r$ is measured from the centre of the disc. Find the COM of the disc.

1. $(0,\frac{3R}{2\pi })$
2. $(0,\frac{4R}{3\pi })$
3. $(0,\frac{2R}{\pi })$
4. $(0,\frac{R}{2})$

Q.

A uniform disc of radius R is put over another uniform disc of radius 2R of the same thickness and density.The peripheries of the two discs touch each other.Locate the centre of mass of the system.

Q. In the figure shown below, a quarter ring of radius $r$ is placed in the first quadrant of a cartesian co-ordinate system, with centre at origin. Find the co-ordinates of $COM$ of the quarter ring.

1. $(\frac{2R}{\pi },0)$
2. $(0,\frac{2R}{\pi })$
3. $(0,0)$
4. $(\frac{2R}{\pi },\frac{2R}{\pi })$

Q. The position of the centre of mass of a uniform semicircular wire of radius $R$ placed in $x-y$ plane with its centre at the origin and the line joining its ends as x-axis is given by$(0,\frac{xR}{\pi }).$ Then, the value of $|x|$ is . (Integer only)

Q. A thin sheet of metal of uniform thickness is cut into the shape bounded by the line $x=a$ and $y=\pm k{x}^2$ as shown.
Find the coordinates of the centre of mass.
[Here, $a\&k$ are positive constant]

1. $(\frac{3a}{2},0)$
2. $(\frac{3a}{4},0)$
3. $(0,\frac{3a}{4})$
4. $(3a,0)$

Q. Seven homogeneous bricks, each of length L, are arranged as shown in figure (9.E2). Each brick is displaced with respect to the one in contact by $\frac{L}{10}$. Find the x-coordinate of the centre of mass relative to the origin shown.

1. $\frac{21}{35}L$
2. $\frac{L}{2}$
3. $\frac{22L}{35}$
4. None of these

Q. A solid cone has a height of $20\text{cm}$. Find the distance of center of mass of the cone from the center of the base of the cone.

1. $5\text{cm}$
2. $10\text{cm}$
3. $15\text{cm}$
4. $12.5\text{cm}$

Q. Find the co-ordinates of the center of mass of a hollow cone having a height of $90\text{cm}$ as shown in the figure.

1. $(0,45)\text{cm}$
2. $(0,30)\text{cm}$
3. $(0,25)\text{cm}$
4. $(0,60)\text{cm}$

Q. A uniform rectangular lamina $\text{ABCD}$ is of mass $M$, length $a$ and breadth $b$, as shown in the figure. If the shaded portion $\text{HBGO}$ is cut-off, the coordinates of the centre of mass of the remaining portion will be

1. $(\frac{2a}{3},\frac{2b}{3})$
2. $(\frac{5a}{3},\frac{5b}{3})$
3. $(\frac{3a}{4},\frac{3b}{4})$
4. $(\frac{5a}{12},\frac{5b}{12})$

Q. Find the height (in cm) of the center of mass of a solid hemisphere of radius $R=8\text{cm}$ from geometric center of its base.

Q.
The figure shows a metallic solid block, placed in a way so that its faces are parallel to the coordinate axes. Edge lengths along axis $x,y$ and $z$ are $a,b$ and $c$ respectively. The block is in a region of uniform magnetic field of magnitude $30mT$. One of the edge length of the block is $25cm$. The block is moved at $4m/s$ parallel to each axis and in turn, the resulting potential difference $V$ that appears across the block is measured.
(1)When the block is moved at $4m/s$ parallel to the $y$- axis, $V=24mV$,
(2)when the block is moved at $4m/s$ parallel to the $z$- axis $V=36mV$,
(3)when the block is moved at $4m/s$ parallel to the $x$- axis, $V=0$
Using the given information, correctly match the dimensions of the block in column-I with the values given in column-II.
$\begin{array}{cc}\text{column I}& \text{Column II}\\ \text{(A) a}& \text{(p) 20 cm}\\ \text{(B) b}& \text{(q) 24 cm}\\ \text{(C) c}& \text{(r) 25 cm}\\ \text{(D)}\frac{bc}{a}& \text{(s) 30 cm}\\ & \text{(t) 26 cm}\end{array}$

1. $A\to p,B\to q,C\to s,D\to t$
2. $A\to s,B\to p,C\to s,D\to t$
3. $A\to q,B\to s,A\to r,D\to p$
4. $A\to r,B\to s,C\to p,D\to q$

Q. Find the position of the center of mass of a hollow hemisphere of radius $R=10\text{cm}$. Assume center $\text{O}$ as origin.

1. $(0,10)\text{cm}$
2. $(0,2.5)\text{cm}$
3. $(0,5)\text{cm}$
4. $(0,0)\text{cm}$

Q. A square of side $2\text{m}$ has one of its corners fixed at the origin as shown in the figure. If the surface density of the square varies with distance $x$ from one side to another as $\sigma =5x{\text{kg/m}}^2$, find the $x$ coordinate of the position of the centre of mass of the square.

1. $\frac{1}{3}\text{m}$
2. $\frac{2}{3}\text{m}$
3. $\frac{4}{3}\text{m}$
4. $1\text{m}$

Q. A uniform circular disc of mass M and radius a is given. A part of the radius b is removed from it. The distance between centre of original disc and circular hole's centre is c. The new position of C.M. from O is:

1. $\frac{bc}{a-b}$
2. $\frac{b^2}{b-a}$
3. $\frac{-b^{2}c}{a^2-b^2}$
4. $\frac{b^{2}c}{a^2+b^2}$

Q. Surface mass density of a semicircular disk varies with position as $\sigma ={\sigma }_0\frac{r}{R}$ where ${\sigma }_0$ is constant, $R$ is the radius of the disc and $r$ is measured from the centre of the disc. Find the COM of the disc.

1. $(0,\frac{R}{2})$
2. $(0,\frac{2R}{\pi })$
3. $(0,\frac{3R}{2\pi })$
4. $(0,\frac{4R}{3\pi })$

Q. The variation of density of a cylindrical thick and long rod, is $\rho ={\rho }_0\frac{x^2}{L^2}$ then position of its centre of mass from $x=0$ end is:

Q. For a square sheet of side $1\text{m}$ having uniform surface density, the position of $\text{COM}$ is at $x_1$. On the other hand, if surface density is varying as $\sigma =2x{\text{kg/m}}^2$, $\text{COM}$ is observed to be at position $x_2$. The distance between $x_2$ and $x_1$ is :

1. $\frac{1}{3}\text{m}$
2. $\frac{1}{6}\text{m}$
3. $\frac{2}{3}\text{m}$
4. $\frac{1}{2}\text{m}$

Q. A rectangular lamina in the $x-y$ plane is shown below:


Density of this lamina is constant along $x$ axis and varies along $y$ axis. If the variation in density is given as $\rho \left(y\right)=16{\rho }_0{y}^2$, then

1. $x_{COM}=\frac{l}{2}$
2. $y_{COM}=\frac{b}{2}$
3. $x_{COM}=\frac{l}{4}$
4. $y_{COM}=\frac{3b}{4}$

Q. The disc of mass $M$ with uniform surface mass density $\sigma$ is shown in the figure. The centre of mass of the quarter disc $($the shaded area$)$ is at the position $(\frac{x}{3}\frac{R}{\pi },\frac{x}{3}\frac{R}{\pi })$, then, $x$ is _______.

$($Round off to the nearest integer$)$
$(R$ is the radius of disc$)$

Q. Find the co-ordinates of the center of mass of a hollow cone having a height of $90\text{cm}$ as shown in the figure.

1. $(0,45)\text{cm}$
2. $(0,60)\text{cm}$
3. $(0,30)\text{cm}$
4. $(0,25)\text{cm}$

Q. Surface mass density of a semicircular disk varies with position as $\sigma ={\sigma }_0\frac{r}{R}$ where ${\sigma }_0$ is constant, $R$ is the radius of the disc and $r$ is measured from the centre of the disc. Find the COM of the disc.

1. $(0,\frac{R}{2})$
2. $(0,\frac{2R}{\pi })$
3. $(0,\frac{3R}{2\pi })$
4. $(0,\frac{4R}{3\pi })$

Q. $1000{\text{m}}^3$ of sand is being poured on the ground. It accumulates on the ground in the shape of a conical pyramid. If the radius of the base of pyramid is $5\text{m}$, where is the centre of mass of the pile of sand located?

1. $\frac{12}{\pi }\text{m}$ from the top of the pyramid.
2. $\frac{3}{\pi }\text{m}$ from the top of the pyramid.
3. $\frac{90}{\pi }\text{m}$ from the centre of the base.
4. $\frac{30}{\pi }\text{m}$ from the centre of the base.

Q. Two bodies of masses $m_1$ and $m_2$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

1. ${\left[\frac{r}{2G\left(m_1{m}_2\right)}\right]}^{1/2}$
2. ${\left[\frac{2G}{r}{m}_1{m}_2\right]}^{1/2}$
3. ${\left[2G\frac{(m_1-m_2)}{r}\right]}^{1/2}$
4. ${\left[\frac{2G}{r}\right(m_1+m_2\left)\right]}^{1/2}$

Q. An iron rod of length $2m$ and cross-sectional area of $50m{m}^2$ stretched by $0.5mm$, when a mass of $250kg$ is hung from its lower end. Young's modulus of iron rod is

1. $19.6\times {10}^{10}N/m^2$
2. $19.6\times {10}^{15}N/m^2$
3. $19.6\times {10}^{20}N/m^2$
4. $19.6\times {10}^{18}N/m^2$

Q. A long straight wire along the Z-axis carries a current ${}^{\prime }{I}^{\prime }$ in the negative Z-direction. The induced magnetic field $B$ at a point having coordinates $(x,y)$ is:

1. $\frac{{\mu }_{0}I}{2\pi }\frac{(x\hat{i}-y\hat{j})}{(x^2+y^2)}$
2. $\frac{{\mu }_{0}I}{2\pi }\frac{(x\hat{j}-y\hat{i})}{(x^2+y^2)}$
3. $\frac{{\mu }_{0}I}{2\pi }\frac{(x\hat{i}+y\hat{j})}{(x^2+y^2)}$
4. $\frac{{\mu }_{0}I}{2\pi }\frac{(y\hat{i}-x\hat{j})}{(x^2+y^2)}$

Q. A solid cone of height $2R$ and base radius $R$ is placed over a solid hemisphere as shown in figure. Cone and hemisphere are made up of the same material. Height of center of mass of system from the ground is

1. $\frac{5}{8}$$R$
2. $\frac{3}{2}$$R$
3. $\frac{17}{16}$$R$
4. $R$

Q. Find the co-ordinates of the center of mass of a hollow cone having a height of $90\text{cm}$ as shown in the figure.

1. $(0,45)\text{cm}$
2. $(0,60)\text{cm}$
3. $(0,30)\text{cm}$
4. $(0,25)\text{cm}$

Q. Assuming gravitational potential energy ${}^{\prime }{U}^{\prime }$ at ground level to be zero.

All objects are made up of same material.
$U_P=$ gravitational potential energy of solid sphere
$U_Q=$ gravitational potential energy of solid cube
$U_R=$ gravitational potential energy of solid cone
$U_S=$ gravitational potential energy of solid cylinder

1. $U_S>U_P$
2. $U_Q>U_S$
3. $U_P>U_Q$
4. $U_S>U_R$

Q.

Match the Columns:

Column 1	Column 2
Point in a body	$-F_{BA}$
$F_{AB}$	Geometric centre
Spherical body	Centre of mass

Q. Infinite number of masses, each of mass m, are placed along a straight line at distance of r, 2r, 4r, 8r, etc. from a reference point O

1. $\frac{5Gm}{4r^2}$
2. $\frac{4Gm}{3r^2}$
3. $\frac{3Gm}{2r^2}$
4. $\frac{2Gm}{r^2}$