COM of Center of Masses

Q. Find the the $y-$ coordinate of the centre of mass of the system of three rods, each of length $6\text{m}$ and two rods, each of length $3\text{m}$ as arranged in the figure shown below. (Assume all rods to be of uniform density)

1. $\frac{9\sqrt{3}}{8}\text{m}$
2. $\frac{9\sqrt{3}}{16}\text{m}$
3. zero
4. $8\sqrt{3}\text{m}$

Q. Position of two particles are given by $x_1=2t$ and $x_2=2+3t$. The velocity of centre of mass at $t=2\text{s}$ is $2\text{m/s}$. Velocity of centre of mass at $t=4\text{s}$ will be

1. $2\text{m/s}$
2. $4\text{m/s}$
3. $1\text{m/s}$
4. zero

Q. Four squares $P,Q,R$ and $S$ each of side $4\text{cm}$ and uniform thickness are kept together as shown in the figure. Then the position of the centre of mass of the combination of squares with respect to the centre of mass of $P$ will be:

1. ($2.4\text{cm},2.8\text{cm})$
2. ($4\text{cm},4\text{cm})$
3. ($4\text{cm},2\text{cm})$
4. ($2\text{cm},2\text{cm})$

Q. A circular portion of radius $R/4$ has been removed from the uniform disc of radius $R$, centred at $A$ as shown in figure. Then, centre of mass of the remaining portion of the uniform disc is:

1. $\frac{R}{20}$ to the left of $A$
2. $\frac{R}{12}$ to the left of $A$
3. $\frac{R}{20}$ to the right of $A$
4. $\frac{R}{12}$ to the right of $A$

Q. Four uniform rods of different densities are connected to each other to form a square as shown in figure.


Linear mass density of $\text{A, B, C}$ and $\text{D}$ are given as $\lambda ,2\lambda ,2\lambda$ and $5\lambda$. Find out the distance of $COM$ of the square from the rod $\text{A}$ if length of each rod is $l$.

1. $l$
2. $\frac{3}{4}l$
3. $0.55l$
4. $0.45l$

Q. A circular plate of uniform thickness has a radius, $a=10\text{cm}$. A circular portion of radius, $b=1\text{cm}$ is removed from the plate as shown in figure. Find the position of center of mass of the remaining part of the circular plate. (Take $\text{O}$ to be the origin)

1. $(0,-\frac{1}{11})$
2. $(0,-\frac{9}{88})$
3. $(0,-\frac{8}{99})$
4. $(0,-\frac{10}{99})$

Q. Three uniform bricks each of length $60\text{cm}$ and mass $m$ are arranged as shown in the figure. Considering only the length of the bricks, find the distance of centre of mass of the system of bricks from the wall.

1. $45\text{cm}$
2. $55\text{cm}$
3. $65\text{cm}$
4. $35\text{cm}$

Q. Four spheres of radii $R$, $2R$, $2R$ and $3R$ respectively, are placed such that their respective centres lie at $x=2\text{m}$, $x=6\text{m}$, $x=10\text{m}$ and $x=20\text{m}$ respectively. If density of each sphere is the same, then find the $x$ coordinate of $\text{COM}$ of the system of spheres.

1. $x=10\text{m}$
2. $x=16\text{m}$
3. $x=16.44\text{m}$
4. $x=15.22\text{m}$

Q. Three hollow spheres of same mass and radius are placed as shown in figure. Find the $COM$ of the system.

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

Q. Two discs of radii $4\text{cm}$ and $2\text{cm}$ respectively are attached as shown in the figure. The distance of the new centre of mass of the system from ${\text{C}}_1$ is

1. $1.2\text{cm}$
2. $2.4\text{cm}$
3. $5.2\text{cm}$
4. $3.2\text{cm}$

Q. A smaller disc is removed from a uniform circular lamina as shown in the figure. Find the position of $COM$ of the uniform lamina, if diameter of the smaller disc is $12\text{m}$.

1. $(-6,0)\text{m}$
2. $(-6,-6)\text{m}$
3. $(-2,0)\text{m}$
4. $(-2,-2)\text{m}$

Q.

A circular disc of radius R has a uniform thickness. A circular hole of diameter equal to the radius of the disc has been cut out as shown in figure. Find the centre of mass of the remaining disc.

1. $\frac{R}{6}$ units towards left

2. $\frac{R}{6}$ units towards Right

3. $\frac{R}{3}$ units towards left

4. $\frac{R}{3}$ units towards right

Q. An infinite number of bricks are placed one over the other as shown in the figure. Each succeeding brick having half the length and breadth of its preceding brick and the mass of each succeeding brick being ${\frac{1}{4}}^{\text{th}}$ of the preceding one. Take $O$ as the origin. The $x-$coordinate of centre of mass of the system of bricks is:

1. $-\frac{a}{7}$
2. $\frac{3a}{7}$
3. $-\frac{3a}{7}$
4. $-\frac{2a}{7}$

Q. A cart of mass $M$ is at rest on a frictionless horizontal surface and a pendulum bob of mass $m$ hangs from the roof of the cart as given in the figure. The string breaks, the bob falls on the floor, makes several collision on the floor and finally lands up in a small slot made in the floor. The horizontal distance between the string and the slot is $L$ . Find the displacement of the cart during this process.

1. $\frac{mL}{M+m}$ towards right.
2. $\frac{mL}{M+m}$ towards left
3. $\frac{ML}{M+m}$ towards left
4. $\frac{ML}{M+M}$ towards right

Q. If the origin of co-ordinate system lies at the centre of mass, the sum of the moments of the masses of the system about the centre of mass

1. may be greater than zero
2. may be less than zero
3. may be equal to zero
4. is always zero

Q. Four uniform rods of mass $m\text{kg}$ each form a rectangle as shown in figure. The rods have negligible area of cross-section. Find the position of center of mass of the rectangle made up of the four uniform rods.

1. $(2,2)\text{m}$
2. $(2,0)\text{m}$
3. $(2,1)\text{m}$
4. $(4,2)\text{m}$

Q. From a circular disk of radius ($R=50\text{cm}$), a square is cut with one of its radii as the diagonal of the square (as shown in figure). The distance of the center of mass of the remaining part from the geometrical center of the disc is

1. $7.5\text{cm}$
2. $9.4\text{cm}$
3. $6.8\text{cm}$
4. $4.73\text{cm}$

Q. Three identical carrom coins, each of radius $5\text{cm}$ are placed touching each other on a horizontal surface such that an equilateral triangle is formed when the centres of the coins are joined. Find the coordinates of this centre of mass, if the origin is fixed at the centre of coin A, as shown in figure.

1. $(5\text{cm},\frac{5}{\sqrt{3}}\text{cm})$
2. $(5\sqrt{3}\text{cm},\frac{5}{\sqrt{3}}\text{cm})$
3. $(2\text{cm},\frac{5}{\sqrt{3}}\text{cm})$
4. $(5\text{cm},5\text{cm})$

Q. Four spheres of radii $R$, $2R$, $2R$ and $3R$ respectively, are placed such that their respective centres lie at $x=2\text{m}$, $x=6\text{m}$, $x=10\text{m}$ and $x=20\text{m}$ respectively. If density of each sphere is the same, then find the $x$ coordinate of $\text{COM}$ of the system of spheres.

1. $x=10\text{m}$
2. $x=16\text{m}$
3. $x=16.44\text{m}$
4. $x=15.22\text{m}$

Q. Centre of mass of three particles of masses $1\text{kg},2\text{kg}$ and $3\text{kg}$ lies at the point $(1,2,3)$ and centre of mass of another system of particles $3\text{kg}$ and $2\text{kg}$ lies at the point $(-1,3,-2)$. Where should we put a particle of mass $5\text{kg}$ so that the centre of mass of entire system lies at the centre of mass of $1^{st}$ system?

1. $(0,0,0)$
2. $(1,3,2)$
3. $(-1,2,3)$
4. $(3,1,8)$

Q. You are supplied with three identical rods of same length and mass . If the length of each rod is $2\pi$. Two of them are converted into rings and then placed over the third rod as shown in figure. If point A is considered as origin of the coordinates system, the coordinate of the centre of mass will be (you may assume AB as x - axis of the coordinate system)

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

Q. A circular portion of radius $R/4$ has been removed from the uniform disc of radius $R$, centred at $A$ as shown in figure. Then, centre of mass of the remaining portion of the uniform disc is:

1. $\frac{R}{20}$ to the left of $A$
2. $\frac{R}{12}$ to the left of $A$
3. $\frac{R}{20}$ to the right of $A$
4. $\frac{R}{12}$ to the right of $A$

Q. A system consists of mass M and m (<< M). The centre of mass of the system is

1. at the middle
2. nearer to M
3. nearer to m
4. at the position of larger mass

Q.

Find the centre of mass of uniform L-shaped lamina (a thin flat plate) with dimensions as shown in figure. The mass of lamina is 3 kg

1. $(\frac{5}{6}m,\frac{5}{6}m)$

2. $(\frac{6}{5}m,\frac{6}{6}m)$

3. $(\frac{5}{2}m,\frac{5}{6}m)$

4. $(\frac{5}{3}m,\frac{6}{5}m)$

Q. Find the co-ordinates of the centre of mass of the lamina, shown in figure.

1. $(0.75,1.75)$
2. $(0.75,1.5)$
3. $(0.5,1.75)$
4. $(0.5,1.5)$

Q. Two particles of equal mass have coordinates $(2\text{m},4\text{m},6\text{m})$ and $(6\text{m},2\text{m},8\text{m})$. Of these one particle has a velocity $\overrightarrow{v_1}=\left(2\hat{i}\right)\text{m/s}$ and another particle has velocity $\overrightarrow{v_2}=\left(2\hat{j}\right)\text{m/s}$ at time $t=0$. The coordinate of their centre of mass at time $t=1\text{s}$ will be

1. $(4\text{m},4\text{m},7\text{m})$
2. $(5\text{m},4\text{m},7\text{m})$
3. $(2\text{m},4\text{m},6\text{m})$
4. $(4\text{m},5\text{m},4\text{m})$

Q. A circular disc of radius $R$ is removed from a bigger circular disc of radius $2R$ such that their circumferences coincide. The centre of mass of the new disc is at a distance of $\alpha R$ from the centre of the bigger disc. The value of $\alpha$ is

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

Q. Two squares of size $(a\times a)$ each are removed from a bigger square of size $(2a\times 2a)$ as shown in the figure. Removed parts are unshaded in the figure given below. Find the coordinates of the $COM$ w.r.t the origin $\text{O}$.

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

Q. In the figure, the L-shaped shaded piece is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the L-shaped piece is

1. 1
2. 2
3. 3
4. 4

Q. A circular plate of uniform density has a diameter of $56\text{cm}$. A circular portion of diameter $42\text{cm}$ is removed from the original circular plate as shown in figure. Find the position of the $\text{COM}$ of the remaining portion, from the centre of the original plate.

1. $7\text{cm}$ towards left
2. $7\text{cm}$ towards right
3. $9\text{cm}$ towards left
4. $9\text{cm}$ towards right