COM of Center of Masses

Q. Two bullets are fired horizontally and simultaneously towards each other from the rooftops of two buildings located $200\text{m}$ apart, ot the same height of $300\text{m}$ with the same velocity of $50\text{m/s}$. Which of the following is/are correct? [Take $g=10{\text{m/s}}^2$]

1. They collide after $2\text{s}$.
2. They collide after $1\text{s}$.
3. They collide at a height of $290\text{m}$.
4. They collide at a height of $280\text{m}$.

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

Q. A uniform ${}^{\prime }{T}^{\prime }$ shaped object with dimensions as shown in the figure, is lying on a smooth floor. A force $\overrightarrow{F}$ is applied at the point P parallel to limb $AB$, such that the object has only the translational motion without rotation. Find the distance of point $P$ from $C$

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

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

Q. A circular plate of uniform thickness has a diameter of $56\text{cm}$. A circular portion of diameter $42\text{cm}$ is removed from the edge of the plate. The position of centre of mass of the remaining portion is

1. $8\text{cm}$ from centre of bigger circle to the right.
2. $9\text{cm}$ from centre of bigger circle to the left.
3. $10\text{cm}$ from centre of bigger circle to the left.
4. $5\text{cm}$ from centre of bigger circle to the left.

Q.

What is the dimensional formula for the radius of gyration?

Q. From a uniform square plate, one fourth part is removed as shown in figure. The centre of mass of remaining part will lie on line joining

1. $OB$
2. $OA$
3. $OC$
4. $OD$

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 steel and a brass wire, each of length 50 cm and cross-sectional area $0.005c{m}^2$ hang from a ceiling and are 15 cm apart. Lower ends of the wires are attached to a light horizontal bar. A suitable downward load is applied to the bar so that each of the wires extends in length by 0.1 cm. At what distance from the steel wire the load must be appied?

[Young's modulus of steel is $2\times {10}^{12}dynes/c{m}^2$] and that of brass is $1\times {10}^{12}dynes/c{m}^2$

1. 7.5 cm
2. 10 cm
3. 3 cm
4. 5 cm

Q. The position of the centre of mass of the $\text{T}$ shaped plate from reference point $\text{O}$ as shown in the figure will be:

1. $7.14\text{cm}$
2. $5.28\text{cm}$
3. $4\text{cm}$
4. $6.25\text{cm}$

Q. A circular plate of diameter $a$, is kept in contact with a square plate of side $a$ as shown in figure. The density of the material and thickness are same everywhere. The centre of mass of the composite system will be :

1. Inside the circular plate
2. Inside the square plate
3. At the point of contact
4. Outside the system

Q.

Four persons K, L, M and N are initially at the corners of a square of side of length d. If every person starts moving, such that K is always headed towards L, L towards M, M is headed directly towards N and N towards K, then the four persons will meet after (assume that they move with a constant speed v)

1. 
2. 
3. 
4. 

Q. A uniform rectangular thin sheet ABCD of mass M has length a and breadth b, as shown in the figure. If the shaded portion 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. $(\frac{5a}{12}.\frac{5b}{12})$
3. $(\frac{3a}{4}.\frac{3b}{4})$
4. $(\frac{5a}{3}.\frac{5b}{3})$

Q. In the letter $E$ whose dimensions are given in the figure, origin is at the bottom left corner. Area of each rectangle is $12{\text{cm}}^2$ and area of the square is $4{\text{cm}}^2$. Assume that the weights are proportional to areas. Then, centre of gravity of the $E$ frame will be at:

1. $(1\text{cm},3\text{cm})$
2. $(2\text{cm},2\text{cm})$
3. $(2.4\text{cm},5\text{cm})$
4. $(3\text{cm},6\text{cm})$

Q. Find the position of the centre of mass of $T$ - shaped lamina of negligible thickness as shown in figure. Assume the origin to be at the intersection of axes and take the mass density of the lamina, $\sigma =1{\text{kg/m}}^2$

1. $(0,6.5)\text{m}$
2. $(0,6)\text{m}$
3. $(0,5.28)\text{m}$
4. $(0,4.55)\text{m}$

Q. When a block is placed on a wedge as shown in the figure. The block starts sliding down and the wedge also starts sliding on ground. All surfaces are rough. The centre of mass of (wedge + block) system will move

1. Leftward and downward
2. Rightward and downwards.
3. Leftward and upwards.
4. Only downward.

Q. Look at the drawing given in the figure, which has been drawn with ink of uniform line thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6m$. The coordinates of centres of the different parts are: outer circle ($0,0$), left inner circle ($-a,a$), right inner circle ($a,a$), vertical line ($0,0$), and horizontal line $(0,-a)$. The $y-$ coordinate of centre of mass of the ink in this drawing is:

1. $\frac{a}{10}$
2. $\frac{a}{8}$
3. $\frac{a}{12}$
4. $\frac{a}{3}$

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 circular plate of uniform thickness has a diameter of $28\text{cm}$. A circular portion of diameter $21\text{cm}$ is removed from the plate as shown. $O$ is the centre of mass of the complete plate. The position of centre of mass of the remaining portion will shift towards left from $O$ by

1. $9\text{cm}$
2. $4.5\text{cm}$
3. $5.5\text{cm}$
4. $5\text{cm}$

Q. Five uniform circular plates, each of diameter $b$ and mass $m$, are placed together to form a pattern as shown in the figure. Find the $y$ coordinate of the centre of mass of the system.

1. $\frac{4b}{5}$
2. $\frac{b}{3}$
3. $\frac{b}{5}$
4. $\frac{2b}{5}$

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 carpenter has constructed a toy as shown in the adjoining figure. If the density of the material of the sphere is $12$ times that of cone, the position of the centre of mass of the toy is given by

1. At a distance of $2R$ from O
2. At a distance of $3R$ from O
3. At a distance of $4R$ from O
4. At a distance of $5R$ from O

Q. A wire of uniform cross-section is bent in the shape shown in the figure. The coordinates of the centre of mass of each side are shown in the figure. Origin is taken at $O$. Find the coordinates of the center of mass of the given system.

1. $(\frac{15l}{14},\frac{6l}{7})$
2. $(\frac{15l}{14},l)$
3. $(l,\frac{l}{2})$
4. $(l,l)$

Q. As shown in the figure, a triangular sheet is removed from a uniform rectangular sheet. Then, shift of the centre of mass for the new shape will be:

1. $3.2\text{cm}$
2. $5.67\text{cm}$
3. $6.67\text{cm}$
4. $4.2\text{cm}$

Q. In the figure shown $v=2\text{m/s}$, $\omega =5\text{rad/s}$ and $\text{CP}=1\text{m}$.


In terms of $\hat{i}$ and $\hat{j}$ , find linear velocity of point $\text{P}$.

1. $4\hat{i}-5\hat{j}$
   
2. $5\hat{i}-4\hat{j}$
3. $-5\hat{i}-4\hat{j}$
4. $5\hat{i}-5\hat{j}$

Q. Three identical circular plates of radius $R$ each are placed on a horizontal surface touching one another as shown in the figure. The y coordinate of the center of mass of the system is [Consider $\text{A}$ to be at the origin]

1. $\frac{R}{3}$
2. $\sqrt{3R}$
3. $\frac{R}{\sqrt{3}}$
4. $R$

Q. A mortar fires a shell of mass $M$ which explodes into two pieces of mass $\frac{M}{5}$ and $\frac{4M}{5}$ at the top of the trajectory. The smaller mass falls very close to the mortar. In the same time the bigger piece lands a distance $D$ from the mortar. The shell would have fallen at a distance $R$ from the mortar if there was no explosion. The value of $D$ is
(Neglect air resistance)

1. $\frac{3R}{2}$
2. $\frac{4R}{3}$
3. $\frac{5R}{4}$
4. None of these

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

Q. The disk of radius $5\text{cm}$ is cut from the uniform disc of radius $10\text{cm}$ in such a way that the edge of the hole touches the edge of the disc. What is the distance of the center of mass of the remaining portion w.r.t the point $\text{O}$ ?

1. $2.66\text{cm}$
2. $6\text{cm}$
3. $1.66\text{cm}$
4. $2\text{cm}$

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