COM of Center of Masses

Q. A wedge of mass 2m and a cube of mass m is shown in figure. Between cube and wedge there iso friction. The minimum coefficient of friction between wedge and ground so that wedge does notmove isLOS2m45\ast(1)0.20(2)0.25(3) 0.10(4) 0.50

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. The position of centre of mass of a system consisting of two particles of masses $m_1$ and $m_2$ separated by a distance L apart, from $m_1$ will be

1. $\frac{m_{1}L}{m_1+m_2}$
2. $\frac{m_{2}L}{m_1+m_2}$
3. $\frac{m_2}{m_1}\text{L}$
4. $\frac{L}{2}$

Q. The position of the centre of mass of a uniform bar made up of three rods each having length $2\text{m}$ and negligible area of cross-section as shown in figure is
[Assume masses of the rods are equal]

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

Q. An isosceles triangle is cut out of a square sheet with given dimensions as shown in figure. Find the distance of centre of mass of system from the $x$ axis.


Consider the material to be of uniform density.

1. $7.29\text{cm}$
2. $3.81\text{cm}$
3. $5.25\text{cm}$
4. $9.5\text{cm}$

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. One quadrant of a circular disc is removed from the original disc of radius $5\text{cm}$. Take reference axes and origin $O$ as shown in the figure. Find the sum of the magnitudes (in $\text{cm}$) of the $x$ and $y$ coordinates of the $\text{COM}$ of the new shape. Consider the material to be of uniform density.

1. $6\text{cm}$
2. $5.12\text{cm}$
3. $4.24\text{cm}$
4. $1.41\text{cm}$

Q. From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

Q. Four homogeneous brick, each of length $L=4\text{m}$, are arranged as shown in the figure. Each brick is displaced with respect to one in contact by $\frac{L}{8}.$ Find the $x-$ coordinate of centre of mass relative to origin $O$.

1. $2.5\text{m}$
2. $1.75\text{m}$
3. $2\text{m}$
4. $2.75\text{m}$

Q. Two identical uniform rods $\text{EF}$ and $\text{GH}$, each of length $L=6\text{m}$ are joined as shown in figure. Locate the centre of mass of the frame.
[Point $\text{O}$, which is the point where the axes of the rods meet is the origin]

1. $(0\text{m},0\text{m})$
2. $(0\text{m},1\text{m})$
3. $(0\text{m},1.5\text{m})$
4. $(0\text{m},2\text{m})$

Q. Two bullets are fired horizontally and simultaneously towards each other from the rooftops of two buildings located $200\text{m}$ apart, at 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. a force F= Mg is applied on the top of a circular ring of mass M and radius Rkept on a rough horizontal surface.if the ring rolls without slipping then minimum coefficient of friction required

Q. Three point masses $1\text{kg},1.5\text{kg},2.5\text{kg}$ are placed at the vertices of a triangle with sides $3\text{cm},4\text{cm}$ and $5\text{cm}$ as shown in the figure. The location of the centre of mass with respect to the $1\text{kg}$ mass is:

1. $0.6\text{cm}$ to the right of $1\text{kg}$ and $2\text{cm}$ above $1\text{kg}$ mass
2. $0.9\text{cm}$ to the right of $1\text{kg}$ and $2\text{cm}$ above $1\text{kg}$ mass
3. $0.9\text{cm}$ to the left of $1\text{kg}$ and $2\text{cm}$ above $1\text{kg}$ mass
4. $0.9\text{cm}$ to the right of $1\text{kg}$ and $1.5\text{cm}$ above $1\text{kg}$ mass

Q. From an uniform circular sheet of radius $a$ units, a circular portion of diameter $a$ units has been removed as shown in the figure. Find, the coordinates of centre of mass of the remaining part.

1. $(-\frac{a}{6},0)$
   
2. $(-\frac{a}{3},0)$
   
3. $(-\frac{a}{4},0)$
   
4. $(-\frac{a}{6},\frac{a}{2})$
   

Q. A square frame is made using four thin uniform rods of length $l$ each having linear mass densities $\lambda ,2\lambda ,3\lambda$ and $4\lambda$ as shown in the figure. Which of the following is correct?

1. $x_{cm}=0,y_{cm}=0$
2. $x_{cm}>0,y_{cm}>0$
3. $x_{cm}<0,y_{cm}<0$
4. $x_{cm}=0,y_{cm}<0$

Q. When a circular portion of radius $\beta$ has been removed from a disc of uniform mass and radius $r$ such that centre of the hole is at a distance $\eta$ from the centre of the original disc. Find the position of centre of mass of the remaining part from the centre $\text{C}$ as shown in figure.

1. $x=\frac{\beta {\eta }^2}{r^2+{\beta }^2}$
2. $x=\frac{\eta \beta }{r+\beta }$
3. $x=\frac{-\eta {\beta }^2}{r+\beta }$
4. $x=-\frac{\eta {\beta }^2}{(r^2-{\beta }^2)}$

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. Two uniform rectangular plates having mass density ($\text{mass/area}$) $2{\text{kg/m}}^2$ and $1{\text{kg/m}}^2$ are joined together to form the $\text{L}$- shaped lamina as shown in the figure. Find the coordinates of centre of mass of the $\text{L}$-shaped lamina.

1. $(\frac{17}{11},\frac{83}{22})\text{m}$
2. $(\frac{26}{7},\frac{43}{14})\text{m}$
3. $(\frac{26}{7},\frac{83}{22})\text{m}$
4. $(\frac{13}{7},\frac{43}{14})\text{m}$

Q. Three uniform bricks each of length $100\text{cm}$ and mass $m$ are arranged as shown in figure. The distance of centre of mass of the system of bricks from the wall is

1. $91\text{cm}$
2. $100\text{cm}$
3. $75\text{cm}$
4. $125\text{cm}$

Q. A machinist starts with three identical square plates, but cuts one corner from one of them, two corners from the second and three corners from the third. Rank the three according to the x-coordinate of their centre of mass, from smallest to largest. (Assuming that the side of each square removed is less then the half of the side of the square plate)

1. 1, 2, 3
2. 1, 3, 2
3. 3, 2, 1
4. Either a or b

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 art. The string breaks, the bob falls on the floor, makes several collisions 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 the process ?

1. 

2. 

3. 

4. 

Q. A sphere of mass m is given some angular velocity about a horizontal axis through its centre, and gently placed on a plank of mass m. The coefficient of friction between the two is μ. The plank rests on a smooth horizontal surface. The initial acceleration of the sphere relative to the plank will be

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. 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. 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. 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. A uniform $\text{L}$ shaped lamina with given dimensions is shown in the figure. Origin is taken at $\text{O}$.


If the mass of the lamina is $6\text{kg}$, then the centre of mass of the lamina is located at:

1. $(\frac{5}{3}\text{m},\frac{5}{6}\text{m})$
2. $(\frac{5}{4}\text{m},\frac{5}{6}\text{m})$
3. $(\frac{5}{3}\text{m},\frac{5}{3}\text{m})$
4. $(\frac{5}{8}\text{m},\frac{5}{3}\text{m})$
   

Q. Find the centre of mass of a uniform $L$ shaped lamina (a thin flate plate) with dimensions as shown in figure. The mass of the lamina is $6\text{kg}$.

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

Q. A T shaped object, having uniform linear mass density, with dimension shown in the figure is lying on a smooth floor. A force $F$ is applied at the point P parallel to AB, such that the object has only translational motion without rotation. The distance of P with respect to C is

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

Q. What is the $y$ coordinate of the centre of mass of a semi-circular disc as shown in the figure? The smaller disc has been cut from the larger one. Consider the discs to have uniform density. Consider the axes and origin as shown in figure.

1. $19.8\text{cm}$
2. $25\text{cm}$
3. $29.3\text{cm}$
4. $8\text{cm}$