COM of Other Bodies

Q.

The density of a linear rod of length 'L' varies as p = A+Bx where x is the distance from the left end. Then the position of the centre of mass from the left end is

1. $\frac{3AL+2B{L}^2}{3(2A+BL)}$

2. $\frac{AL+3B{L}^2}{(2A+BL)}$

3. $\frac{3AL+2B{L}^2}{(2A+BL)}$

4. $\frac{3AL+2B{L}^2}{3(2A+BL)}$

Q. The centre of mass of a hollow cylinder and a solid cylinder lie at different points along the axis of the cylinder.

1. True
2. False

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. 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. The length of a steel rod is $5cm$ greater than that of a brass rod. If this difference in their lengths is to remain the same at all temperatures, then the length of brass rod will be:

1. $15cm$
2. $20cm$
3. $5cm$
4. $10cm$

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

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 COM of the body shown below is close to

1. A
2. B
3. C
4. D

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. $1000{\text{m}}^3$ 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. $\frac{90}{\pi }\text{m}$ from the centre of the base.
2. $\frac{30}{\pi }\text{m}$ from the centre of the base.
3. $\frac{12}{\pi }\text{m}$ from the top of the pyramid.
4. $\frac{3}{\pi }\text{m}$ from the top of the pyramid.

Q. A machinist starts with three identical square plates but cuts one corner from one of them, two corners from the second and three comers from the third. Rank the three according to the x - coordinate of their of mass, from smallest to largest.

Arrange in ascending order. For eg if the order is $x_1<x_2<x_3$, answer is 123

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

Q. If this aluminum rod measure a length of steel as 88.42 cm at ${35}^{\circ }C$, what is the correct length of the steel at ${35}^{\circ }C$ ?

1. $44.49cm$
2. $58.49cm$
3. $88.49cm$
4. $77.49cm$

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

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