COM of Other Bodies

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

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

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. The COM of the body shown below is close to

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

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. 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. $U_S>U_R$