Com of Solid Bodies

Q. The displacement of a particle along x axis is given by x = a sin(square)wt. Is the motion SHM ? If yes then what is its frequency?

Q.

Define rectangular components of vector.

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.

Water rises to a height of 10cm in a capillary tube and mercury falls to a depth of 3.42 cm in the same capillary tube. If the density of mercury is 13.6g/cc and the angle of contact of mercury and water are ${135}^o$ and $0^o$ respectively, the ratio of surface tension of water and mercury is

1. 1 : 3

2. 1 : 6.5

3. 1.5 : 1

4. 1 : 0.15

Q. A thin circular plate of mass $M$ and radius $R$ has its density varying as $\rho \left(r\right)={\rho }_{0}r$ with ${\rho }_0$ as constant and $r$ is the distance from its centre. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I=aM{R}^2$. The value of the coefficient $a$ is

1. $\frac{3}{2}$
2. $\frac{1}{2}$
3. $\frac{3}{5}$
4. $\frac{8}{5}$

Q. Distance of the centre of mass of a solid uniform cone from its vertex is $z_0$. If the radius of its base is $R$ and its height is $h$, then $z_0$ is equal to.

1. $\frac{3h}{4}$
2. $\frac{h^2}{4R}$
3. $\frac{5h}{8}$
4. $\frac{3h^2}{8R}$

Q.

Two blocks of masses 10 kg and 30 kg are placed along a vertical line.The first block is raised through a height of 7 cm.By what distance should the second mass be moved to raise the centre of mass by 1 cm ?

Q. A thin wire of uniform density is bend in the form of an arc of a circle which subtends an angle $\alpha$ at the centre of the circle as shown in the figure. Where is the centre of mass of this circular arc located?

1. $(0,0)$
2. $(\frac{2R\sin \frac{\alpha }{2}}{\alpha },0)$
3. $(\frac{2R\cos \frac{\alpha }{2}}{\alpha },\frac{2R\cos \frac{\alpha }{2}}{\alpha })$
4. $(0,\frac{2R\sin \frac{\alpha }{2}}{\alpha })$

Q. The respective radii of the two spheres of a spherical condenser are $12cm$ and $9cm$. The dieletric constant of the medium between them is $6$. The capacity of the conderser will be

1. $240\mu F$
2. $240F$
3. None of the above
4. $240pF$

Q. A disc has mass $9M$ . A small disc of radius $R/3$ is cut from it as shown in the figure. The moment of inertia of remaining part about an axis passing through the centre ${}^{\prime }{O}^{\prime }$ of the disc and perpendicular to the plane of the disc is :

1. $8M{R}^2$
2. $4M{R}^2$
3. $\frac{40}{9}M{R}^2$
4. $\frac{37}{9}M{R}^2$

Q. Masses 8, 2, 4, 2 kg are placed at the corners A, B, C, D respectively of a square ABCD of diagonal 80 cm. The distance of centre of mass from A will be

1. 60 cm
2. 20 cm
3. 30 cm
4. 40 cm

Q.

Three particles, each of mass 200 g, are kept at the corners of an equilateral triangle of side 10 cm. Find the moment of intertia of the system about an axis

(a) joining two of the particles and

(b) passing through one of the particles and perpendicular to the plane of the particles.

Q.

A ring has a greater moment of inertia than a circular disc of the same mass and radius, about an axis passing through its center of mass perpendicular to its plane, why?

Q. Find the centre of mass of a rectangular lamina having sides $8\text{m}$ and $6\text{m}$ as shown in figure. [Take $\text{O}$ as origin]

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

Q. Four particles $A,B,C$ and $D$ with masses $m_A=m,m_B=2m,m_C=3m$ and $m_D=4m$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is

1. $\frac{a}{5}(\hat{i}-\hat{j})$
2. $\frac{a}{5}(\hat{i}+\hat{j})$
3. Zero
4. $a(\hat{i}+\hat{j})$

Q.

One end of a cylindrical glass rod $(\mu =1.5)$ of radius 1.0 cm is rounded in the shape of a hemisphere. The rod is immersed in water $(\mu =\frac{4}{3})$ and an object is placed in the water along the axis of the rod at a distance of 8.0 cm from the rounded edge. Locate the image of the object.

Q. The centre of mass of a uniform circular ring or a disc lies at its geometric centre.

1. False
2. True

Q.

A uniform metal sphere of radius a and mass M is surrounded by a thin uniform spherical shell of equal mass and radius 4a (figure 11-E2). The centre of th shell on the surface of the inner sphere. Find th egravitational field at the points $P_1$ and $P_2$ shown in the figure.

Q.

A paperweight in the form of a hemisphere of radius 3.0 cm is used to hold down a printed page. An observer looks at the page vertically through the paperweight. At what height above the page will the printed letters near the centre appear to the observer ?

Q. A rod of length L has non-uniform linear mass density given by $\rho \left(x\right)=a+b{\left(\frac{x}{L}\right)}^2$ where a and b are constants and $0\le \text{x}\le \text{L}.$ The value of $x$ for the centre of mass of the rod is at

1. $\frac{3}{2}\left(\frac{a+b}{2a+b}\right)L$
2. $\frac{3}{2}\left(\frac{2a+b}{3a+b}\right)L$
3. $\frac{4}{3}\left(\frac{a+b}{2a+3b}\right)L$
4. $\frac{3}{4}\left(\frac{2a+b}{3a+b}\right)L$

Q. A uniform disc of diameter R/2 is put over another uniform disc of diameter R of a same thickness and density.the peripheries of the 2 discs touch each other. the centre of mass of system from centre of big disc is
1)R/10
2)R/20
3)9R/10
4)11R/10

Q. Blocks A and B are resting on a smooth horizontal surface given equal speeds of $2\text{m/s}$ in opposite sense as shown in the figure.

At $t=0$, the position of blocks are shown, then the co-ordinates of centre of mass at $t=3\text{s}$ will be

1. $(1,0)$
2. $(3,0)$
3. $(5,0)$
4. $(2.25,0)$

Q.

Mass of a ring is non-uniformly distributed around its geometric centre. If

R is radius of the ring, then

a. centre of mass does not coincide with geometric centre

b. Position centre of mass from the geometric centre will be x (0<x<R)

c. centre of mass will be nearer to the greater mass distribution

d. centre of mass may lie outside the periphery

1. All the correct

2. Only a & b are correct

3. Only b & c are correct

4. a, b & c are correct

Q.

A cone falling with a speed V0strikes and penetrates the block of a packaging material. The acceleration of cone after impact is a=g-cx^{2. Where c is a positive constant and x is the penetration distance . If maximum penetration depth is x metre then find c.}

Q.

Area of a rectangle and the area of a circle are equal. If the dimensions of the rectangle are $14cm\times 11cm$, then radius of the circle is

1. $21cm$

2. $10.5cm$

3. $14cm$

4. $7cm$

Q. Find the distance of $\text{COM}$ of a solid hemisphere of radius $16\text{cm}$ from its geometric centre.

1. $36\text{cm}$
2. $12\text{cm}$
3. $6\text{cm}$
4. $4\text{cm}$

Q. A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?

1. Angular velocity
2. Moment of inertia
3. Rotational kinetic energy
4. Angular momentum

Q. A circular plate of uniform thickness has a diameter $56\text{cm}$. A circular portion of diameter $42\text{cm}$ is removed from one edge as shown in the figure. The centre of mass of the remaining portion from the centre of plate will be :

1. $5\text{cm}$
2. $9\text{cm}$
3. $11\text{cm}$
4. $7\text{cm}$

Q.

In a TV picture tube electrons are ejected from the cathode with negligible speed and reach a velocity of $5\times {10}^{6}m/s$ in travelling one centimeter. Assuming straight line motion, find the constant force exerted on the electron. The mass of the electron is $9.1\times {10}^{-31}kg.$

Q. If mass is uniformly distributed, find the distance $\left(\text{in m}\right)$ of the centre of mass of the T-shaped plate from point $O$.

1. $\frac{19}{7}$
2. $\frac{9}{7}$
3. $\frac{7}{9}$
4. $\frac{7}{19}$