A Half Disc

Q. The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass $4\text{kg}$. (The coordinates of the same are shown in figure) are:

1. $\text{1.25 m, 1.50 m}$
2. $\text{0.75 m, 1.75 m}$
3. $\text{0.75 m, 0.75 m}$
4. $\text{1 m, 1.75 m}$

Q. In the figure shown, a hole of radius $2\text{cm}$ is made in a semicircular disc of radius $6\pi$ at a distance $8\text{cm}$ from the centre $\text{C}$ of the disc. The distance of the centre of mass of this system from point $\text{C}$ is: (COM of semicircle of radius $r$ is at distance $\frac{4r}{3\pi }$ from the base)

1. $8\text{cm}$
2. $6\text{cm}$
3. $4\text{cm}$
4. $12\text{cm}$

Q. In the figure shown, find out centre of mass of a system of a uniform circular plate of radius $3R$ from $\text{O}$ in which a hole of radius $R$ is cut whose centre is at $2R$ distance from the centre of large circular plate.

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

Q. Find the height of the center of mass of a semi-circular disk of radius $R=20\text{cm}$ from point $\text{O}$ (base center).

1. $\frac{60}{3\pi }\text{cm}$
2. $\frac{40}{3\pi }\text{cm}$
3. $\frac{20}{3\pi }\text{cm}$
4. $\frac{80}{3\pi }\text{cm}$

Q.

Three identical metal balls each of radius "R” are placed on a horizontal plane touching each other such that their centres form the vertices of an equilateral triangle. The distance of the centre of mass of the system from the centre of any one of the balls is

1. 

2. R/2

3. 

4. R

Q. Two long parallel conductors are placed at right angles to a metre scale at the $2cm$ and $4cm$ marks, as shown in the figure. They carry currents of $1A$ and $3A$ respectively. They will produce zero magnetic field at the

1. $1cm$ mark
2. $8cm$ mark
3. $0.5cm$ mark
4. $2.5cm$ mark

Q. A sonometer wire of length 114 cm is fixed at the both the ends. Where should the two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 1:3:4?

1. at 36 cm and 84 cm from one end
2. at 24 cmand 72 cm from one end
3. at 48 cm and 96 cm from one end
4. at 72 cm and 96 cm from one end

Q. A circular hole of radius $1cm$ is cut off from a disc of radius $6cm$. The centre of hole is $3m$ from the centre of the disc. The position of centre of mass of the remaining disc from the centre of disc is:

1. $-\frac{3}{35}cm$
2. $\frac{1}{35}cm$
3. $\frac{3}{10}cm$
4. $Noneofthese$

Q. Natural length of a spring is $60cm$ and its spring constant is $4000N/m$. A mass of $20kg$ is hung from it. The extension produced in the spring is (take g = $9.8m/s^2$):

1. $4.9cm$
2. $0.49cm$
3. $0.94cm$
4. $9.4cm$

Q. A body sliding on asmooth inclined plane requires $4$sec to reach the bottom after starting from rest at the top.How much time does it take to cover one fourth the distance starting from the top.

Q. An elongation of $0.1$% in a wire of cross-section ${10}^{-6}{m}^2$ causes a tension of $100N$. $Y$ for the wire is

1. ${10}^{12}N/m^2$
2. ${10}^{11}N/m^2$
3. ${10}^{10}N/m^2$
4. $100N/m^2$

Q. A long wire hangs vertically with its upper end clamped. A torque of $8Nm$ applied to the free end twists it through ${45}^{\circ }$. The potential energy of the twisted wire is

1. $\pi J$
2. $\frac{\pi }{2}J$
3. $\frac{\pi }{8}J$
4. $\frac{\pi }{4}J$

Q. The moment of inertia of a uniform circular disc of radius 'R' and mass 'M' about an axis passing from the edge of the disc and normal to the disc is-

1. $\frac{1}{2}M{R}^2$
2. $\frac{7}{2}M{R}^2$
3. $\frac{3}{2}M{R}^2$
4. $M{R}^2$

Q. A spherical body of area $A$ and emissivity $e=0.6$ is kept inside a perfectly black body. Total heat radiated by the body at temperature T is :

1. $0.4eA{T}^4$
2. $0.8eA{T}^4$
3. $0.6eA{T}^4$
4. $1.0eA{T}^4$

Q. Vector $\overline{A}$ has $x$ and $y$ components of -8.70cm and 15.0 cm respectively. $\overline{B}$ vector has $x$ and $y$ components of 13.2 $cm$ and -6.60cm respectively. lf $\overline{A}-\overline{B}+3\overline{C}=0$ then $x$ and $y$ components of $\overline{C}$ are:

1. 7.30cm and -7.20cm
2. -7.20cm and 7.30cm
3. -7.30cm and 7.20cm
4. -7.30cm and -7.20cm

Q. A uniform rod is hinged as shown in the figure and is released from a horizontal position.The angular velocity of the rod as it passes the vertical position is:

1. $\sqrt{\frac{12g}{3l}}$
2. $\sqrt{\frac{2g}{3l}}$
3. $\sqrt{\frac{24g}{7l}}$
4. $\sqrt{\frac{3g}{7l}}$

Q. Three identical spheres A, B, C each of radius $R$ are placed touching one another on a horizontal table. Where is the centre of mass of the system located relative to A (say)?

Q. From a uniform circular disc of mass $M$ and radius $R$ a small circular disc of radius $R/2$ is removed in such a way that both have a common tangent. Find the distance of centre of mass of remaining part from the centre of original disc.

Q. Find the height of the center of mass of a semi-circular disk of radius $R=20\text{cm}$ from point $\text{O}$ (base center).

1. $\frac{40}{3\pi }\text{cm}$
2. $\frac{60}{3\pi }\text{cm}$
3. $\frac{20}{3\pi }\text{cm}$
4. $\frac{80}{3\pi }\text{cm}$

Q. Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_1$ and $R_2$ (figure 9 - E5)

1. $=\frac{4}{\pi }\frac{(R_2^2+R_1^2+R_1{R}_2)}{R_1+R_2}$ above the centre
2. $=\frac{2}{3\pi }\frac{(R_2^2+R_1^2+2R_1{R}_2)}{R_1+R_2}$ above the centre
3. $=\frac{4}{3\pi }\frac{(R_2^2+R_1^2+R_1{R}_2)}{R_1+R_2}$ above the centre
4. None of these

Q. A uniformly charged disk of radius $35.0cm$ carries charge with a density of $7.90\times {10}^{-3}C/m^2$. Calculate the electric field on the axis of the disk at (a) $5.00cm$, (b) $10.0cm$, (c) $50.0cm$, and (d) $200cm$ from the center of the disk.

Q. A thin rod of negligible mass and area of cross-section $2\ast {10}^{-5}{m}^2$, suspended vertically from one end, has a length of 0.5 m at ${200}^{\circ }$C. The rod is cooled to $0^{\circ }$C, but length is kept same by attaching a mass at the lower end. The value of this mass is (Young's modulus $={10}^{11}N/m^2$, Coefficient of linear expansion ${10}^{-5}{K}^{-1}$).

1. 20 kg
2. 30 kg
3. 50 kg
4. Answer Required
5. 40 kg

Q. A wire of length 60 cm is bent into a circle with a gap of 1 cm. At its ends, on heating it by ${100}^{\circ }$ C, the length of the gap increases to 1)02 cm. $\alpha$ of material of wire is

1. $6\times {10}^{-4}{/}^{\circ }C$
2. $1\times {10}^{-4}{/}^{\circ }C$
3. $2\times {10}^{-4}{/}^{\circ }C$
4. $4\times {10}^{-4}{/}^{\circ }C$

Q. A tube $1c{m}^2$ in cross section is attached to the top of a vessel $1cm$ high and of cross section $100c{m}^2$. Water is poured into the system filling it to a depth of $100cm$ above the bottom of the vessel as shown in the figure. Take $g=10m{s}^{-2}$. Find the correct statement.

1. The force exerted by water against the bottom of the vessel is $100N$
2. The weight of water in the system is $1.99N$
3. Neither (a) nor (b) is correct
4. Both (a) and (b) are correct

Q. In the figure shown, the distance of centre of mass of a system of a uniform circular plate of radius $3R$ from O in which a hole of radius $R$ is cut whose centre is at $2R$ distance from centre of large circular plate is :

1. $R/4$
2. $R/5$
3. $R/2$
4. none of these

Q. The breaking strength of a rod diameter $2cm$ is $2\times {10}^{-5}N$. Then that for rod of same material but diameter $1cm$ will be

1. $2\times {10}^{5}N$
2. $1\times {10}^{5}N$
3. $8\times {10}^{5}N$
4. $0.5\times {10}^{5}N$

Q. A unifonn rod of length $L$ and mass $2m$ rests on a smooth horizontal table, A point mass $m$ moving horizontally at right angles to the rod with velocity v collides with one end of the rod and sticks it Then

1. angular velocity of the system after collision is $\frac{2}{5}\frac{v}{l}$
2. angular velocity of the system after collision is $\frac{v}{2l}$
3. the loss, in kinetic energy of the system as a whole as a result of the collision is $\frac{3}{10}m{v}^2$
4. the loss, in kinetic energy of the system as a whole as a result of the collision is $\frac{7m{v}^2}{24}m{v}^2$

Q. Two spheres of mass $M$ and $7M$ are connected by a rod whose mass is negligible, and the distance between the centers of each sphere is $d$. How far from the center of the $7M$ sphere is the Center of Mass for this object?

1. $d/8$
2. $d/7$
3. $d/2$
4. $6d/7$
5. $7d/8$

Q. A thin bar of mass $M$ and length $L$ is free to rotate about a fixed horizontal axis through a point at its end. The bar is brought to a horizontal position and then released. The axis is perpendicular to the rod. The angular velocity when it reaches the lowest point is

1. directly proportional to its length and inversely proportional to its mass
2. independent of mass and inversely proportional to the square root of its length
3. dependent only upon the acceleration due to gravity and the length of the bar
4. directly proportional to its length and inversely proportional to the acceleration due to gravity

Q. If unwrapped length of the thread between the cylinder and block $B$ is $20m$ at the beginning, when the system was released from rest, what would it be 2s later?

1. $28m$
2. $30m$
3. $22m$
4. $32.5m$