Consider a disc of mass 5 kg , radius 2 m , rotating with angular velocity of $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$ about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is $\_\_\_\_$ J.

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be $\_\_\_\_$ hours 30 minutes.

The identical spheres each of mass $2M$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4 m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $\frac { 4 \sqrt { 2 } } { x }$, where the value of $x$ is $\_\_\_\_$.

A string is wrapped around the rim of a wheel of moment of inertia $0.40 \mathrm { kgm } ^ { 2 }$ and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is $x \mathrm { rad } / \mathrm { s }$, where $x$ is $\_\_\_\_$

Two metallic wires $P$ and $Q$ have same volume and are made up of same material. If their area of cross sections are in the ratio $4 : 1$ and force $F _ { 1 }$ is applied to $P$, an extension of $\Delta l$ is produced. The force which is required to produce same extension in Q is $F _ { 2 }$. The value of $\frac { F _ { 1 } } { F _ { 2 } }$ is $\_\_\_\_$ .

Each of three blocks $P , Q$ and $R$ shown in figure has a mass of 3 kg . Each of the wire $A$ and $B$ has cross-sectional area $0.005 \mathrm {~cm} ^ { 2 }$ and Young's modulus $2 \times 10 ^ { 11 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$. Neglecting friction, the longitudinal strain on wire $B$ is $\_\_\_\_$ $\times 10 ^ { - 4 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ )

Two blocks of mass 2 kg and 4 kg are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is $4.0 \times 10^{-5}$ m and Young's modulus of the metal is $2.0 \times 10^{11}$ N m$^{-2}$. The longitudinal strain developed in the wire is $\dfrac{1}{\alpha \pi}$. The value of $\alpha$ is $\_\_\_\_$. [Use $g = 10$ m s$^{-2}$]

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of big drop is $\frac { 10 } { x }$. The value of $x$ is $\_\_\_\_$

A plane is in level flight at constant speed and each of its two wings has an area of $40 \mathrm {~m} ^ { 2 }$. If the speed of the air is $180 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ over the lower wing surface and $252 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ over the upper wing surface, the mass of the plane is $\_\_\_\_$ kg. (Take air density to be $1 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ and $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)

An elastic spring under tension of 3 N has a length $a$. Its length is $b$ under tension 2 N. For its length $( 3 a - 2 b )$, the value of tension will be $\_\_\_\_$ N.

The density and breaking stress of a wire are $6 \times 10^4\mathrm{~kg/m}^3$ and $1.2 \times 10^8\mathrm{~N/m}^2$ respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is $\frac{1}{3}^{\text{rd}}$ of the value on the surface of earth. The maximum length of the wire without breaking is $\_\_\_\_$ m (take, $\mathrm{g} = 10\mathrm{~m/s}^2$).

Two persons pull a wire towards themselves. Each person exerts a force of 200 N on the wire. Young's modulus of the material of wire is $1 \times 10 ^ { 11 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$. Original length of the wire is 2 m and the area of cross section is $2 \mathrm {~cm} ^ { 2 }$. The wire will extend in length by $\_\_\_\_$ $\mu \mathrm { m }$.

A tuning fork resonates with a sonometer wire of length 1 m stretched with a tension of 6 N. When the tension in the wire is changed to 54 N, the same tuning fork produces 12 beats per second with it. The frequency of the tuning fork is $\_\_\_\_$ Hz.

A soap bubble is blown to a diameter of 7 cm. 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne $/ \mathrm { cm }$ then the new radius is $\_\_\_\_$ cm. Take $\left( \pi = \frac { 22 } { 7 } \right)$

Three capacitors of capacitances $25\mu\mathrm{F}, 30\mu\mathrm{F}$ and $45\mu\mathrm{F}$ are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored energy is $\frac{9}{x}\mathrm{E}$. The value of $x$ is $\_\_\_\_$.

The distance between charges $+q$ and $-q$ is $2l$ and between $+2q$ and $-2q$ is $4l$. The electrostatic potential at point $P$ at a distance $r$ from centre $O$ is $-\alpha \dfrac{ql}{r^2} \times 10^9$ V, where the value of $\alpha$ is $\_\_\_\_$. (Use $\dfrac{1}{4\pi\varepsilon_0} = 9 \times 10^9$ N$^2$ m$^2$)

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $1.5 \mathrm {~g/cc}$, the dielectric constant of water will be $\_\_\_\_$. (Take density of water $= 1 \mathrm {~g/cc}$)

The electric field between the two parallel plates of a capacitor of $1.5\mu\mathrm{F}$ capacitance drops to one third of its initial value in $6.6\mu\mathrm{s}$ when the plates are connected by a thin wire. The resistance of this wire is $\_\_\_\_$ $\Omega$. (Given, $\log 3 = 1.1$)

At the centre of a half ring of radius $\mathrm { R } = 10 \mathrm {~cm}$ and linear charge density $4 \mathrm { nCm } ^ { - 1 }$, the potential is $x \pi \mathrm {~V}$. The value of $x$ is $\_\_\_\_$

In the following circuit, the battery has an emf of 2 V and an internal resistance of $\dfrac{2}{3}\,\Omega$. The power consumption in the entire circuit is $\_\_\_\_$ W.

A wire of resistance $R$ and radius $r$ is stretched till its radius became $r / 2$. If new resistance of the stretched wire is $x R$, then value of $x$ is $\_\_\_\_$

The current in a conductor is expressed as $I = 3t ^ { 2 } + 4t ^ { 3 }$, where $I$ is in Ampere and $t$ is in second. The amount of electric charge that flows through a section of the conductor during $t = 1 \mathrm {~s}$ to $t = 2 \mathrm {~s}$ is $\_\_\_\_$ C.

The current flowing through the $1 \Omega$ resistor is $\frac { n } { 10 } \mathrm {~A}$. The value of $n$ is $\_\_\_\_$
[Figure]

Two circular coils $P$ and $Q$ of 100 turns each have same radius of $\pi$ cm. The currents in $P$ and $Q$ are 1 A and 2 A respectively. $P$ and $Q$ are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic field induction at the center of the coils is $\sqrt{x}$ mT, where $x =$ $\_\_\_\_$. [Use $\mu_0 = 4\pi \times 10^{-7}$ T m A$^{-1}$]

A regular polygon of 6 sides is formed by bending a wire of length $4\pi$ meter. If an electric current of $4\pi\sqrt{3}$ A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $x \times 10 ^ { - 7 } \mathrm {~T}$. The value of $x$ is $\_\_\_\_$.