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The potential difference across a cell of emf E and internal resistance r is

Two wires P and Q of same cross-sectional area $A=10m{m}^2$ and the same length $l_0=1$ m but made of different materials are joined together and their ends are rigidly clamp between two walls with negligible tension. Their respective Young's modulus and the coefficients of linear expansion are as $Y_1,{\alpha }_1$ and $Y_2,{\alpha }_2$ respectively.
Given $Y_1=2\times {10}^{9}N/m^2,{\alpha }_1=3\times {10}^{-4}{/}^{\circ }C,Y_2={10}^{9}N/m^2,{\alpha }_2=6\times {10}^{-4}{/}^{\circ }C$,

Displacement of joint O is

We wish to make a plano convex lens of focal length 16 cm from glass having refractive index 1.5. It is to be used in air. What should be the radius of curvature of the curved surface?

A body is projected vertically upwards at time $t=0$ and it is seen at a height $H$ at times $t_1$ and $t_2$ seconds during its flight. The maximum height attained is ($g$ is the acceleration due to gravity)

For a certain organ pipe three successive resonance frequencies are observed at 425 Hz, 595 Hz and 765 Hz respectively. If the speed of sound in air is 340 m/s, then the length of the pipe is ___ m.

Two semi-circular discs of mass density $1{\text{kg/m}}^2$ and $2{\text{kg/m}}^2$, radius $r=1\text{m}$ each are joined to form a complete disc. Find the moment of inertia ($\text{MI}$) of complete disc about an axis passing through it's centre.

The coefficient of $x^m$ in $(1+x{)}^m+(1+x{)}^{m+1}+.....+(1+x{)}^n$, m £ n is

A heavy uniform chain lies on a horizontal table top. If the co efficient of friction between the chain and the table surface is 0.25 , then the maximum fraction of the length of the chain that can hang over one edge of the table is

The temperatures of equal masses of three different liquids A, B and C are ${12}^{\circ }C,{19}^{\circ }C$ and ${28}^{\circ }C$ respectively. The temperature when A and B are mixed is ${16}^{\circ }C,$ and when B and C are mixed, it is ${23}^{\circ }C$. What will be the temperature when A and C are mixed ?

A wire is wound on a long rod of material of relative permeability ${\mu }_r=4000$ to make a solenoid. If the current through the wire is $5A$ and number of turns per unit length is $1000$ per metre, then the magnetic field inside the solenoid is:

A ball is projected upwards from the top of the tower with a velocity $50{\text{ms}}^{-1}$ making an angle of ${30}^{\circ }$ with the horizontal. The height of tower is $70\text{m}$. After how many seconds the ball will strike the ground?

Find the total energy of the capacitors present in the circuit given below

(a)Distinguish between ‘Analog’ and ‘Digital’ forms of communication.

(b)Explain briefly two commonly used applications of the ‘Internet’.

Our universe has been continuously expanding since its formation at the Big Bang, thus gradually reducing the density of its major constituents - hydrogen and helium, and the average temperature to 2.7K (associated with the so-called microwave background radiation left over from the Big Bang). What is the rms speed of hydrogen molecules at this temperature?

The speed of a projectile u reduces by 50 percent on reaching maximum height. What is the range on the horizontal plane ?

A person $1.6\text{m}$ tall is standing at the centre between two walls $3\text{m}$ high. What is the minimum size of a plane mirror fixed on the wall in front of him, if the person wants to see the image of the full height of the wall behind him?

A block of mass $m$ is put on a rough inclined plane of inclination $\theta$, and is tied with a light thread shown. Inclination $\theta$ is increased gradually from $\theta =0^{\circ }to{90}^{\circ }.$ Match the columns according to corresponding curve.

A weightless spring which has a force constant oscillates with frequency n when a mass m is suspended from it. The spring is cut into two equal halves and a mass 2m is suspended from it. The frequency of oscillation will now become

Monochromatic light is incident on a glass prism of angle A. If the refractive index of the material of the prism is μ, a ray, incident at an angle q, on the face AB would get transmitted through the face AC of the prism provided:

A ball is thrown vertically upwards with a velocity of $10\text{m/s}$. It returns to the ground with velocity of $9\text{m/s}$. If $g=9.8{\text{m/s}}^2$, then the maximum height attained by the ball is nearly (assume air resistance to be uniform)

A man of mass M stands at one end of a plank of length L which lies at rest on a frictionless surface. The man walks to the other end of the plank. If the mass of the plank is $\frac{M}{3}$, the distance that the man moves relative to the ground is

A glass full of water has a bottom of area 20 cm², top of area 20 cm², height 20 cm and volume half a liter. Considering the equilibrium of the water, find the resultant force exerted by the sides of the glass on water. Atmospheric pressure = 105 N/m², density of water = 1000 kg/m³ and g = 10 m/s².

A square loop of uniform conducting wire is as shown in figure. A current $I$ (in
amperes) enters the loop from one end and leaves the loop from opposite end as
shown in figure. The length of one side of square loop is $l$ meter. The wire has
uniform cross section area and uniform linear mass density. In four situations of
column-I, the loop is subjected to four different uniform and constant magnetic field.
Under the conditions of column I, match the column-I with corresponding results of
column-II. (in column I is a positive non-zero constant).

A longer solenoid of diameter $0.1m$ has $2\times {10}^4$ turns per meter. At the center of the solenoid, a coil of $100$ turns and radius $0.01m$ is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to $0A$ from $4A$ in $0.05s$. If the resistance of the coil is $10{\pi }^2\Omega$, the total charge flowing through the coil during this time is

Find the dimensions of

(a) electric dipole moment p and

(b) magnetic dipole moment M.

The defining equations are p = q,d and M = IA;

where d is distance, A is area, q is charge and I is current.

The magnetic flux () linked with a coil is related to the number (N) of turns of the coil as

If $|\overrightarrow{A}+\overrightarrow{B}|$ =$|\overrightarrow{A}-\overrightarrow{B}|$ then the angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is

Explain how does the Lenz's law show the conservation of energy in the phenomenon of electromagnetic induction.

Air (ideal gas with $\gamma =1.4)$ at $1\text{bar}$ and $300\text{K}$ is compressed till the final volume is one-sixteenth of the original volume, following a polytropic process $P{V}^{1.25}=\text{const}.$
Calculate the (a) work done and (b) the energy transferred as heat per mole of the air.

Two bodies are thrown up at angles of ${45}^{\circ }$ and ${60}^{\circ }$, respectively, with the horizontal. If both bodies attain same vertical height, then the ratio of velocities with which these are thrown is

A T shaped object, having uniform linear mass density, with dimension shown in the figure is lying on a smooth floor. A force $F$ is applied at the point P parallel to AB, such that the object has only translational motion without rotation. The distance of P with respect to C is

A small body of mass $m=0.10kg$ moves in the reference frame rotating about a stationary axis with constant angular velocity $\omega =5.0rad/s$. What work does the centrifugal force of inertia perform during the transfer of this body along an arbitrary path from point $1$ to point $2$ which are located at the distance $r_1=30cm$ and $r_2=50cm$ from the axis of rotation?

A small solid cylinder of radius r and mass M slides down a smooth hill of height h from rest and gets onto the plank of mass M lying on the smooth horizontal plane at the base of the hill as shown in the figure. Due to friction between the cylinder and plank, the cylinder slows down and starts rolling without friction over the plank. While the cylinder is rolling (without sliding) over the plank, the velocity of the plank is $v_p$, the velocity of centre of mass of the cylinder is $v_c$ and angular velocity of the cylinder is “$\omega$”. The coefficient of friction between the cylinder and plank is $\mu$ , while gravity (g) is uniform in the space.
[Assume that height of the plank is negligible.]

The net work performed by friction M ground frame and in plank frame

A bomb of mass $12\text{kg}$ is dropped by a fighter plane moving horizontally with a speed of $100{\text{ms}}^{-1}$ from a height of $1\text{km}$ from the ground. The bomb exploded after $10\text{s}$ into two pieces of masses in the ratio $1:5$. If the small part started moving horizontally with a speed of $600{\text{ms}}^{-1},$ the speed of bigger part will be $(g=10{\text{ms}}^{-2})$

Consider the figure. Each capacitor has capacitance C.

The capacitance between 1 and 3 is

If the position vector of a point $P$ relative to the origin varies with time as, $\overrightarrow{r}=2\alpha t\hat{i}-\beta t^2\hat{j}$ , where $\alpha$ and $\beta$ are constants, then the equation for trajectory of the particle is

A rectangular loop of sides of length ‘l’ = 2m and ‘b’ = 0.5m is placed in X – Y plane inside a uniform but time varying magnetic field of strength $\overrightarrow{B}=20t\hat{i}+10t^2\hat{j}+50\hat{k}$ where t is time elapsed. The magnitude of induced e.m.f at time t is :

The displacement of the particle is given by $x=3t^3+6t^2+2t+5$ in $\text{m}$. The instantaneous speed of the particle at $4\text{sec}$ is

Question:You are given a rod of length 'L'. The linear mass density as ${}^{\prime }{\lambda }^{\prime }$. $\lambda =a+bx$. Here a & b are constants and the mass of the rod increases as x increases. Find the mass of the rod.