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A parallel-plate capacitor is connected to a battery, which maintains a potential difference $V_0$ between its plates. A slab of dielectric constant $K$ is slowly inserted between the plates, completely filling the space between them. If $Q_0$ is the charge on the plates before the dielectric is inserted, how much work is done by mechanical forces on the slab in the whole process ?

Two particles $A$ and $B$ are moving in a uniform circular motion in concentric circles of radius $r_A$ and $r_B$ with speed $v_A$ and $v_B$ respectively. The time period of rotation is the same. The ratio of angular speed of A to that of B will be

A spring block system is kept on a frictionless surface. Match the respective entries of column (I) with column (II) assuming minimum potential energy for the system as zero.

($K$ = Spring constant, $A$ = Amplitude, $m$ = Mass of block)

$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A) If mass of the block is doubled}& \text{(p) Time period increases}\\ \text{(keeping}K\text{,}A\text{unchanged)}& \\ \text{(B) If the amplitude of oscillation is doubled}& \text{(q) Time period decreases}\\ \text{(keeping}K\text{,}m\text{unchanged)}& \\ \text{(C) If force constant is doubled}& \text{(r) Energy of oscillation increases}\\ \text{(keeping}m\text{,}A\text{unchanged)}& \\ \text{(D) If another spring of same force constant}& \text{(s) Energy of oscillation decreases}\\ \text{is attached parallel to the previous one}& \\ \text{(keeping}m\text{,}A\text{unchanged)}& \text{(t) Energy of oscillation remain constant}\end{array}$

Velocity of a car as a function of time is given by $v\left(t\right)=(t^3-\cos t)$ in $\text{m/s}$. The acceleration in $\left(m/s^2\right)$ of the car as a function of time $t$ is

A particle of mass $m$ moving with speed $v$ collide inelastically with the stationary ring and gets embedded in it. Find the angular velocity with which the system rotates after the collision. Assume that the thickness of the ring is much smaller than its radius.

A monochromatic source of light operating at 200 W emits $4\times {10}^{20}$ photons per second. The wavelength of this light will be __ nm.

Pipe $1$ has the radius and length of $0.2\text{m}$ and $50\text{m}$ respectively connected with the pipe $2$ having radius and length of $0.4\text{m}$ and $100\text{m}$ respectively. Find the total resistance offered to the flow of water by the system of pipes, Ignoring the resistance offered by the joint.
($\eta$ is the viscosity of water)

Two elastic rods are joined between fixed supports as shown in figure. Condition for no change in the lengths of the individual rods with increase in temperature -

[Given $\alpha \to$ linear expansion coefficient, $A\to$ area of cross-section of rod, $Y\to$ Young's modulus and $L\to$ length of rod]

Linear mass density (mass per unit length) of a rod depends on the distance from one end (say A) as ${\lambda }_x=ax+\beta .Hereaand\beta .$ are constants. The moment of inertia of this rod about an axis passing through A and perpendicular to the rod. (Length of the rod is l) is

If $W_1,W_{2}and{W}_3$ represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between $W_1,W_{2}and{W}_3$ [IIT JEE 2003]

Two blocks of mass 10g and 30g are fastened over a frictionless pulley . The 10g block rests on the table what minimum force must be applied on the 10g block to keep it on table ? What will be the tension in the cord ? What will be the acceleration of the system and the tension in the cord when force is withdrawn?

The first overtone frequency of a closed organ pipe $P_1$ is equal to the fundamental frequency of an open organ pipe $P_2$. If the length of the pipe $P_1$ is $60\text{cm}$, what will be the length of $P_2$?

A closed organ pipe, of length $1.5\text{m}$ and filled with a gas, is resonated in its fundamental mode with the help of a tuning fork. Another open pipe of same length but filled with air resonates with the same fork. The room temperature is ${30}^{\circ }\text{C}$. If the speed of sound in air at ${30}^{\circ }\text{C}$ is $340\text{m/s}$, then the speed of sound in the first gas at ${30}^{\circ }\text{C}$ will be

A plank of mass “m” is lying on a smooth surface in situation as shown in figure. Assume that the plank slips over the surface and surface of plank exposed to radiation is black. Find the force on the plank due to radiation.

The light beam is incident on the black body at an angle $\theta$ to the normal.

Two cylinders contain same amount of ideal monatomic gas. Same amount of heat is given to two cylinders. If temperature rise in cylinder A is $T_o$ then temperature rise in cylinder B will be

A Liquid is filled in a vessel of height $\frac{2H}{3}.$ At the bottom of the vessel there is a spot $\text{P}$ and a hole from which liquid is coming out. Let $d$ be the distance of image of $\text{P}$ from an eye at height $H$ from bottom at an instant when level of liquid in the vessel is $x$. Which of the following represents the correct graph between $d$ and $x$,

1. 2

There is a hollow sphere of charge Q and radius R. The self-potential energy of the sphere is proportional to (charge)and (radius).

A solid sphere of uniform density and radius $R$ applies a gravitational force of attraction equal to $F_1$ on a particle placed at a distance $3R$ from the centre of the sphere. A spherical cavity of radius $\frac{R}{2}$ is now made in the sphere as shown in the figure. The sphere with cavity now applies a gravitational force $F_2$ on the same particle. The ratio $\frac{F_2}{F_1}$ is:

The electric field of a plane electromagnetic wave is given by $\overrightarrow{E}=E_0\hat{i}\cos \left(kz\right)\cos \left(\omega t\right)$

The corresponding magnetic field $\overrightarrow{B}$ is then given by:

A wire carrying a current i is placed in a uniform magnetic field in the form of the curve $y=asin\left(\frac{\pi x}{L}\right)0\le x\le 2L.$ The force acting on the wire is

A uniform magnetic
field of 1.5 T exists in a cylindrical region of radius10.0 cm, its
direction parallel to the axis along east to west. A wire carrying
current of 7.0 A in the north to south direction passes through this
region. What is the magnitude and direction of the force on the wire
if,

(a) the wire
intersects the axis,

(b) the wire is
turned from N-S to northeast-northwest direction,

(c) the wire in
the N-S direction is lowered from the axis by a distance of 6.0 cm?

The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29:

Which of the following formulae are correct for describing the motion of the particle over the time-interval t₂ to t₁?

(a) x(t₂) = x(t₁) + v(t₁)(t₂–t₁) + ()a(t₂–t₁)²

(b) v(t₂)= v(t₁) + a(t₂–t₁)

(c) v_Average = (x(t₂) – x(t₁)) / (t₂ – t₁)

(d) a_Average = (v(t₂) – v(t₁)) / (t₂ – t₁)

(e) x(t₂) = x(t₁) + v_Average(t₂ – t₁) + ()a_Average(t₂ – t₁)²

(f) x(t₂) – x(t₁) = area under the v–t curve bounded by the t-axis and the dotted line shown.

Two blocks of masses $10\text{kg}$ and $20\text{kg}$ are connected by a light string passing over a frictionless pulley as shown in the figure, the angle of inclination is ${30}^{\circ }$. If the system is released, the blocks move with an acceleration of

A rigid body of mass $m$ and radius $R$ rolls without slipping on a rough surface. A force is acting on the rigid body at distance $x$ from the centre as shown in figure. Find the value of $x$ so that static friction will be zero.