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Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volumes is $1:4$, the ratio of their diameters is,

Bulk modulus of an incompressible fluid

1. 6

A projectile can have the same range $R$ for two angles of projection. If $T_1$ and $T_2$ be times of flight in two cases, then the product of the two times of flight is directly proportional to

A pipe closed at one end produces a fundamental frequency of $375\text{Hz}$. If it is cut into two pipes of equal length, then the fundamental frequency produced by the open and closed pipes respectively are

A
gas in equilibrium has uniform density and pressure throughout its
volume. This is strictly true only if there are no external
influences. A gas column under gravity, for example, does not have
uniform density (and pressure). As you might expect, its density
decreases with height. The precise dependence is given by the
so-called law of atmospheres

n₂
= n₁
exp [-mg
(h₂
–
h₁)/
k_BT]

Where
n₂,
n₁
refer to number density at heights h₂
and h₁
respectively. Use this relation to derive the equation for
sedimentation equilibrium of a suspension in a liquid column:

n₂
= n₁
exp [-mg N_A(ρ
- P′)
(h₂
–h₁)/
(ρRT)]

Where
ρ is the density of the suspended particle, and ρ’ that
of surrounding medium. [N_A
is Avogadro’s number, and R
the
universal gas constant.] [Hint:
Use Archimedes principle to find the apparent weight of the suspended
particle.]

A uniform rod AB of length $l$ is rotating with an angular velocity $\omega$ while its centre moves with a linear velocity $v=\frac{\omega l}{6}$. If the end A of the rod is suddenly fixed, the angular velocity of the rod will be:

In below figure two blocks are placed on fixed inclined plane and friction coefficient between the blocks and inclined plane are ${\mu }_1$ and ${\mu }_2$ respectively as shown in figure. Value of ${\mu }_1$ and ${\mu }_2$ are given in column 1 and value of $a_1$ and $a_2$ are given in column 2 and column 3 respectively.

$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \left(i\right)& {\mu }_1=0.8& \left(i\right)& a_1=2m/s^2& \left(P\right)& a_2=2m/s^2\\ & {\mu }_2=0.8& & & & \\ \left(ii\right)& {\mu }_1=0.5& \left(ii\right)& a_1=0.8m/s^2& \left(Q\right)& a_2=0.8m/s^2\\ & {\mu }_2=0.5& & & & \\ \left(iii\right)& {\mu }_1=0.5& \left(ii\right)& a_1=4.4m/s^2& \left(R\right)& a_2=4.4m/s^2\\ & {\mu }_2=0.2& & & & \\ \left(iv\right)& {\mu }_1=0.5& \left(iv\right)& a_1=0& \left(S\right)& a_2=0\\ & {\mu }_2=0.8& & & & \end{array}$

In which of the following combination both blocks will move together and normal force between them is non-zero?

A particle slides down from rest on an inclined plane of angle $\theta$ with horizontal. The distances are as shown. The particle slides down from top to position A where its velocity is v. Match the options of two columns.

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& (v^2-2gh)\text{will}& \text{(p)}& \text{Increase}\\ \text{(b)}& (v^2-2gssin\theta )\text{will}& \text{(q)}& \text{Decrease}\\ \text{(c)}& (v^2-2g\frac{H}{p})\text{will}& \text{(r)}& \text{Remains constant}\\ \text{(d)}& [v^2-\frac{2gs(H-h)}{(p-s)}]\text{will}& \text{(s)}& \text{May increase or decrease}\end{array}$

In the shown figure, block moves downwards with velocity $v_1$, the wedge moves rightwards with velocity $v_2$. The correct relation between $v_1$ and $v_2$ is

Consider the nuclear reaction

$X^{200}\to A^{110}+B^{90}$

If the binding energy per nucleon for $X,A$ and $B$ are $7.4\text{MeV},8.2\text{MeV}$ and $8.2\text{MeV}$ respectively, the energy released in the reaction is

A conducting wire $ABC$ has been bent with its two arms $AB$ and $BC$ making ${60}^{\circ }$. Length of each arm is $l$. The bent wire is moving with a velocity $v$ along the bisector of the angle $ABC$ and there exists a uniform magnetic field $\left(B\right)$ in the region, directed perpendicular to plane of the wire. Find the emf between the two ends $A$ and $C$ of wire.

A circular loop of wire having a radius of $8.0\text{cm}$ carries a current of $0.2\text{A}$. A vector of unit length and parallel to the dipole moment $\overrightarrow{\mu }$ of the loop is given by $0.60\hat{i}-0.80\hat{j}$. If the loop is located in uniform magnetic field given by $\overrightarrow{B}=\left(0.25\right)\hat{i}+\left(0.30\right)\hat{k}$. Find the torque acting on the loop.

As shown in figure, a wheel $(r=1\text{m})$ is performing combined translational and rotational motion with angular velocity $2\text{rad/s}$ and the velocity of its centre as $\overrightarrow{V_O}=4\hat{i}\text{m/s}$. Find the velocity of point $B$ in $\text{m/s}$ on the wheel.

Calculate the amount of heat required to convert 1.00 kg of ice at $-{10}^{\circ }$C into steam at ${100}^{\circ }$C at normal pressure. $(S_{ice}=2100Jk{g}^{-1}{K}^{-1},S_{water}=4200Jk{g}^{-1}{K}^{-1}L{f}_{ice}=3.36\times {10}^{5}Jk{g}^{-1}L{v}_{water}=2.25\times {10}^{6}Jk{g}^{-1})$

Four identical capacitors are connected as shown in the figure. When a battery of $9\text{V}$ is connected across $\text{A}$ and $\text{B},$ the charge stored is found to be $3\mu \text{C}$. The value of capacitance $C_1$ is

A body of mass $2kg$ is kept on a rough horizontal surface. If ${\mu }_s=0.5$, find the contact force between the body and the surface. Take $g=10m/s^2$.

Solve : $65÷13+(7\times 6)-7$

A gas absorbs 120 J of heat and expands against the external pressure of 1.10atm from a volume of 0.5 litre to 2.0 litre .What is the change in internal energy ? (1Latm =101.3J)

An oil drop falls through air with a terminal velocity $5\times {10}^{-4}\text{m/s}$. Find the radius of the drop.
Neglect the density of air as compared to that of oil. (Take viscosity of air $=3.6\times {10}^{-5}{\text{N-s/m}}^2,g=10{\text{m/s}}^2$, density of oil ${\rho }_o=900{\text{kg/m}}^3)$

Two parallel wires 1 m apart carry currents of 1 A and 3 A respectively in opposite directions. The force per unit length acting between two wires is

Figure shows an over-head view of a corridor with a plane mirror $M$ mounted at one end. A burglar $B$ sneaks along the corridor directly toward the centre of the mirror. If $d=3.0\text{m}$, how far from the mirror will he be, when the security guard $S$ first observes him in the mirror ?

When we have to apply relative velocity in a given question?

A beam of light consisting of red, green and blue color is incident on a right angled prism. The RI's of the material of the prism for above red, green and blue wavelengths are $1.39,1.44$ and $1.47$ respectively. The prism will

Figure 17.16 shows the P - V diagram for a fixed mass of an ideal gas undergoing cyclic process. AB represents isothermal process and CA represents adiabatic process. Which of the graphs shown in Fig. 17.17 represents the P - T diagram of the cyclic process ?

A particle is moving three times as fast as an electron. The ratio of the de Broglie wavelength of the particle to that of the electron is $1.813\times {10}^{-4}.$ Calculate the particle’s mass. (Given mass of electron is $9.109\times {10}^{-31}kg)$

For the two similar interfering waves, $y_1=A_1\sin (\omega t-k{x}_1)$ and $y_2=A_2\sin (\omega t-k{x}_2+\theta )$, the phase $\left(\alpha \right)$ of the resultant wave is -

The angular position of a particle (in radians), along a circle of radius 0.8 m is given by the function in time (seconds) by θ = t² + 2.3t + 1 . The linear velocity of the particle