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A long uniform rod of length L and mass M is free to rotate in a vertical plane about a horizontal axis through its one end 'O'. A spring of force constant k is connected vertically between one end of the rod and ground. When the rod is in equilibrium it is parallel to the ground.

What is the period of small oscillations that result when the rod is rotated slightly and released?

A mass m, lying on a horizontal, frictionless surface, is connected to one end of a spring of spring constant $k$. The other end of the spring is connected to a wall, as shown in the figure. At $t=0$, the mass is given an impulse $J$ as shown. Taking left direction as positive, the time dependence of the displacement, acceleration and the velocity of the mass are:

A source of sound is travelling with a velocity of 30 m/s towards a stationary observer. If actual frequency of source is 1000 Hz and the wind is blowing with velocity 20 m/s in a direction at ${60}^{\circ }$ with the direction of motion of source, then the apparent frequency heard by observer is (speed of sound is 340 m/s)

When a metal rod is heated and then prevented from contracting as it cools, a considerable force is exerted at the ends of the rod, this can be found using

What is the ratio of electrostatic force and gravitational force between a proton and electron at 1mm distance?

Mass of proton – $1.67\times {10}^{-27}$ kg

Mass of electron – $9.11\times {10}^{-31}$ kg

Charge of electron – $1.6\times {10}^{-19}$ kg

Volume charge density of a non conducting solid sphere of radius $R=30cm$ is given as $\rho ={\rho }_o(1-\frac{r}{R}).$ Electric field inside sphere is minimum at

A square loop $\text{ABCD}$ of side $L$ carrying a current $i_1$ is placed near and co-planar with a long straight conductor $\text{XY}$ carrying a current $i_2$ as shown in the figure. Find the net force on the loop is-

A short linear object of length $l$ lies along the axis of a concave mirror at a distance $u$ from it. If $v$ is the distance of the image from the mirror, then the size of the image is

Calculate the work done by the gas in the state diagram shown

A uniform wire of length $L=1\text{m}$ is bent into the shape $V$ as shown in figure. The distance of its centre of mass from the vertex $A$ is

Figure (16-E12) shows a source of sound moving along the X-axis at a speed of $22m{s}^{-1}$ continuously emitting a sound of frequency 2.0 kHz which travels in air at a speed of $330m{s}^{-1}$. A listener Q stands on the Y-axis at a distance of 330 m from the origin. At t = 0, the source crosses the origin P. (a) When does the sound emitted from the source at P reach the listener Q ? (b) What will be the frequency heard by the listener at this instant? (c) Where will the source be at this instant ?

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Two object are placed on the principal axis of a thin convergnig lens. One is $10\text{cm}$ from the lens and the other is on the other side of the lens at a distance of $40\text{cm}$ from the lens. If he image of both objects coincide, the focal length of the lens (in $\text{cm}$) is

Which of the following is the correct orientation of the given dice?

Find the magnitude of the electric field at a point 4 cm away from a line charge od density $2\times 10-11C{m}^{-1}$

Two thin discs each of mass 4.0 kg and radius 0.4 m are attached as shown in figure to form a rigid body. The rotational inertia of this body about an axis perpendicular to the plane of disc $B$ and passing through its center is.

Three closed vessels A, B and C are at the same temperature T and contain gases which obey the Maxwellian distribution of velocities.Vessel A contains only$O_2$ ,B only $N_2$. and C a mixture of equal quantities of $O_2$ and $N_2$. If the average speed of the $O_2$ molecules in vessel A is $V_1$ , that of the$N_2$ molecules in vessel B is $V_2$, the average speed of the $O_2$ molecules in vessel C is

Find the capacitance between $\text{P}$ and $\text{O}$, if each capacitor has capacitance $C$.

A body moving along a straight line with uniform acceleration, starts from point $O$ and through points $A,B,C$ and $D$ at distances $x_1,x_2.x_3$ and $x_4$ from $O$. It covers distances $AB,BC$ and $CD$ in equal intervals of time. Then $(x_4-x_1)$ is equal to

A particle of mass $5\times {10}^{-6}\text{g}$ is kept over a large horizontal sheet of charge density $4\times {10}^{-6}{\text{C/m}}^2$. What charge should be given to this particle so that if released, it remains in equilibrium? [Take ${ϵ}_0=9\times {10}^{-12}{\text{C}}^2{\text{/Nm}}^2$]

A spherical ball A of surface area 20 $c{m}^2$ is kept at the centre of a hollow spherical shell B of area 80 $c{m}^2$. The surface of A and the inner surface of B emit as blackbodies. Assume that the thermal conductivity of the material of B is extremely poor and that of A is very high and that the air between A and B has been pumped out. The heat capacities of A and B are 42 J/${}^{\circ }$C and 82 J/ ${}^{\circ }$ C respectively. Initially the temperature of A is 100 ${}^{\circ }$ C and that of B is 20 ${}^{\circ }$ C.Find the rate of change of temperature of A and that of B at this instant.

Seven resistors are connected as shown in the diagram.





The equivalent resistance in ohms of this network between A and B is

Find the second order
derivatives of the function.

An LCR circuit contains resistance of $110\Omega$ and a supply of $220\text{V}$ at $300\text{rad/s}$ angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by ${45}^{\circ }$. If on the other hand, only inductor is removed the current leads by ${45}^{\circ }$ with the applied voltage. The rms current flowing in the circuit will be:

A particle moving along x-axis is being acted upon by one dimensional conservative force F. In the f - x curve shown, four points J, K, L, M are marked on the curve. Column II gives different type of equilibriums for the particle at different positions. Column I gives certain positions on the force position graph. Match the position in Column - I with the corresponding nature of equilibriums in Column - II.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A) Point J is position of}& \text{(p) Neutral equilibrium}\\ \text{(B) Point K is position of}& \text{(q) Unstable equilibrium}\\ \text{(C) Point L is position of}& \text{(r) Stable equilibrium}\\ \text{(D) Point M is position of}& \text{(s) No equilibrium}\end{array}$

A uniform steel wire of length 4 m and area of cross-section $3\times {10}^{-6}{m}^2$ is extended by 1 mm by the application of a force. If Young's modulus of steel is $2\times {10}^{11}N{m}^{-2}$, the energy stored in the wire is

Two Perfect gases at absolute temperatures T1 and T2 are mixed. There is no loss of energy. Find the temperature of the mixture if the masses of the molecules are m1 and m2 and the number of molecules in the gases are $n_1$ and $n_2$.