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Moment of inertia of a cylinder of mass $M,$ length $L$ and radius $R$ about an axis passing through its centre and perpendicular to the axis of the cylinder is $I=M(\frac{R^2}{4}+\frac{L^2}{12})$. If such a cylinder to be made for a given mass of a material, the ratio $\frac{L}{R}$ for it to have minimum possible $I$ is-

Which of the following graphs correctly represents the variation of $\log {\lambda }_m$ with $\log T$?

[${\lambda }_m$ is the wavelength corresponding to maximum intensity of radiation and $T$ represents absolute temperature.]

A metal bar 70 cm long and 4kg in mass is supported on two knife edges placed 10cm from each end. A 6 kg weight is suspended at 30cm from one end. Find the reaction at knife edges. (assume the bar to be uniform cross section and homogeneous). Take g=9.8m/s2

A capillary tube of radius ${}^{\prime }{r}^{\prime }$ is immersed in a liquid and the liquid level falls in it to height $h$. The contact angle $\theta$ can be

In sound wave, $V_w$ is the wave velocity and $V_p$ is particle velocity. Then:

A copper disc of radius $0.1\text{m}$ is rotated about its centre at $20$ revolutions per second in a uniform magnetic field of $0.1\text{T}$ with its plane perpendicular to the field. The emf induced across the radius of the disc is -

A beam is of thin tubular section with diameter D and wall thickness t . If V is the transverse shear force on beam section , then what is maximum shear stress developed in section?

A hollow cylinder is rolling without slipping on a horizontal surface as shown in figure. Then, the angular acceleration of the cylinder will be -

A thin prism of glass is placed in air and water respectively. If $n_g=\frac{3}{2}$ and $n_w=\frac{4}{3}$, then the ratio of deviation produced by the prism for a small angle of incidence when placed in air and water separately is:

A mass $m$ is placed at point $\text{P}$ at a distance $h$ along the normal through the centre $\text{O}$ of a thin circular ring of mass $M$ and radius $r$ as shown in figure. If the mass is moved further away such that, $\text{OP}$ becomes $2h$, by what factor, the force of gravitation will decrease, if $h=r$

An AC voltage source, $V=200\sqrt{2}\sin 100t$, is connected across a circuit containing an AC ammeter and capacitor of capacity $1\mu F$. Reading of the ammeter is:

The work done in increasing the potential of a capacitor from $V$ volt to $2V$ volt is $W$. Then, the work done in increasing the potential of the same capacitor from $2V$ volt to $4V$ volt will be

Two stones are projected simultaniously from a tower at different angles of projection with same speed $u$. The distance between two stones is increasing at constant rate $u$. Then the angle between the initial velocity vectors of the two stone is:

Paragraph for below question

नीचे दिये गये प्रश्न के लिए अनुच्छेद

Q. The shortest wavelength of Balmer series of H-atom spectrum is given by

प्रश्न - H-परमाणु के स्पेक्ट्रम की बामर श्रेणी की लघुतम तरंगदैर्ध्य का व्यंजक है

Because of the friction between the water in oceans with the earth's surface, the rotational kinetic energy of the earth is continuously decreasing. If the earth's angular speed decreases by 0.0016 rad/day in 100 years, find the average torque of the friction on the earth. Radius of the earth is 6400 km and its mass is 6.0 $\times {10}^{24}$kg.

What makes the wheels of a steam engine move? Explain in detail. [3 MARKS]

Calculate the increase in the internal energy of $20\text{g}$ of water when it is heated from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$ and converted into steam at $100\text{kPa}$. The density of steam $=0.50{\text{kg/m}}^3$, specific heat capacity of water $=4200{\text{J kg}}^{-1}{}^{\circ }{\text{C}}^{-1}$ and latent heat of vaporization of water $=2.3\times {10}^6\text{J/kg}$

A
spherical capacitor consists of two concentric spherical conductors,
held in position by suitable insulating supports (Fig. 2.36). Show

that
the capacitance of a spherical capacitor is given by

where
r₁
and r₂
are the radii of outer and inner spheres, respectively.

A bar magnet of length $14\text{cm}$ is placed in the magnetic meridian with its north pole pointing towards the geographic north pole. A neutral point is obtained at a distance of $18\text{cm}$ from the center of the magnet. If $B_H=0.4G$, the magnetic moment of the magnet is $(1G={10}^{-4}\text{T})$

A bob of a simple pendulum, which is initially at rest at its mean position, is now raised to a vertical height of $1.8\text{cm}$ and released from rest. The velocity of the bob at the mean position is

As shown in the figure a metal rod makes contact and complete the circuit. The circuit is perpendicular to the magnetic field with B=0.15 tesla If the resistance is $3\Omega$ , force needed to move the rod as indicated with a constant speed of 2m/sec is

A bobbin is pushed along on a rough stationary horizontal surface as shown in the figure. The board is kept horizontal and there is no slipping at any contact points. Find the distance moved by the board when distance moved by the axis of the bobbin is ℓ.

A velocity vector is given as $\overrightarrow{v}=5xy\hat{i}+2y^2\hat{j}+3y{z}^2\hat{k}.$ The divergence of this velocity vector at $(1,1,1)$ is

A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with at a speed of $10m{s}^{-1}$. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of $10m{s}^{-1}$ ? The speed of sound in still air can be taken as $340m{s}^{-1}$

A block $m_2$ is loaded onto another block $m_1$ placed on a spring of stiffness $k$ as shown in figure. If block $m_2$ is suddenly removed, then magnitude of acceleration of block $m_1$ just after the removal of $m_2$ will be:

1. 0.115

The potential energy between two atoms in a molecule is given by $U=a{x}^3-b{x}^2$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when $x$ is equal to

Three
electrolytic cells A,B,C containing solutions of ZnSO₄,
AgNO₃
and CuSO₄,
respectively are connected in series. A steady current of 1.5 amperes
was passed through them until 1.45 g of silver deposited at the
cathode of cell B. How long did the current flow? What mass of copper
and zinc were deposited?

A vernier calliper has 25 divisions on its vernier scale for every 35 divisions on its main scale. If 1 cm on the main scale is split into 12 units, the least count of the calliper will be.

The curve between $v_{rms}^2$ and pressure $P$ for constant volume is

In the given figure, a body of mass $M$ is held between two massless spring on a smooth inclined plane. The free end of the spring are attached to firm supports. If each spring has spring constant $K$, the frequency of oscillation of the given body:

One twirls a circular ring (of mass $M$ and radius $R$ ) near the tip of one's finger as shown in Figure $1.In$ the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is $r$. The finger rotates with an angular velocity ${\omega }_0$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2 ). The cocficieicient of friction between the ring and the finger is $\mu$ and the acceleration due to gravity is $g$



The minimum value of ${\omega }_0$ below which the ring will drop down is