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The ends of a rod of length $l$ and mass $m$ are attached to two identical springs as shown in figure. The rod is free to rotate about its centre ${}^{\prime }{\text{O}}^{\prime }$. The rod is depressed slightly at end $\text{A}$ and released. The time period of the resulting oscillation is

A uniform chain of mass $m$ and length $l$ hangs on a thread and touches the surface of the table by its lower end. The thread breaks suddenly, find the force exerted by the table on the chain when half of its length has fallen on the table. The fallen part does not form a heap.

A screen is at a distance $D=80\text{cm}$ from a diaphragm having two narrow slits $S_1$ and $S_2$ which are $d=2\text{mm}$ apart. Slit $S_1$ is covered by a transparent sheet of thickness $t_1=2.5\mu \text{m}$ and slit $S_2$ is covered by another sheet of thickness $t_2=1.25\mu \text{m}$ as shown in figure





Both sheets are made of same material having refractive index $\mu =1.40$. Water is filled in the space between diaphragm and screen. A monochromatic light beam of wavelength $\lambda =5000\dot{\text{A}}$ is incident normally on the diaphragm. Assuming intensity of beam to be uniform, calculate ratio of intensity at $C$ to the maximum intensity of interference pattern obtained on the screen $({\mu }_w=\frac{4}{3})$.

A block of mass 4 kg rests on a rough inclined plane making an angle of 30° with the horizontal. The coefficient of static friction between the block and the plane is 0.7. The frictional force on the block is: $(g=9.8m/s^2)$

If $y=x^2\sin \left(x\right)$, then $\frac{dy}{dx}$ will be

A fly flies from a point $P(0,2,5)$ to the point $Q(3,1,6)$. The magnitude of displacement covered by it is

The pulley arrangements of figures (a) and (b) are identical. The mass of the rope is negligible. In Fig. (a), the mass m is lifted up by attaching a mass 2m to the other end of the rope. In Fig. (b) , m is lifted up by pulling the other end of the rope with a constant downward force F=2mg. The acceleration of m is the same in both cases.

A cylindrical vessel filled with water is released on an inclined surface of angle $\theta$ as shown in figure. The friction coefficient of surface with vessel is $\mu (<tan\theta )$. Then the constant angle made by the surface of water with the incline will be

Two masses $m_1=5\text{kg}$ and $m_2=10\text{kg},$ connected through an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of horizontal surface is 0.15 (${\mu }_k={\mu }_s=\mu )$ . The minimum weight $m$ that should be put on top of $m_2$ to stop the motion is:

A small block of mass $2\text{kg}$ having a velocity of $50\text{m/s}$ hits a larger block of mass $10\text{kg}$ initially at rest. The collision is perfectly inelastic and the smaller block sticks to the larger block. Find the velocity of the combined mass after the collision.

What is the difference between null vector and zero vector?

A particle executes simple harmonic motion with an amplitude of $5\text{cm}$. When the particle is at $4\text{cm}$ from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is

An object is placed at $30cm$ in front of the convex lens of focal length $10cm$. Find out the image distance from lens?

For the given circuit, comment on the type of transformer used.

1. Step down transformer

2. Auxiliary transformer

3. Step-up transformer

4. Autotransformer

Electric current has both magnitude and direction . But still ,it is referred to as a scalar quantity. I mean,how's that possible??!!

How many unique solutions does the equation $4x^2+4x+1=0$ have?

Find the charge on $9\mu \text{F}$ capacitor in the given capacitor network.





[Assume that, the capacitors are in steady state]

An electric dipole is formed by two equal and opposite charges q with separation d. The charges have same mass m. It is kept in a uniform electric field E. If it is slightly rotated from its equilibrium orientation, then its angular frequency $\omega$ will be -

A block attached to a spring, pulled by a constant horizontal force, is kept on a smooth surface as shown in fig. Initially, the spring is in the natural length state. Then the maximum positive work that the applied force F can do is (given that string does not break)

The radioactive sources $A$ and $B$ of half lives of $2\text{hr}$ and $4\text{hr}$ respectively, initially contain the same number of radioactive atoms. At the end of $2\text{hours}$, their rates of disintegration are in the ratio

A conducting sphere of radius R, and carrying a charge q is joined to a conducting sphere of radius 2R, and carrying a charge – 2q. The charge flown between them will be?

A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m, which can slide freely along the rod. Initially the two beads are at the centre of the rod and the system is rotating with angular velocity ${\omega }_o$ about its axis perpendicular to the rod and passing through its mid point (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is

A fish is rising up vertically inside a pond with velocity $4\text{cm/s}$, and notices a bird, which is diving downward and its velocity appears to be $16\text{cm/s}$ (to the fish). The real velocity of the diving bird is $[{\mu }_{\text{water}}=4/3]$

A loop of 6.28 cm long thread is put gently on a soap film in a wire loop. The film is pricked with a needle inside the soap film enclosed by the thread. If the surface tension of soap solution is 0.030 N/m, then the tension in the thread is

Two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ have equal magnitudes. If the magnitude of $\overrightarrow{P}+\overrightarrow{Q}$ is $n$ times the magnitude of $\overrightarrow{P}-\overrightarrow{Q}$, then angle between $\overrightarrow{P}$and $\overrightarrow{Q}$ is :

Figure (8-E15) shows a smooth track, a part of which is a circle of radius r. A blocks of mass m is pushed aginst a spring of spring constant k fixed at he left end and is then released.Find the initial compression of the spring so that the block presses the track with a force mg when it reaches the point P, where the radius of the track is horizontal.

Area of muzzle is $A$. Due to ejection of water (density $=\rho$) at a velocity $v$, the spring is compressed as shown in figure. The compression in spring, in equilibrium, is -

A moving-coil galvanometer has $100\text{turns}$ and each turn has an area $2.0{\text{cm}}^2$. The magnetic field produced by the magnet is $0.01\text{T}$. The deflection in the coil is $0.05\text{radian}$ when a current of $10\text{mA}$ is passed through it. Find the torsional constant of the suspension wire.

A battery of internal resistance $4\Omega$ is connected to the network of resistances as shown in figure. In order that the maximum power can be delivered to the network, the value of $R($in $\Omega )$ should be

Two waves passing through a region are represented by

$y=1.0cmsin\left[\right(3.14c{m}^{-1})x-(157s^{-1}\left)t\right]$

and $y=1.5cmsin\left[\right(1.57c{m}^{-1})x-(314s^{-1}\left)t\right]$

Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms.