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A wire is bent at an angle $\theta$. A rod of mass $m$ can slide along the bent wire without friction as shown in the figure. A soap film is maintained in the wire frame kept in a vertical position and the rod is in equilibrium as shown in the figure. If rod is displaced slightly in vertical direction, then the time period of small oscillations of the rod is :-

An object is thrown horizontally with a velocity of $10\text{m/s}$. Find the radius of curvature of its trajectory $3\text{s}$ after the motion has begun. Take $g=10{\text{m/s}}^2$

For the situation shown in the figure, if the spring is elongated by ${}^{\prime }{x}^{\prime }$. The condition for no slipping on the inclined plane is given by

A non-uniform rod $AB$ has a mass $M$ and length $2l$.The mass per unit length of the rod is $mx$ at a point of the rod distant $x$ from $A$. Find the moment of inertia of this rod about an axis perpendicular to the rod through $A$.

The displacement of a body at a particular second $\left(n\right)$ is given by the expression $s_{n^{th}}=u+\frac{a}{2}(2n-1)$. The dimensional formula of $s_{n^{th}}$ in this equation is

A bullet with mass $m_b=400\text{g}$ and velocity $v_b=100\text{m/s}$ hits a ballistic pendulum of mass $m_a=9.6\text{kg}$ and lodges into it.It makes pendulum swing up from equilibrium position and rises to height $h$ as shown in figure. Determine the value of $h$. (Take $g=10m/s^2$)

If $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$, also $|\overrightarrow{R}|=25\text{N}$ and $|\overrightarrow{P}|=10\text{N}$. Find the angle $\theta$ made by $\overrightarrow{Q}$ with the positive x-axis.

Waves from two sources superpose on each other at a particular point, amplitude and frequency of both the waves are equal. The ratio of intensities when both waves reach in the same phase and when they reach with the phase difference of ${90}^{\circ }$ will be

A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4 m/s due east. The river is 800 m wide. How far south of his starting point will be reach the bank?

When a body moves with a constant speed along a circle

Two wooden blocks are moving on smooth horizontal surface such that the mass $m$ remains stationary with respect to the block of mass $M$ as shown. Find the force exerted by the $M$ on $m$.

The retardation experienced by a moving motor boat after its engine is cut off, is given by $\frac{dv}{dt}=-8v^2$. If the magnitude of velocity at cut off is $v_0=3\text{m/s}$, the magnitude of the velocity $1\text{sec}$ after the cut off is (in $\text{m/s}$)

${\int }_0^{10}se{c}^2(3x+6)dx$

Figure shows two rods $A$ and $B$ of same length $L$ and same cross-sectional area $S$ but of different material having coefficient of linear expansion ${\alpha }_1$ and ${\alpha }_2$ respectively. They are clamped between two rigid walls, separated by a distance $2L$. This all refers to temp $t^{\circ }\text{C}$. Find the tension in each rod at temp $2t^{\circ }\text{C}$ (Take the young’s modulus for the two rods to be $Y_1$ and $Y_2$ respectively).

An ideal gas is taken along the process $AB$ as shown in the $P-V$ diagram. Find the volume of the gas where the temperature becomes maximum.

A plane simple harmonic progressive wave given by the equation, $y=A\sin (\omega t-kx)$ of wavelength $120\text{cm}$ is incident normally on a plane surface which is a perfect reflector (acts as fixed end). If a stationary wave is formed, then the ratio of amplitudes of vibrations at points $10\text{cm}$ and $30\text{cm}$ from the reflector is



(Here $x$ is measured from reflector)

A rectangular film of liquid is extended from $(4cm\times 2cm)$ to $(5cm\times 4cm)$. If the work done is $3\times {10}^{-4}J$, the value of the surface tension of the liquid is

What is the SI unit of the coefficient of thermal conductivity?

A particle moves on xy plane. Its position vector at any time t is $\overrightarrow{r}=\left\{\right(2t)\hat{i}+(2t^2\left)\hat{j}\right\}m$. The rate of change of $\theta$ at time t=2 second. (Where $\theta$ is the angle which its velocity vector makes with positive x-axis) is

The wavelength of the first line in Balmer series in the hydrogen spectrum is $\lambda$. What is the wavelength of the second line?

In a simple pendulum, the breaking strength of the string is double the weight of the bob. The bob is released from rest when the string is horizontal. The string breaks when it makes an angle $\theta$ with the vertical. Then

Which of the following represents an impossible situation?

As shown in the figure a ball is moving with a velocity $u=6\hat{i}+\hat{j}$ and it collides with a vertical wall which is parallel to the vector $\hat{j}$. If the coefficient of restitution between the ball and the wall is $0.5$ and mass of the ball is $m=1\text{kg}$. Find the impulse that acts on the ball.

Two boys are standing at the ends A and B of a ground where $AB=a$. The boy at B starts running in a direction perpendicular to $AB$ with velocity $v_1$. The boy at $A$ starts running simultaneously with velocity $v$ and catches the other boy in a time $t$, where $t$ 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

A satellite is launched into a circular orbit of radius ‘R’ around earth while a second satellite is launched intoan orbit of radius 1.02 R. The percentage difference in the time periods of the two satellites is

A string of length $1\text{m}$ is fixed at one end with a bob of mass $100\text{g}$ and the string makes $\left(\frac{2}{\pi }\right)\text{rev/s}$ around a vertical axis through a fixed point. The tension in the string is

Two springs A and B have force constants $k_{1}and{k}_2$ respectively. The ratio of the work done on A to that done on B in increasing their length by the same amount is

A certain ideal gas undergoes a polytropic process represented by equation $P{V}^n=\text{constant}$. If Molar heat capacity for this process is negative and $\gamma$ is a standard constant which denotes the ratio of the molar heat capacities of a gas, then the range of values of $n$ will be

Two identical cylindrical vessels with their bases at same level each contains a liquid of density d. The height of the liquid in one vessel is $h_1$ and that in the other vessel is $h_2$. The area of either bases is A. The work done by gravity in equalizing the levels when the two vessels are connected, is

The velocity time graph of a lift moving upwards has been shown below. Let $T_1,T_2$ and $T_3$ be the tensions in the elevator cable during the three time intervals $\Delta t_1,\Delta t_2$ and $\Delta t_3$, then $T_1:T_2:T_3$

A particle leaves the origin with an initial velocity $\overrightarrow{v}$ = ( $3.00\hat{i}$ ) m/s and a constant acceleration $\overrightarrow{a}$ = ( $-1.00\hat{i}-0.500\hat{j}$ ) m/$s^2$. When it reaches its maximum x coordinate, match its position vector and velocity vector(Coloumn A) with vectors (Coloumn B)?

A block of mass $m$ is placed on another block of mass $M$ which itself is lying on a horizontal surface. The coefficient of friction between two blocks is ${\mu }_1$ and that between the block of mass $M$ and horizontal surface is ${\mu }_2$. What maximum horizontal force can be applied to the lower block so that the two blocks move without separation?

The blocks $(m_2>m_1)$ are held in such a manner that spring is unstretched initially. Consider the situation at the time of release of the block $(t=0)$ and short time after releasing the block $(t>0)$. Accelerations of the blocks of masses $m_1$ and $m_2$ are $a_1$ and $a_2$ respectively

A glass capillary tube of inner diameter $0.28\text{mm}$ is lowered vertically into water in a vessel. The pressure to be applied on the water in the capillary tube so that water level in the tube is same as that in the vessel $\left({\text{in N/m}}^2\right)$ is

(Surface tension of water $=0.07\text{N/m}$ and atmospheric pressure $={10}^5{\text{N/m}}^2)$

In an instrument, there are 25 divisions on the vernier scale which coincides with 24th division of the main scale. 1 cm on main scale is divided into 20 equal parts. The least count of the instrument is

A block of mass of $10kg$ is in contact with the inner wall of a hollow cylindrical drum of radius $1m.$ The coefficient of friction between the block and the inner wall of the cylinder is $0.1.$ The minimum angular velocity needed for the cylinder to keep the block stationary when the cylinder is vertical and rotating about its axis, will be $\left(g=10m/s^2\right)$

A ball is dropped from a bridge $122.5\text{m}$ high. After the first ball has fallen for $2$ second, a second ball is thrown straight down after it, what must be the initial velocity of the second ball, so that both the balls hit the surface of water at the same time?

If at same temperature and pressure, the densities for two diatomic gases are $d_{1}and{d}_2$ respectively, then the ratio of velocities of sound in these gases will be

If the linear mass density of a rod of length $2\text{m}$ varies as $\lambda =a+bx\text{kg/m}$, where $x$ is the distance (in metres) from its one end, then its centre of mass (in metres) is given by

Find the terminal velocity of a free falling water drop of radius $0.04\text{mm}$:-

The coefficient of viscosity of air is $1.9\times {10}^{-5}{\text{Ns/m}}^2$ and its density is $1.2{\text{kg/m}}^3$. Density of water is $1000{\text{kg/m}}^3$. Take $g=10{\text{m/s}}^2$.

A block of mass $m_1$ is pushed towards the movable wedge of mass $m_2$ and height $h$, with a velocity $v_0$. All surfaces are smooth. The minimum value of $v_0$ for which the block will reach the top of the wedge is

If the arcs of the same lengths in two circles subtend angles ${65}^o$ and ${110}^o$ at the center, find the ratio of their radii.