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In figure, the block moves downwards with velocity $u_1$, the wedge moves rightwards with velocity $u_2$. The correct relation between $u_1$ and $u_2$ is

A screw gauge having $100$ equal divisions and a pitch length of $1\text{mm}$ is used to measure the diameter of a wire of length $5.6\text{cm}$. The main scale reading is $1\text{mm}$ and ${47}^{\text{th}}$ circular division coincides with the main scale. The curved surface area of wire (in ${\text{cm}}^2$) to appropriate significant figures is

A taut string for which $\mu =15\times {10}^{-2}\text{kg/m}$ is under a tension of $240\text{N}$. How much power must be supplied to the string to generate sinusoidal waves at a frequency of $18.0\text{Hz}$ and an amplitude of $18\text{cm}$?

In the diagram particle A moves along the line y = 25$\sqrt{3}$ m with a constant velocity $\overrightarrow{v}$ of magnitude 5 m/s and parallel to the x axis. At the instant particle A passes the y axis, particle B leaves the origin with a zero initial speed and a constant acceleration $\overrightarrow{a}$ of magnitude 4 m/$s^2$. What angle $\theta$ between $\overrightarrow{v}$ and the positive direction of the y axis would result in a collision?

What is the magnitude of $\overrightarrow{v}$ ?

The corresponding displacement time graph will be ? The body starts from rest.

Find the loss of energy as heat when the key is swithed from position A to B.

Velocity of block B in the given arrangement is $300mm/sec$ towards right. Then velocity of A will be

Two identical hollow spheres having mass $m=3\text{kg}$ and radius $r=1\text{m}$ are performing combined translational and rotational motion as shown in the figure. If $\omega =2\text{rad/s}$,$v_{com}=3\text{m/s}$ and ${\omega }^{\prime }=3\text{rad/s},v_{com}^{{}^{\prime }}=2\text{m/s}$, find the ratio of their kinetic energies.

Newton's law of cooling is a special case of:

A satellite which is geostationary in a particular orbit is taken to another orbit. Its distance from the centre of earth in new orbit is 2 times that of the earlier orbit. The time period in the second orbit is

The pulleys and strings shown in the figure are smooth and of negligible mass. For the system to remain in equilibrium, the angle $\alpha$ should be

A particle is moving on a circular path of radius $3cm$ with a time varying speed of $v=3tcm/s$, where $t$ is in seconds. The magnitude of tangential acceleration and the total acceleration of the particle at $t=1s$ is

A thin uniform vertical rod of mass $m$ and length $l$ is hinged about the point $O$ as shown in the figure. The combined stiffness of springs is equal to $k$. Considering the springs to be light, find the angular frequency of small oscillations.

A wire of cross-section area $2{\text{mm}}^2$ is stretched by $0.1\text{mm}$ by a certain force. How far (length) will the wire of same material and length but area $8{\text{mm}}^2$ stretch under the action of same force?

Mohit travels $10m$ over a circular path having radius $10m$. Angle substended by Mohit at the center of circle is

A tap is attached to a water tank as shown in figure. Water level above the tap is maintained at $h=0.2\text{m}$. The cross-sectional area of the tap is $\sqrt{2}\times {10}^{-4}{\text{m}}^2$. Assuming constant pressure throughout the stream of water, find the velocity & cross-sectional area of the stream $0.2\text{m}$ below the opening of the tap :-

$[\text{Assume}g=10{\text{m/s}}^2]$

A ball is in a hot air balloon that starts from rest at t = 0 and accelerates upwards at 2m/s². After 10s, the ball is dropped from the balloon.Draw ball's v - t graph:

A spring balance hanging in steady state inside a train moving on a straight track shows a reading of double the actual mass of an object. Calculate the acceleration of the train.

A capillary tube of radius $R$ is immersed in water and water rises in it to a height $H$. Mass of water in the capillary tube is $M$. If the radius of the tube is doubled, then the mass of water that rises in the capillary tube will be:

Two bodies of masses $m_1$ and $m_2$ are initially at rest at an infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

If isothermal bulk modulus is $x$ times adiabatic bulk modulus, then value of $x$ is equal to

Calculate the energy released when $1000$ small water drops each of same radius ${10}^{-7}\text{m}$ coalesce to form one large drop. The surface tension of water is $5\times {10}^{-2}\text{N/m}$.

In the arrangement shown in the figure, masses of the blocks $A$ and $B$ are $\text{m}$ and $2\text{m}$ respectively. Surface between block $B$ and floor is smooth. The block $B$ is connected to the block $C$ by means of a string pulley system. If the whole system is released, then find the minimum value of mass of block $C$ so that block $A$ remains stationary w.r.t. $B$. Coefficient of friction between $A$ and $B$ is $\mu$.

The insulated metal spheres of radii R₁ and R₂ having charges Q₁ and Q₂ respectively are connected to each other. There is

An earth satellite of mass m revolves in a circular orbit at a height h from the surface of the earth. R is the radius of the earth and g is acceleration due to gravity at the surface of the earth. The velocity of the satellite in the orbit is given by

Flint glass prism is joined by a crown glass prism to produce dispersion without deviation. The refractive indices of these for mean rays are 1.602 and 1.500 respectively. Angle of prism of flint prism is 10^o, then the angle of prism for crown prism will be

A uniform wire of resistance $72\Omega$ is bent in the form of a hexagon (regular), then the effective resistance between A & B is

The displacement of a body in $8\text{s}$, starting from rest, with an acceleration of $20{\text{cm/s}}^2$ is

Vectors A and B are $6\hat{i}-7\hat{j}+6\hat{k}$ and $3\hat{i}+4\hat{j}+6\hat{k}$ respectively. Their vector difference in unit vector notation will be

In photoelectric effect, the K.E. of electrons emitted from the metal surface depends upon

If M is the mass of a nucleus and A its atomic mass, then the packing fraction is

A body is projected upwards with a velocity of $98m/s$. The second body is projected upwards with the same initial velocity but after $4s$. Both the bodies will meet after

The velocity of a body depends on time according to equation V=20 + 0.1 $t^2$. the body is undergoing

A stone is dropped into a lake from a tower 500 metre high. Taking the speed of sound as 330 m/s, the sound of the splash will be heard by the man approximately after

If the power of a body is given by $P=xv$, where $v$ is the velocity of the body then the dimension of $x$ is

A sample of ${}_{19}{K}^{40}$ disintegrates into two nuclei Ca and Ar with decay constant ${\lambda }_{ca}=4.5\times {10}^{-10}{s}^{-1}$ and ${\lambda }_{Ar}=0.5\times {10}^{-10}{s}^{-1}$ respectively. The time after which $99\%$ of ${}_{19}{K}^{40}$ get decayed is

A uniform metre scale of length $1\text{m}$ is balanced on a fixed semi-circular cylinder of radius $30\text{cm}$ as shown in figure. One end of the scale is slightly depressed and released. The time period (in seconds) of the resulting simple harmonic motion is $(\text{Take}g=10{\text{ms}}^{-2})$

A thin rod of length f/3 lies along the axis of a concave mirror of focal length f One end of its magnified image touches an end of the rod. The length of the image is

[MP PET 1995]

Which of the following displacement time (x-t) graphs can possibly represent the one-dimensional motion of a particle?

Three liquids of densities ${\rho }_1,{\rho }_2$ and ${\rho }_3$(with ${\rho }_1>{\rho }_2>{\rho }_3$) having the same value of surface tension $\mathrm{T\; and}$ rise to the same height in three identical capillaries. The angles of contact ${\theta }_1,{\theta }_2$ and ${\theta }_3$ obey

A bullet of mass a and velocity b is fired into a large block of mass c. The final velocity of the system is

If $y=x^5,then\frac{dy}{dx}$=?

If $a=(3t^2+2t+1)m/s^2$ is the expression according to which the acceleration of a particle moving along a straight line varies, then find displacement of the particle 2 seconds after starting.

The maximum velocity and acceleration of a particle in $\text{SHM}$ are $100\text{cm/s}$ and $157{\text{cm/s}}^2$ respectively. The time period in seconds will be :

The acceleration of a particle is defined by the relation, $a=-4x(-1+\frac{1}{4}{x}^2)$ , where $x$ is displacement along $x-$ axis. All the quantities are in $SI$ units. If velocity of the particle $\left(v\right)=17\text{m/s}$ when $x=0$, then the velocity of the particle when $x=4\text{m}$ is

Power supplied to a particle of mass $2\text{kg}$ varies with time as $P=\frac{3t^2}{2}\text{W}$, where $t$ is in seconds. If velocity of particle at $t=0$ is $v=0$, the magnitude of velocity of particle at time $t=2\text{s}$ will be