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A boy wants to climb down a rope. The rope can withstand a maximum tension equal to two-thirds the weight of the boy. If $g$ is the acceleration due to gravity, the minimum acceleration with which the boy should climb down the rope is equal to

A wire of length $L_0$ is supplied heat to to raise its temperature by T. If $\gamma$ is the coefficient of volume ex the wire and Y is the young's modulus of the wire then the energy density stored in the wire is

A capillary tube $A$ is dipped in water. Another identical capillary tube $B$ is dipped in soap- water solution. Which of the following shows the relative nature of the liquid columns in the two tubes? Consider the surface tension of soap solution and water to be $0.025\text{N/m}$ and $0.073\text{N/m}$ respectively at room temperature.

A mass M, attached to a horizontal spring, executes SHM with a amplitude $A_1$. When the mass M passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude $A_2$. The ration of

A bullet of mass $20\text{g}$ strikes a pendulum of mass $5\text{kg}$ which is initially at rest. The center of mass of the pendulum rises a vertical distance of $10\text{cm}$. If the bullet gets embedded into the pendulum, calculate the initial speed of the bullet.

A body of mass $2\text{kg}$ is sliding on an inclined plane as shown





Mass of the wedge is $4\text{kg}$. Find the velocity of the wedge just before the body reaches the ground.

The root mean square speed of the molecules of a gas at absolute temperature T is proportional to

A gas flushes out of a container through a pin hole four times quickly as methane. The molecular weight of the gas is

The S.I. unit of electric flux is

A ball crosses point p with speed 6 $m/s$ and returns same point p with speed 8 $m/s$ in 5 seconds. Assuming acceleration($a$) is uniform, find the magnitude of $a$

The equation of a travelling wave is, $y=60\cos (1800t-6x)),$ where $y$ is in $\text{microns}$, $t$ in $\text{sec}$ and $x$ in $\text{metres}$. The ratio of maximum particle velocity to velocity of wave propagation is

The resultant of two forces having magnitude $3p$ and $2p$ is $R$. If the force having magnitude $3p$ is doubled keeping the same direction, then the resultant is also doubled. The angle between two forces is

By a monochromatic wave, we mean

[AFMC 1995]

Find the expression for the time period of a pendulum using dimensional analysis.

A monkey of mass $40\text{kg}$ climbs up a rope of breaking load $600\text{N}$, hanging from a ceiling. If it climbs up the rope with maximum possible acceleration, then the time taken by the monkey to climb up is:

(Length of the rope is $10\text{m}$ and $g=10{\text{m/s}}^2$).

A projectile thrown with a speed v at an angle $\theta$ has a range R on the surface of earth. For same v and $\theta$, its range on the surface of moon will be

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

A motorcyclist starts from the bottom of a slope of angle ${45}^{\circ }$ to cross the valley $PR$ as shown in the figure. The width of the valley is $90\text{m}$ and length of the slope is $80\sqrt{2}\text{m}$. The minimum velocity at point $O$ required to clear the valley will be

Find $\overrightarrow{A}\times \overrightarrow{B}$, if $\overrightarrow{A}=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{B}=-2\hat{i}+3\hat{j}+4\hat{k}$

What will be the derivative of a constant?

On an inclined plane of inclination ${30}^{\circ }$, a ball is thrown at an angle of ${60}^{\circ }$ with the horizontal, from the foot of the incline, with a velocity of $10\sqrt{3}{\text{ms}}^{-1}$. Then, the ball will hit the inclined plane in time: (take $g=10m{s}^{-2}$)

A system consists of two idential cubes each of mass $m$ linked together by a massless spring of spring constant $k$. The spring is compressed by $x$ connecting cubes by thread. Find minimum value of $x$ for which lower cube will bounce up after the thread has been burnt.

If the line of action of the applied force intersects the Axis of rotation, then the torque of this force about the axis of rotation is

A rectangular tank of depth 8 meter is full of water (), the bottom is seen at the depth [MP PMT 1987

A particle of mass 1 kg is projected upwards with velocity 60 m/s. Another particle of mass 2 kg is just dropped from a certain height. Find out the acceleration and the velocity of COM after 2 s when neither of the particles has collided with the ground.

Two sound waves move in the same diretion in the same medium. The pressure amplitudes of the waves are equal but the wavelength of the first wave is double the second. Let the average power transmitted across a cross saection by the first wave be $P_1$ and that by the second wave be $P_2$ . Then

A body weighs $200N$ on the surface of the earth. How much will it weight halfway down to the centre of the earth ?

The heart of a man pumps 5 litres of blood through the arteries per minute at a pressure of 150 $mm$ of mercury. If the density of mercury is $13.6\times {10}^{3}kg/m^3$ and $g=10m/s^2$ , then the power of heart in Watts is

In a bicycle competition called Tour De France, two riders; A and B are riding bicycles at constant acceleration of magnitude $1{\text{m/s}}^2$ and $3{\text{m/s}}^2$ respectively, in the North direction. Their initial speeds are $15\text{m/s}$ and $10\text{m/s}$ respectively while they were heading in the North direction. Find the separation between them after $t=10\text{s}$.

In the figure, all pulleys are massless and strings are light.

$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(a) 1 kg block}& \text{(p)}\text{will remain stationary}\\ \text{(b) 2 kg block}& \text{(q)}\text{will move down}\\ \text{(c) 3 kg block}& \text{(r)}\text{will move up}\\ \text{(a) 4 kg block}& \text{(s)}5{\text{m/s}}^2\\ & \text{(t)}10{\text{m/s}}^2\end{array}$

Two spherical rain drops with radii in the ratio $1:2$ fall from a great height through the atmosphere. The ratio of their momenta after they have attained terminal velocity is:

Two bodies $A$ and $B$ are travelling with constant velocities of $20\text{m/s}$ and $30\text{m/s}$ in opposite direction. The relative speed of $A$ with respect to $B$ is

Velocity $\left(V\right)$ versus displacement $\left(S\right)$ graph of a particle moving in a straight line is as shown in the figure. Given $V_o=4m/s$ and $S_o=20m.$

Corresponding acceleration $\left(a\right)$ versus displacement $\left(S\right)$ graph of the particle would be

A cubical block is subjected to three forces $F_1=20\text{N}$ acting at an angle of ${45}^{\circ }$ with horizontal and lying in the diagonal plane of the cube, $F_2=30\text{N}$ along $y-$axis and $F_3=40\text{N}$ acting in the negative $x-$direction as shown. Find the magnitude of friction force acting on the body.

In the ideal gas equation $PV=nRT$, dimensional formula of $R$ is

Six particles situated at the corners of a regular hexagon of side a move at a constant speed v. Each particle maintains a direction towards the particle at the next corner. Calculate the time the particles will take to meet each other.

A thin uniform rod of length $3\text{m}$ and mass $M$ is held horizontally by two vertical strings attached to the two ends, as shown in figure. One of the strings is cut. Find the angular acceleration of the rod at the moment it is cut. (Take $g=10{\text{m/s}}^2$).

A body of mass m is accelerated to velocity v in time $t_1$. The work done by the force as a function of time t will be

Helium in a piston/cylinder at ${27}^{\circ }C,$ $100\text{kPa}$ is brought to $400\text{K}$ in a reversible polytropic process with exponent $n=1.25$. Helium can be assumed to be an ideal gas with constant specific heat capacity. Find the work done per unit mass of gas.

Assume $R$ for helium $=2.07\text{kJ/kg K}$

The pointer of a dead-beat galvanometer gives a steady deflection because [MP PMT 1994]

Two blocks of masses $4\text{kg}$ and $2\text{kg}$ are placed on a frictionless surface and connected by a spring. An external kick gives a velocity of $12\text{m/s}$ to the heavier block in the direction towards the lighter one. The magnitude of the velocities of two blocks in the center of mass frame, after the kick, are respectively

A particle has displacement of 12 m towards east and 5 m towards north then 6 m vertically upward. The sum of these displacements is

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

A train travels from a city $A$ to city $B$ with a constant speed of $10\text{m/s}$ and returns back to city $A$ in half of the time taken to travel from $A$ to $B$. Find its average speed during the entire journey.

For a uniform square sheet from which another smaller square is cut out as shown in the figure, find out the distance of centre of mass of the remaining part from the $y-$axis. Consider the origin as shown in figure.

In a young’s double slit experiment the resultant intensity at a point on the screen with two sources with intensities 2I and 8I is 14I. Calculate the phase difference between the two sources.

If in a medium wave's wavelength is decreased, what will happen to its frequency?