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A lead ball at $30{}^{\circ }C$ is dropped from a height of $6.2km$. The ball is heated due to the air resistance and it completely melts just before reaching the ground. The molten substance falls slowly on the ground. Calculate the latent heat of fusion of lead. Specific heat capacity of lead = $126Jk{g}^{-1}{}^{\circ }{C}^{-1}$ and melting point of lead = $330{}^{\circ }C$. Assume that any mechanical energy lost is used to heat the ball. Use $g$ = $10m{s}^{-2}$.

Convert $\frac{\pi }{6}$,$5\pi$ radians to degrees

A particle is dropped from a tower in a uniform gravitational field at $t=0$. The particle is blown over by a horizontal wind with constant velocity.

Slope (m) of trajectory of the particle with horizontal and its kinetic energy vary according to the curves.

Point masses of $1,2,3$ and $4\text{kg}$ are lying at the points $(0,0,0),(2,0,0),(0,3,0)\text{and}(-2,-2,0)$ respectively. What will be the moment of inertia of the system about $x$ - axis?

What is the efficiency of a Carnot engine operating between higher temperature $T_H$ and lower temperature $T_L$?

A balloon starts rising from the surface of the Earth. The ascension rate is constant and equal to v_o. Due to the wind the balloon gathers the horizontal velocity component v_x=ay, where a is a constant and y is the height of ascent. Find the total, tangential, and normal accelerations of the balloon.

Find derivative of $f\left(x\right)=\frac{\cos x}{1+\sin x}$

[Hint: ${\sin }^2\theta +{\cos }^2\theta =1$]

Evaluate: ${\int }_0^2(e^x+\frac{1}{2x+1})dx$

A body of mass m = 1 kg is dropped from a height h = 40 cm on a horizontal platform fixed to one end of an elastic spring, the other being fixed to a base as shown in the figure. As a result, the spring is compressed by an amount x = 10 cm. What is the force constant of the spring? Take $g=10m{s}^{-2}$.

What is the distance of the centre of mass from end $\text{P}$of a non-uniform rod of length $L$, whose mass per unit length $\lambda =\frac{K{x}^2}{L}$ ($K$ is a constant) and $x$ is the distance from end $\text{P}$ ?

An ant moves from A to B and then comes back to point A and reaches C. Find the distance (d) and displacement (s) of the ant in this entire journey.

Four identical plates are connected as shown. Find ceff of the system.

Two uniform spherical balls of the same density and radii $r$ and $2r$ are dropped in air and are falling vertically downward. If the terminal velocity of the ball of radius $r$ is $1\text{cm/s}$, then the terminal velocity of the ball of radius $2r$ will be

If the velocity-time graph has the shape $AMB$, what would be the shape of the corresponding acceleration-time graph?

A right circular cone of density $\rho ,$ floats just immersed with its vertex downwards, in a vessel containing two liquids of densities ${\sigma }_1$ and ${\sigma }_2$ respectively. The plane of separation of the two liquids cuts off from the axis of the cone at a fraction $z$ (taken from vertex) of its length. Find $z$.

A parallelogram has diagonals expressed as $\overrightarrow{A}=5\hat{i}-4\hat{j}+3\hat{k}$ and $\overrightarrow{B}=3\hat{i}+2\hat{j}-\hat{k}$. Area of parallelogram is

Phase difference between a particle undergoing compression and a particle undergoing successive rarefaction is

If the rms speed of nitrogen molecules is $490m{s}^{-1}$ at 273K, find the rms speed of hydrogen molecules at the same temperature -

Threshold frequency for a metal is ${10}^{15}Hz$. Light of $\lambda =4000\stackrel{\circ }{A}$ falls on its surface. Which of the following statements is correct

If difference of pressure between inside and outside of the spherical drop is $70{\text{N/m}}^2$, then what is the radius of spherical drop ?

Given: Surface tension of water is $70\times {10}^{-3}\text{N/m}$.

A vessel contains $1$ mole of $O_2$ gas (molar mass $32$) at a temperature $T$. The pressure of the gas is $P$. An identical vessel containing one mole of $He$ gas (molar mass $4$) at a temperature $2T$ has a pressure of

The frictional force acting on $1kg$ block will be:

A man of mass 'M' having a bag of mass 'm' slips from the roof of a tall building of height 'H' and starts falling vertically (figure). When at a height 'h' from the ground, he notices that the ground below him is pretty hard, but there is a pond at a horizontal distance 'x' from the line of fall. In order to save himself he throws the bag horizontally (with respect to himself) in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in the water.

What is the time period of a pendulum formed by pivoting a meter stick so that it is free to rotate about a horizontal axis passing through the $75\text{cm}$ mark?

Find the potential drop across $4\mu \text{F}$ capacitor and $6\Omega$ resistor as shown in the following circuit.

A thick uniform rubber rope of density $1.5gc{m}^{-3}$ and Young's modulus $5\times {10}^{6}N{m}^{-2}$ has a length of 8 m. When hung from the ceiling of a room, the increase in length of the rope due to its own weight will be equal to

Which of the following is/are a valid configuration for an electric field?

A $0.2\text{kg}$ object at rest is subjected to a force $(0.3\hat{i}-0.4\hat{j})\text{N}$. What is the velocity of the object (in $m/s$) after $6\text{s}$

A small sphere initially at rest, falls in to a viscous liquid. Due to drag, heat is produced. Then the relation between rate of production of heat and the radius of the sphere at terminal velocity, is best represented by:

A uniform rod of mass $M$ and length $L$ is hanging from point $P$ as shown in the figure. Find the elongation in its length due to the self weight of the rod. The Young's modulus of elasticity of the rod is $Y$ and cross-sectional area of the rod is $A$.

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

Two blocks $A$ and $B$ of masses $2m$ and $m$, respectively, are connected by a massless and inextensible string. The whole system is suspended by a massless spring as shown in the figure. The magnitude of the acceleration of $A$ and $B$, immediately after the string is cut, are respectively

System shown in figure is released from rest. Pulley and spring is massless and friction is absent everywhere. The speed of $5\text{kg}$ block when $2\text{kg}$ block leaves contact with ground is (Take force constant of $5\text{kg}$ spring $k=40\frac{\text{N}}{\text{m}}$ and $g=10\frac{\text{m}}{{\text{s}}^2}$

[CPMT 1990]

A coil of area $80$$c{m}^2$ and 50 turns is rotating with 2000 revolutions per minute about an axis perpendicular to a magnetic field of $0.05$ $Tesla$. The maximum value of the e.m.f. induced in it is

A body is allowed to slide on a frictionless track from rest position under gravity. The track ends into a circular loop of diameter $D$. What should be the minimum height of the curved track in terms of $D$ so that it may complete the loop (successfully) ?

Three infinite long plane sheets carrying uniform charge densities

${\sigma }_1=-\sigma ,{\sigma }_2=+2\sigma$ and ${\sigma }_3=+3\sigma$ are placed parallel to the x-z plane at y =a, y=3a and y=4a as shown in Fig. 20.18. The electric field at point P is

An ideal gas undergoes a thermodynamic process in which the work done by the gas is the same as the increase in the internal energy of the gas. The molar heat capacity of the gas for the process is

(Given, $\gamma =1.5$ for the gas)

Four identical rods of same material are joined end to end to form a square. If the temperature difference between the ends of a diagonal is ${100}^{\circ }$ C , then the temperature difference between the ends of other diagonal will be

A particle P is moving in a circle of radius $R$ with a uniform speed $v$. C is the centre of the circle and AB is the diameter. The angular velocity of the particle when it is at point B about A and about C are in the ratio of

The pressure at the bottom of a tank of water is $3P$ where $P$ is the atmospheric pressure. If the water is drawn out till the level of water is lowered by one-fifth, the pressure at the bottom of the tank will now be:

An iron rod of length 50 cm is joined at an end to an aluminum rod of length 100cm. All measurement refer to ${20}^{\circ }C$. Find the length of the composite system at ${100}^{\circ }C$ and its average coefficient of linear expansion. The coefficient of linear expansion of iron and aluminum are $12\times {10}^{-6}{/}^{\circ }C$ and $24\times {10}^{-6}{/}^{\circ }C$ respectively.

The work function for metals A, B and C are respectively 1.92 eV, 2.0 eV and 5 eV. According to Einstein’s equation, the metals which will emit photo electrons for a radiation of wavelength $4100\stackrel{\circ }{A}$ is/are

A solid sphere is at rest on a horizontal surface. A horizontal impulse $J$ is applied at a height $h$ from the centre. The sphere starts rolling just after the application of impulse. The ratio $\frac{h}{R}$ will be

The maximum horizontal range of a projectile is 400 m. The maximum value of height attained by it will be

A solid sphere of mass $2\text{kg}$ and radius $1\text{m}$ is performing combined rotational and translational motion as shown in figure. Find the total kinetic energy of the solid sphere.

A cylindrical wire of length $0.1\text{m}$ has specific gravity $11$. What is the maximum diameter of the wire that can float on the surface of water, if surface tension of water is $0.07\text{N/m}$. (Take $g=10{\text{m/s}}^2$)