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A body of mass $m$ accelerates uniformly from rest to $v_1$ in time $t_1$. As a function of time $t$, the instantaneous power delivered to the body is [2004 AIEEE]

A bullet of mass $50\text{gm}$ is fired with a velocity of $90\text{m/s}$ towards a ball of mass $500\text{gm}$, initially at rest. When the bullet strikes the ball, it gets embedded into it and the system starts moving. What is the ratio of the kinetic energy of the bullet to the kinetic energy of the $\text{(bullet + ball)}$ system?

In a space, potential $\left(V\right)$ is given by $V={\log }_e\left(xy\right)$, where $V,x,y$ are in SI units. The angle between $\overrightarrow{E}$ & $X$-axis at point $(6,6\sqrt{3})\text{m}$

A particle moves on a circular arc of radius $r$, the distance covered by particle when it travels from $A$ to $B$ on the arc subtending an angle ${60}^{\circ }$ at centre of circular path.

A metal rod of $30\text{cm}$ length expands by $0.075\text{cm}$ when its temperature is raised from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$. Another rod of different metal of length $45\text{cm}$ expands by $0.045\text{cm}$ for the same rise in temperature. A composite rod $C$ made by joining $A$ and $B$ end to end expands by $0.040\text{cm}$ when its length is $45\text{cm}$ and it is heated from $0^{\circ }\text{C}$ to ${50}^{\circ }\text{C}$. Find the length of each portion of the composite rod.

In a particular system, the unit of length, mass and time are chosen to be $10\text{cm}$,$10\text{gm}$ and $0.1\text{s}$ respectively. The unit of force in this system will be equivalent to

A body of mass $1kg$, traveling with a velocity of $3m/s$, accelerates uniformly to acquire a velocity of $8m/s$ in $2s$. Find the force acting on the body.

A box contains $N$ molecules of a perfect ideal gas at temperature $T_1$ and pressure $P_1$. The number of molecules in the box is doubled keeping the total kinetic energy of the gas same as before. If the new pressure is $P_2$ and new temperature is $T_2$, then

The figure shows the $P-V$ diagram of a thermodynamic cycle for an ideal gas. Which of the following graphs for the corresponding $P-T$ diagram is correct ?

The values of maximum current and impedance in the given circuit are

What is the angular velocity in $\text{rad/s}$ of a wheel making $1200$ revolutions in $50$ minutes ?

If the force constant of the spring is $50\text{N/m}$. Find the mass of the block if it rests in the given situation. Take $g=10m/s^2$

The distance of the centre of mass of the T - shaped plate from O is nearly equal to

A cylindrical rod has a hot end at ${70}^{\circ }\text{C}$ and cold end at ${30}^{\circ }\text{C}$. The rate of heat flow is $9\text{W}$. If the dimensions of the cylindrical rod are tripled keeping the temperature difference constant, then find the value of new rate of heat flow.

A ring of radius R is moving with a speed $\theta$ in a uniform field B. Find $V_A-V_B$.

Three vectors $\overrightarrow{A},\overrightarrow{B}\&\overrightarrow{C}$ are given as $\overrightarrow{A}=2\hat{i}+3\hat{j}+4\hat{k},\overrightarrow{B}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{C}=m\hat{i}+2\hat{j}+\hat{k}$ . Find the value of $m$, for which these vectors are coplanar.

A metal plate of mass $200\text{gm}$ is balanced in mid – air by throwing 40 balls /sec, each of mass $2\text{gm}$, vertically upwards from below. The balls rebound with the same speed with which they strike the plate. Find the velocity with which the balls strike the plate. (Take $g=9.8m/s^2$)

A battery is charged at a potential of $15V$ for $8h$ when the current flowing is $10A$. The battery on discharge supplies a current of $5A$ for $15h$. The mean terminal voltage during discharge is $14V$. The watt-hour efficiency of the battery is

A cricketer hits a ball from the ground level with a velocity of $\left(10\sqrt{2}\right)m{s}^{-1}$ at an angle of ${45}^{\circ }$ with the horizontal. What should be the minimum speed of the fielder standing [Along the direction in which ball is hit] at a distance of $5m$ from the batsman so as to catch the ball at ground level. Assume the direction of motion of the ball and the fielder are same.

A force $F=-K(y\hat{i}+x\hat{j})$ (where $K$ is a positive constant) acts on a particle moving in the xy-plane. Starting from the origin, the particle is taken along the positive x-axis to the point $(a,0)$ and then parallel to the y-axis to the point $(a,a)$. The total work done by the force $F$ on the particles is

Two balls of masses $3\text{kg}$ and $5\text{kg}$ are moving towards each other with speeds $5\text{m/s}$ and $3\text{m/s}$ respectively on a frictionless surface as shown in figure. After collision, the $5\text{kg}$ ball retraces its path with speed $2\text{m/s}$. Find the coefficient of restitution.

Two coherent sources $S_1$ and $S_2$ which emit sound of wavelength $\lambda$ in phase are separated by $2\lambda$. A circular wire of large radius is placed in such a way that $S_1{S}_2$ lies in its plane and the middle point of $S_1{S}_2$ is at the centre of the wire as shown below. Find the angular position $\theta$ (in first quadrant) on the wire for which destructive interference takes place.

[Assume the radius of ring $R>>2\lambda$]

Acceleration due to gravity on moon is $\frac{1}{6}$ of the acceleration due to gravity on earth. If the ratio of densities of earth $\left({\rho }_m\right)$ and moon $\left({\rho }_e\right)$ is $\left(\frac{{\rho }_e}{{\rho }_m}\right)=\frac{5}{3}$ then radius of moon $R_m$ in terms of $R_e$ will be

Which of the following predictions of the classical wave model of light (Maxwell's EM theory) were contradicted by the observations during the photoelectric effect experiment?

A body is executing simple harmonic motion. At a displacement x, its potential energy is $E_1$ and at a displacement y, its potential energy is$E_2$ The potential energy E at a displacement $(x+y)$ is

Find the moment of inertia of ring of mass $M$ and radius $R$ about the axis passing through the ring diametrically. If moment of inertia about the axis passing through its center of mass and perpendicular to the plane is $M{R}^2$.

When a semiconductor is heated, its resistance

[KCET 1992; MP PMT 1994; MP PET 1992, 2002;RPMT 2001; DCE 2001]

A man with normal near point $25cm$ reads a book with small print using a magnifying glass, a thin convex lens of focal length $5cm$. The closest and farthest distances from the magnifying glass at which he can read the book when viewing through the magnifying glass, respectively are -

If the pulley is massless and moves with an upward acceleration $a_o$, find the acceleration of block $m_1$ w.r.t elevator.

Calculate the speed (in $\text{mm/s}$) of flow of glycerin at the inlet of the conical section of a pipe. If its inlet and outlet radius are $0.2\text{m}$ and $0.05\text{m}$ respectively. The pressure drop across its length is $10{\text{N/m}}^2$ and density of glycerin is $1.25\times {10}^3{\text{kg/m}}^3$.

A boat is moving with a velocity of $3\hat{i}+4\hat{j}$ with respect to the ground. The water in the river is moving with a velocity of $-3\hat{i}-4\hat{j}$ with respect to the ground. The relative velocity of the boat with respect to water is

A feather is dropped on the moon from a height of $2m$. The acceleration of gravity on the moon is $1.69{\text{m/s}}^2$. Determine the time for the feather to fall to the surface of the moon.

If $F$ represents the strength of cohesive force, then what will be the correct order for strength of cohesive force for solid, liquid and gas ?

Find the MOI of a uniform square plate of mass $m$ and edge $a$ about one of its diagonals.

Two ends of a rod of non-uniform area of cross-section are maintained at temperature $T_1$ and $T_2(T_1>T_2)$ as shown in the figure. If $I$ is the heat current through the cross - section of the conductor at distance $x$ from its left face then the variation of $I$ with time $t$ is best represented by

Particle 'A' moves with speed $10\text{m/s}$ in a frictionless circular fixed horiozontal pipe of radius $5$ m and strikes with ${}^{\prime }{B}^{\prime }$ of double mass that of A. Coefficient of resitution is $\frac{1}{2}$ and particle 'A' starts its journey at $t=0$. The time at which second collision occurs is:

A gas expands with temperature according to the relation $V=k{T}^{2/3}$. What is the work done, when the temperature changes by ${30}^{\circ }\text{C}$?

A block of mass $4\text{kg}$ is kept on a rough horizontal surface with coefficient of friction ${\mu }_s=0.4$ and ${\mu }_k=0.3$ and a time varying horizontal force $F=4t$ is applied on it. Then the acceleration - time graph of the particle is $(g=10{\text{m/s}}^2)$

Calculate the Q-value in the following decay:

${}_8^{19}O{\to }_9^{19}F+e^{-}+\overline{v}.$

These may look like long calculations; use the unit amu or u for the masses, and $c^2=931\frac{MeV}{u},$ and it will become much simpler.

$Atomicmasses{:}_8^{19}O:19.003576u;{}_9^{19}F:18.998403u.$

A body of mass $5\text{kg}$ is moving along a straight line with a velocity of $2{\text{ms}}^{-1}$. A variable force $F$ started acting on it for a time period of $0.5s$ as shown in force - time curve. The final velocity of the block is

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

A car moving at a speed u is stopped in a certain distance when the brakes produce a deceleration a. If the speed of the car is nu, what must be the deceleration of the car to stop it in the same distance?

The potential energy of a particle varies with distance $x$ as $\frac{A{x}^{1/2}}{x^2+B}$, where $A$ and $B$ are constants. The dimensional formula for $A\times B$ is

The work function of metal is 1 eV. Light of wavelength $3000\stackrel{\circ }{A}$ is incident on this metal surface. The velocity of emitted photo-electrons will be