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A 1 kg block is executing simple harmonic motion of amplitude 0.1 m on a smooth horizontal surface under the restoring force of a spring of spring constant $100N{m}^{-1.}$ A block of mass 3 kg is gently placed on it at the instant it passes through the mean position. Assuming that the two blocks move together, find the frequency and the amplitude of the motion.

In the figure shown, find the tension $T$ in the string if the system is in equilibrium.

The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by

$E=\frac{3}{5}\frac{Z(Z-1)e^2}{4\pi {\epsilon }_{0}R}$

The measured masses of the neutron, ${}_1^{1}H,{}_7^{15}N$ and ${}_8^{15}O$ are $1.008665u,1.007825u,15.000109u$ and $15.003065u$ respectively. Given that the radii of both the ${}_7^{15}N$ and ${}_8^{15}O$ nuclei are same, $1u=931.5MeV/c^2$ ($c$ is the speed of light) and $e^2/\left(4\pi {\epsilon }_0\right)=1.44MeVfm$. Assuming that the difference between the binding energies of ${}_7^{15}N$ and ${}_8^{15}O$ is purely due to the electrostatic energy, the radius of either of the nuclei is $(1fm={10}^{-15}m)$

The value of coefficient of volume expansion of glycerin is $5\times {10}^{-4}{\text{K}}^{-1}$. The fractional change in the density of glycerin for a rise of $40\text{K}$ in its temperature will be

An object is placed in front of a convex mirror at a distance of $50\text{cm}$. A plane mirror is introduced covering the lower half of the convex mirror. If the distance between the object and the plane mirror is $30\text{cm}$, it is found that there is no parallax between the images formed by the two mirrors. What is the radius of curvature of the convex mirror?

Doppler shift in frequency does not depend upon

The temperatures of two bodies measured by a thermometer are $t_1={20}^{\circ }C\pm {0.5}^{\circ }C$ and $t_2={50}^{\circ }C\pm {0.5}^{\circ }C$. The temperature difference and the error therein is:

A resistor of resistance R is connected to an ideal battery. If the value of R is decreased, then the power dissipatedb in the resistor will?

A passenger train of length $60\text{m}$ travels at a speed of $80\text{km/hr}$. Another freight train of length $120\text{m}$ travels at a speed of $30\text{km/hr}$. The ratio of times taken by the passenger train to completely cross the freight train when : (i) they are moving in the same direction and (ii) in the opposite direction is

A light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (vertically). If the acceleration of the system is g/8 then the ratio of the masses is

A current of 1.0 A is established in a tigtly wound solenoid of radius 2 cm having 1000 turns / metre. Find the magnetic energy stored in each metre of the solenoid.

If the sum of two unit vectors is a unit vector, then magnitude of their difference is

One mole of a diatomic ideal gas $(\gamma =1.4)$ is taken through a cyclic process starting from point $A$. The process $A\to B$ is an adiabatic compression, $B\to C$ is an isobaric expansion, $C\to D$ is an adibatic expansion and $D\to A$ is an isochoric process. The volume ratios are $\frac{V_A}{V_B}=16,\frac{V_C}{V_B}=2$ and temperature $T_A={27}^{\circ }\text{C}$. Find the efficiency of the cycle.

A particle of mass m is revolving in a circular path of radius r. Its angular momentum is L. The centripetal force acting on the particle is

A small block of mass $m$ is placed over a long plank of mass $M$. Coefficient of friction between them is $\mu$. Ground is smooth. At $t=0$, $m$ is given a velocity $v_1$ and $M$ a velocity $v_2(>v_1)$ as shown in the figure. After this $M$ is maintained at constant acceleration $a(<\mu g)$.

The forward force acting on the plank before and after $t_0$ respectively is

Two ships are $60\text{km}$ apart on North-South vertical at an instant as shown in the figure. The one farther North is streaming South at $30\sqrt{3}\text{km/hr}$ and the other is streaming East at $30\text{km/hr}$. What is the time taken by the ships to reach the closest approach?

Multiple choice questions

1. A ball thrown vertically upward with a velocity u it takes t0 seconds to reach the highest point the distance travelled in the last t seconds of its ascent ( t < t0 second) is

(a) (t0 - t) u (b) (1/2)*(g(t0-t)²

(c) ut-(1/2)*(gt²) (d) (1/2)*(gt²)

Two uniform solids $A$ and $B$ having same volume, float in water. It is observed that $A$ floats with half of its volume immersed and $B$ floats with $\frac{2}{3}rd$ of its volume immersed. Find the ratio of mass of $A$ to $B$.

Find the force of interaction between $2kg$ and $3kg$. (Assume all surface are smooth)

When a projectile is thrown up at an angle θ to the ground, the time taken by it to rise and to fall are related as

Is projectile motion also 1d. Please give eg.

If a feather is thrown at perpedicular to the horizontal surface of moon ..will it be considered as a projectile....

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by 100 Hz than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is:

The position of a body moving along x-axis at t is given by x=(t²-4t+6)m. The distance travelled by body in time interval t=0 to t=3 s is
(A) 5m
(B) 7m
(C) 4m
(D) 3m

The formula to calculate the packing efficiency of the body centred unit cell is:

The electric field and force experienced by a test charge placed at origin is 6 x 10⁵ N/C and 18 x 10^-4 N. The value of test charge is:

Two moles of a monoatomic gas is mixed with three moles of a diatomic gas. The molar specific heat of the mixture at a constant volume is

Consider a special situtation in which both the faces of the block $M_0$ are smooth, as shown in adjoining figure. Mark out the correct statement(s)


(A) If $F=0$, the blocks $M$ and $m$ cannot remain stationary
(B) For one unique value of $F$, the blocks $M$ and $m$ remain stationary with respect to block $M_0$
(C) There exists a range of $F$ for which blocks $M$ and $m$ remain stationary with respect to block $M_0$
(D) Since there is no friction, therefore, blocks $M$ and $m$ cannot be in equillibrium with respect to block $M_0$

The position $x$ of a body as a function of time $t$ is given by the equation:
$x=2t^3-6t^2+12t+16$
The acceleration of the body is zero at time $t$ is equal to

Formula to find the average velocity of a body is given by

In order to measure physical quantities in the sub-atomic world, the quantum theory often employs energy $E$, angular momentum $J$ and velocity $c$ as fundamental dimensions instead of the usual mass, length and time. Then, the dimension of pressure in this theory is

In the given figure two tiny conducting balls of identical mass m and identical charge q hang from non-conducting threads of equal length L. Assume that $\theta$ is so small that $tan\theta \approx sin\theta$, then for equilibrium x is equal to

A hydrogen atom emits ultraviolet radition of wavelen gth 102.5 nm. What are the quantum numbers of the states involved in the transition ?

A cubical block of mass 1.0 kg and edge 5.0 cm is heated to 227^0 C. It is kept in an evacuated chamber maintained at 27^0 C. Assuming that the block emits radiation like a blackbody, find the rate at which the temperature of the block will decrease. Specific heat capacity of the material of the block is 400 J $k{g}^{-1}{K}^{-1}$.

A block is projectd along a rough horizontal road with a speed of 10 m/s. If the coefficient of kinetic friction is 0.10 how far will it travel before coming to rest?

A proton is projected with a velocity ${10}^{7}m/s$ at right angle to a uniform magnetic field of induction 100 mT. The time taken by proton to traverse ${90}^{\circ }$ arc is

A spring -block system is as shown in the figure. Blocks $2\text{kg}$ and $3\text{kg}$ are at rest. Coefficient of friction between $2\text{kg}$ and incline is

a satellite revolves around the earth of radius R in a circular orbit of radius 3R.the % of increase in energy required to lift it to an orbit of radius 5R

Find the ratio of the magnitude of the electric force to the gravitational force acting between two protons.

Water is boiled in a container having a bottom of surface are $25c{m}^2$, thickness 1.0 mm and thermal conductivity 50 $W{m}^{-1\circ }{C}^{-1}$. 100 g of water is converted into steam per minute in the steady state after the boiling starts. Assuming that no heat is lost to the atmosphere, calculate the temperature of the lower surface of the bottom. Latent heat of vaporization of water = $2.26\times {10}^{6}JK{g}^{-1}$.

A copper wire of diameter 1.6 mm carries a current of 20 A. Find the maximum magnitude of the magnetic field $\overrightarrow{B}$ due to this current.

A solid sphere of mass m and radius r is placed inside a hollow thin spherical shell of mass M and radius R as shown in figure (11-E1). A particle of mass m is placed on the line joining the two centres at a distance x from the point o contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if (a) r< x< 2r,
(b) 2r < x < 2R and (c) x> 2R.