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The elevator shown in figure is descending with an acceleration of $2{\text{ms}}^{-2}$. The mass of the block $A=0.5\text{kg}$. The force exerted by the block $A$ on the block $B$ is $(g=10{\text{ms}}^{-2})$

In the $\text{HCl}$ molecule, the separation between the nuclei of the two atoms is about $1.5\stackrel{\circ }{A}(1\stackrel{\circ }{A}={10}^{-10}\text{m})$. The approximate location of the centre of mass from the hydrogen atom assuming the chlorine atom to be about 35.5 times as massive as hydrogen is

A particle is projected upwards from top of a tower of height $40m$ with a speed of $10m/s$. The time it will take to strike ground $(g=10m/s^2)$ will be

Find the acceleration of the particle at $x=2\text{m}$.

An isosceles triangle is cut out of a square sheet with given dimensions as shown in figure. Find the distance of centre of mass of system from the $x$ axis.





Consider the material to be of uniform density.

Find derivative of $f\left(x\right)=e^{\sin x}+\ln x+8^x$

Two paper screens $A$ and $B$ are separated by $150\text{m}$. A bullet pierces $A$ and $B$ such that hole in $B$ is $h=15\text{cm}$ below the hole in $A$. If the bullet is travelling horizontally at the time of hitting $A$, then the velocity of bullet at $A$ is

540 g of ice at $0^{\circ }C$ is mixed with 540 g of water at ${80}^{\circ }C$. The final temperature of the mixture is

Work done on a liquid drop to increase its radius from $3\text{cm}$ to $5\text{cm}$ is $\pi \times {10}^{-4}\text{J}$. Find the surface tension of liquid.

A wave moves with speed 300 m/s on a wire which is under a tension of 500 N. Find how much the tension must be changed to increase the speed to 312 m/s?

The mean lives of a radioactive substance are 1620 and 405 years for $\alpha$-emission and $\beta$-emission respectively. Find out the time during which three fourth of a sample will decay if it is decaying both by $\alpha$-emission and $\beta$-emission simultaneously.

The acceleration of the particle moving in $x-$direction with an initial velocity $8\text{m/s}$ is $3{\text{m/s}}^2$. Find its velocity at $t=2s$.

A wire of length $5\text{m}$ and radius $1\text{mm}$ has resistance of $1\Omega$. What length of the same material at the same temperature, and of radius $2\text{mm}$ will also have a resistance of $1\Omega$ ?

The amount of heat needed to increase the temperature of $1\text{gm}$ of water from ${14.5}^{\circ }\text{C}$ to ${15.5}^{\circ }\text{C}$ at one atmospheric pressure is:

A solid sphere of density $\eta (>1)$ times lighter than that of water is suspended in a water tank by a string tied to its base as shown in figure. If the mass of the sphere is $m$, then the tension in the string is given by

Aliens on the planet Xargon spot a distant dying star, 5 light years far. If we see the star dying today then after how much time will the star's explosion be heard by the Xargonians? (Assume Xargonians to be an immortal peace loving race) speed of sound $3\times {10}^2$ m/s

A train moves towards a stationary observer with a speed of $34\text{m/s}$. The train sounds a whistle and its frequency registered by the observer is $f_1$ . If the train speed is reduced to $17\text{m/s}$, the frequency registered is $f_2$. If the speed of sound is $340\text{m/s}$, then the ratio $\frac{f_1}{f_2}$ is

Find the derivative of $y=(x^2+1)(x^3+3)$.

The potential energy of a system is represented in the first figure. The force acting on the system will be represented by

A disc is rolling without slipping on a horizontal surface. $P$ and $Q$ are two points equidistant from $C$. Then, identify the correct statement regarding the velocity of the points:

Six identical conducting rods are joined as shown in figure. Points A and D are maintained at temperatures 200${}^{\circ }$C and 20 ${}^{\circ }$ C respectively. Find the temperature of junction B.

When radiation of wavelength $\lambda$ is incident on a metallic surface the stopping potential is 4.8 volts. If the same surface is irradiated with radiation of double the wavelength, then the stopping potential becomes 1.6 volts. Then the threshold wavelength for the surface is

The speed of block $B$ in a pulley-block system as shown in figure is

The electric current in a charging R-C circuit is given by $i=i_o{e}^{\frac{-t}{RC}}$ where io, R and C are constant parameter of the circuit and t is time. Find the rate of change of current at (x)t=0, (y)t=RC, (z)t=10 RC



$\left(i\right)\frac{-i_o}{RC}\left(ii\right)0\left(iii\right)\frac{-i_o}{R{C}_e}\left(iv\right)\frac{-i_o}{R{C}_e^{10}}\left(v\right)\frac{-i_o{e}^{10}}{RC}$

In the $CGS$ system of units, the magnitude of the force is $100\text{dyne}$. If in another system of units, the fundamental physical quantities like mass, length and time are expressed in kilogram, metre and minute respectively, the magnitude of the force in this system of units is

The displacement – time graph of a body of mass $2\text{kg}$ is shown. Find the impulse acting on the body in the time interval $t=2\text{sec}$ to $t=6\text{sec}$

If $v_e$ and $v_o$ represent the escape velocity and orbital velocity of a satellite corresponding to a circular orbit of radius R, then

A particle of mass $M$ at rest decays into two particles of masses $m_1$ and $m_2$, having non-zero velocities. The ratio of the de-Broglie wavelengths of the particles, ${\lambda }_1/{\lambda }_2$ is

A man standing on a road has to hold his umbrella at $30°$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10\text{km/h}$. He finds that raindrops are hitting his head vertically. What is the speed of rain with respect to the ground?

A disc of radius $r$ has linear velocity $V$ and angular velocity $\omega$ as shown in figure. If point $B$ is at a distance $\frac{r}{2}$ from centre, then find the velocity of point $A$ w.r.t point $B$.

A particle is moving along positive $x-$ axis with velocity $v=A-Bt$, where $A$ and $B$ are positive constants and '$t$' is in seconds. If the particle is at the origin initially, then find the coordinates of the position of the particle at $t=\frac{3A}{B}$.

A particle moves in a potential region given by $U=8x^2-4x+400J$. Its state of equilibrium will be

A locomotive of mass m starts moving so that its velocity varies according to the law v = a$\sqrt{s}$, where a is a constant, and s is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first t seconds after the beginning of motion.

The maximum height of an oblique projectile is $8\text{m}$. The horizontal range is $24\text{m}$. The vertical component of the velocity of projection is

A block is shot with an initial velocity of $5m/s$ on a rough horizontal plane. Find the distance covered by the block till it comes to rest. Given coefficient of friction between the surfaces of contact is $0.1\text{and}g=10m/s^2$.

A small block of mass $m$ is placed on a body of mass $M$ at rest. If the block is set in horizontal motion with velocity $v$ as shown,



(Assume that the small block leaves from surface of $M$ vertically at A and all surfaces are smooth)

Find velocity of mass $M,$ when the small block leaves the surface of $M.$

A point mass $m$ moving with velocity $v_0$ hits a ball of same mass $m$ at rest elastically. After the collision, point mass moves with velocity $v_0/2$ at an angle $\theta$ from initial direction of motion. Then value of $\theta$ is

Look at the drawing given in the figure,which has been drawn with ink of uniform line thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6m$. The coordinates of centres of the different parts are: outer circle ($0,0$), left inner circle ($-a,a$), right inner circle ($a,a$), vertical line ($0,0$), and horizontal line $(0,-a)$. The $y-$ coordinate of centre of mass of the ink in this drawing is:

A man goes $10\text{m}$ towards North, then $20\text{m}$ towards east. Then, total length of displacement is

Find the velocity of the hanging block if the velocities of the free ends of the rope are as indicated in the figure. Take the upward velocity to be positive.

A particle is revolving in a circular path having radius $‘r^{\prime }=1m$ at a speed of $2m/s$. It is increasing its speed at the rate of $3m/s^2$. The magnitude of total acceleration of the particle at that instant is

A bomber plane moves horizontally with a speed of 500 m/s and a bomb released from it, strikes the ground in 10 sec. Angle at which it strikes the ground will be $(g=10m/s^2)$

A battery of emf $E_0=12\text{V}$ is connected across a $4\text{m}$ long uniform wire having resistance $4\Omega /\text{m}$. The cells of small emfs ${ϵ}_1=2\text{V}$ and ${ϵ}_2=4\text{V}$ having internal resistance $2\Omega$ and $6\Omega$ respectively, are connected as shown in the figure. If galvanometer shows no deflection at the point $N$, the distance of point $N$ from the point $A$ is equal to