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A frictionless track has a curve of radius $90\sqrt{3}\text{m}$. If the speed limit on this road is $30\text{m/s}$, what should be the banking angle for this road to ensure safe driving ? Take $g=10{\text{m/s}}^2$.

An insect is travelling in between the two inclined mirrors as shown in the figure. The number of images formed of the insect by the mirrors are

A block of mass m moves with a speed v towards the right block in equilibrium with a spring. If the surface is frictionless and collistions are elastic, the frequency of collisions between the masses will be :

A body of mass $m$ hangs from a smooth fixed pulley $P_1$ by an inextensible string fitted with the springs of stiffness constants $k_1$ and $k_3$ The string passes over a smooth light pulley $P_2$ , which is connected with another ideal spring of stiffness constant $k_2$. Find the period of small oscillations of the body.

A metallic spherical shell having inner and outer radii $a$ and $b$ respectively has thermal conductivity $K=\frac{K_0}{r^2}\{a\le r\le b\}$, where $r$ is the distance from the center of the sphere. The inner surface is maintained at temperature ${\theta }_1$ and outer surface is maintained at temperature ${\theta }_2$ . Calculate the rate at which heat flows radially through the material $[{\theta }_2>{\theta }_1]$.

Find the time period of small oscillations of the following systems.

(p) A metre stick suspended through the 20 cm mark.

(q) A ring of mass m and radius r suspended through a point on its periphery.

(i)$2\pi \sqrt{\frac{2\sqrt{2}a}{3g}}$ (ii) 1.51 sec (iii) $2\pi \sqrt{\frac{3r}{3g}}$ (iv)$T=2\pi \sqrt{\frac{2R}{g}}$

When the road is dry and coefficient of static friction is $\mu$, the maximum speed of a car in the circular path is $10m{s}^{-1}$. If the road becomes wet and ${\mu }^{\prime }=\frac{\mu }{2}$, what is the maximum permitted speed for the car?

Two particles $A$ and $B$ are moving with uniform velocities as shown in the figure given below at $t=0$

Find out shortest distance between two particles.

A rectangular lamina of mass $6\text{kg}$, breadth $4\text{m}$ and length $6\text{m}$ is taken as shown in figure. If $I_x$ and $I_y$ are the moments of inertia of the rectangular lamina about $x-$ axis and $y-$ axis respectively, then the value $I_y+I_x$ is

A particle is given an initial speed $u$ inside a smooth spherical shell of radius $R=1\text{m}$ and it is just able to complete the circle. Acceleration of the particle when its velocity is vertical is:

During motion, we know distance never decreases(ie. It is a monotonically increasing function of time), this implies

Two identical parallel plate capacitors are connected in series and then joined in series with a battery of 100 V. A slab of dielectric constant K = 3 is inserted between the plates of the first capacitor. Then, the potential difference across the capacitors will be, respectively,

In the figure shown below, $m_A=2\text{kg}$ and $m_B=3\text{kg}$. A force $\text{F}=40\text{N}$ is applied on $2\text{kg}$ block as shown. Find the acceleration of COM of the system. (Neglect friction everywhere and assume the string to be inextensible)

A block of mass 2 kg hangs from the rim of a wheel of radius 0.5 m. On releasing 2 kg from rest the block falls through 5 m height in 2 s. The moment of inertia of the wheel will be

The ratio of radii of two wires of same material is $3:1$. If these wires are stretched by equal force, the ratio of stresses produced in them is

A block of mass 5 kg is lying on a rough horizontal surface. The coefficients of static and kinetic friction between the block and the surface respectively are 0.7 and 0.5. A horizontal force just sufficient to move the block is applied to it. If the force continues to act even after the block has started moving, what will be the acceleration of the block? Take $g=10m{s}^{-2}$.

Two wires are kept tight between the same pair of supports. The tensions in the wires are in the ratio $3:4$. Their radii are in the ratio $1:2$ and the density are in the ratio $3:4$. The ratio of their fundamental frequencies are

An object is displaced from point $A(0,1,2)m$ to a point $B(1,2,3)m$ under the influence of a force $\overrightarrow{F}=2x\hat{i}+3y^2\hat{j}+\hat{k}.$ The workdone by this force is

A pendulum bob of mass $10g$ is attached to a rod of length $2m$ and is allowed to move in the vertical circle. Find the minimum velocity of bob at the bottom of the circle so that the bob completes the circle.

(Take $g=10m/s^2$)

Two bodies of equal masses revolve in circular orbits of radii $R_1$ and $R_2$ with the same period. Their centripetal forces are in the ratio

A container has two immiscible liquids of densities ${\rho }_1$ and ${\rho }_2(>{\rho }_1)$. A capillary tube of radius $r$ is inserted in the liquid so that its bottom reaches upto the denser liquid. The denser liquid rises in the capillary and attains a height $h$ from the interface of the liquids, which is equal to the column length of the lighter liquid. Assuming angle of contact to be zero, the surface tension of heavier liquid is

A thin lens of focal length $f$ and its aperture diameter $d$, forms a real image of intensity $I$. Now the central part of the aperture upto diameter $\left(\frac{d}{2}\right)$ is blocked by an opaque paper. The focal length and image intensity would change to

A car has to move on a level turn of radius $45\text{m}$. If the coefficient of static friction between the tyres and the road is ${\mu }_s=0.5$, find the maximum speed the car can achieve without skidding.

Four identical thin rods each of mass M and length l, from a squre frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is

A $40\text{kg}$ girl is swinging on a swing from rest. Then, the power delivered when moving with a velocity of $2\text{m/s}$ upwards in a direction making an angle ${60}^{\circ }$ with the vertical is

The $FBD$ of mass $M$ in the figure shown is

It is not convenient to use a spherical Gaussian surface to find the electric field due to an electric dipole using Gauss's theorem because

Two discs are rotating about the same axis passing through their centres and perpendicular to both disc's planes. If disc $1$ has mass $5\text{kg}$ and $30\text{cm}$ radius and disc $2$ has mass $7\text{kg}$ and $60\text{cm}$ radius, then find the moment of intertia about an axis $\text{OO'}$.

What is the reading of the spring balance in the following device?

Same spring is attached with 2kg, 3kg and 1 kg blocks in three different cases as shown in the figure. If $x_1,x_2\text{and}{x}_3$ be the extensions in the spring in these three cases then

A 220 V main supply is connected to a resistance of 100k $\Omega$ . The rms current is

Find the correct option among the following -

$[a\to$ absorptive power,$r\to$ reflective power,$t\to$ transmissive power $]$

An air column vibrates in a pipe closed at one end in its third overtone due to a tuning fork of frequency $595\text{Hz}$. $P_o$ is the mean pressure at any point in the pipe and $\Delta P_o$ is the maximum amplitude of pressure variation. Speed of sound in air is $340\text{m/s}$ and end correction has a negligible effect. If the amplitude of pressure variation at the middle of the pipe is $\frac{\Delta P_o}{\sqrt{n}},$ then $n$ is:

[Given $\cos \left(\frac{595\pi }{340}\right)=\frac{1}{\sqrt{2}}$ ]

You have 2 pipes of length 1.20 m. One pipe is open, other has one end closed. First overtone is formed in both the pipes. Find the position of 1^st displacement node from origin

If mean wavelength of light radiated by 100 W lamp is 5000 $\stackrel{\circ }{A}$, then number of photons radiated per second are

In an adiabatic process, $4\text{g}$ of $H_2$ gas expands and temperature changes from ${47}^{\circ }\text{C}$ to ${27}^{\circ }\text{C}$. Find the magnitude of the work done by the gas on the surrounding.

A ball is dropped from a height $h_0$ on a horizontal floor and keeps rebounding after hitting the floor. Find: (i) speed after $n$ rebounds

(ii) Total distance travelled by the ball before stopping

(iii) Height it rises after nth rebound

(iv) Total time before stopping

(coefficient of restitution is $e$)

A solid cylinder of mass $m=4\text{kg}$ and radius $R=10\text{cm}$ has a string wrapped around it as shown in figure.The cylinder is held horizontal by fixing the two free ends of the string to the hook on the ceiling such that both the strings are exactly vertical. The cylinder is released to fall under gravity. Find the linear acceleration $\left(a\right)$ of the cylinder.

John travels $250\text{m}$ to North but then back-tracks to South for $100\text{m}$ to pick up a friend. What is John’s total displacement?

A rectangular lamina of mass $6\text{kg}$ is placed in such a way that the centre of rectangular lamina coincides with the origin($\text{O}$) as shown in figure. If $I_x$ and $I_y$ are the moment of inertia of rectangular lamina about x - axis and y - axis respectively, then find the ratio $\frac{I_x}{I_y}$.

A particle moves from a point $\left(-2\hat{i}+5\hat{j}\right)$ to $\left(4\hat{j}+3\hat{k}\right)$ when a force of $\left(4\hat{i}+3\hat{j}\right)$$N$ is applied. How much work has been done by the force ?

A table with a smooth horizonal surface is placed in a cabin which moves in a horizontal circle of a large radius $R$. A smooth pulley of small radius is fixed to the table. Two masses $m$ and $2m$ placed on the table are connected through a light string going over the pulley. System is released from rest with respect to the cabin. The initial tension in the string will be:

Two springs $P$ and $Q$ of force constants $k_P$ and $k_Q(k_Q=\frac{k_P}{2})$ are stretched by applying forces of equal magnitude. If the energy stored in $Q$ is $E$, then the energy stored in $P$ is

A monkey of mass $20\text{kg}$ is holding a vertical rope. The rope will not break when a mass of $25\text{kg}$ is suspended from it but will break if the mass exceeds $25\text{kg}$. What is the maximum acceleration with which the monkey can climb up along the rope? ($g=10{\text{m/s}}^2)$

Visible light of wavelength $6000\times {10}^{-8}cm$ falls normally on a single slit and produces a diffraction pattern. It is found that the second diffraction minima is at ${60}^o$ from the central maxima. If the first minimum is produced at ${\theta }_1$, then ${\theta }_1$ is close to

A body is projected at ${30}^{\circ }$ with velocity $v_1$ from point $A$ on the ground. At the same time another body is projected vertically upwards from $B$ with velocity $v_2$ from the ground. The point $B$ lies vertically below the highest point of the trajectory of ball $A$. For both the bodies to collide, what should be the ratio of $v_1/v_2$ ?

One end of a nylon of length $4.5m$ and diameter $6mm$ is fixed to a tree-limb. A monkey weighing $100N$ jumps very slowly to catch the rope and stays there. Find the change in the diameter of the rope. Young’s modulus of nylon $=4.8\times {10}^{11}N{m}^{-2}$ and Poisson’s ratio of nylon $=0.2$.

Two point mass $50\text{g}$ and $100\text{g}$ are attached to the ends of a rod of length $2\text{m}$ and of negligible mass. The rod which was initially at rest is set into rotation about an axis perpendicular to the length of the rod. Find the position of axis of rotation from $50\text{g}$ about which the work required to set the rod into rotation with an uniform angular velocity of $2\text{rad/s}$, is minimum.