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The monkey $B$ shown in figure is holding on to the tail of the monkey $A$ which is climbing up a rope. The masses of the monkeys $A$ and $B$ are $5\text{kg}$ and $2\text{kg}$ respectively. If $A$ can tolerate tension of $30\text{N}$ in its tail, what force should it apply on the rope in order to carry the monkey $B$ with it? Take $g=10{\text{m/s}}^2$.

1. When the centre of the wheel ($\text{O}$) is not fixed to any surface.


2. When the centre of the wheel ($\text{O}$) is pivoted to a firm surface.

A ball is thrown vertically upwards with a velocity of $10\text{m/s}$. It returns to the ground with velocity of $9\text{m/s}$. If $g=9.8{\text{m/s}}^2$, then the maximum height attained by the ball is nearly (assume air resistance to be uniform)

Find the magnitude of the electric field $\left(|\overrightarrow{E_{ind}}|\right)$ at point $\text{A}$ due to the induced charges on the sphere.

A submerged scuba diver hears the sound of a boat horn directly above her on the surface of the lake. At the same time, a friend on dry land 22.0 m from the boat also hears the horn (fig). The horn is 1.2 m above the surface of the water. What is the distance (labeled by "?” in fig) from the horn to the diver? Both air and water are at 20${}^{\circ }$C.

$B_{water}=2.2\times 109Pa,{\gamma }_{air}=1.4,M_{air}\approx 30g$

1. 100

A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth :

A magnetic needle of magnetic moment $6.7\times {10}^{-2}A{m}^2$ and moment of inertia $7.5\times {10}^{-6}kg{m}^2$ is performing simple harmonic oscillations in a magnetic field of $0.01T$. Time taken for $10$ complete oscillations is

An AC voltage source described by $V=10\cos \left(\frac{\pi }{2}t\right)$ is connected to a $1\mu \text{F}$ capacitor as shown in the figure. The key $\text{K}$ is closed at $t=0$. The time $t>0$ after which the magnitude of current reaches its maximum value for the first time is -

An electric dipole of length $10\text{cm}$ having charges $6\times {10}^{-3}\text{C}$ is placed at ${60}^{\circ }$ with respect to a uniform electric field. If it experiences a torque of $3\sqrt{3}\text{N-m}$, then the magnitude of electric field is

A whistle giving out 450 Hz approaches a stationary observer at a speed of 33 m/s. The frequency heard by the observer (in Hz) is :

The plates $\text{A}$ and $\text{B}$ of a parallel plate capacitor (PPC) as shown in the figure contain charges $\text{+Q}$ and $\text{-Q}$ respectively. The plate $\text{A}$ is fixed and the plate $\text{B}$ is connected to one end of a spring of spring constant $K$. The other end of the spring is connected to a rigid support. If the system is released from rest, what is the maximum elongation produced in the spring? The area of each plate is $\text{A}$ and the medium between the plates is air.

A current of $1.5A$ is flowing through a triangle, of side $9\text{cm}$ each. The magnetic field at the centroid of the triangle is

(Assume that the current is flowing in the clockwise direction.)

There are four concentric shells $\text{A, B, C}$ and $\text{D}$ of radii $a,2a,3a$ and $4a$ respectively. Shells $\text{B}$ and $\text{D}$ are given charges $+q$ and $-q$ respectively. Shell $\text{C}$ is now earthed. Find the potential difference $V_A-V_C$ (Where $K=\frac{1}{4\pi {ϵ}_0}$)

A square card board of side $l$ and mass $m$ is suspended from a horizontal axis $\text{XY}$ as shown in figure. A single wire is wound along the periphery of board carrying a clockwise current $I$. At $t=0$ a vertically downward magnetic field of induction $B$ is switched ON. What will be the minimum value of $B$ so that the board will be able to rotate upto horizontal level?

A man is coming down an incline of angle ${30}^{\circ }$. When he walks with speed $\sqrt{3}$m/s he has to keep his umbrella vertical w.r.t. ground to protect himself from rain. The actual speed of rain is 5 m/s. At what angle with vertical should he keep his umbrella when he is at rest so that he does not get drenched?

The rate at which the radiant energy reaches the surface of earth from the sun is about 1.4 KW/m². The distance from the earth to the sun is about 1.5 10¹¹ m, and the radius of sun is about 0.7 10⁹m If the sun radiates as ideal black body, what is the temperature of its surface?

Three particles of mass $m_1=1\text{kg}$, $m_2=2\text{kg}$, $m_3=3\text{kg}$ are placed at the corners $A,B,C$ of an equilateral triangle, respectively. The edge length of the triangle is $1\text{m}$. Find the distance of $\text{COM}$ of the system of particles from corner $A$.

A parallel plate capacitor of plate area $a$ and plate separation $d$ is charged to potential difference $V$ and then the battery is disconnected. A slab of dielectric constant $K$ is then inserted between the plates of the capacitor so, as to fill the space between the plates. If $Q,E,$ and $W$ denote, respectively, the magnitude of charge on each plate, the electric field between the plates (after the slab is inserted) and work done on the system. In question, in the process of inserting the slab, then

A 5 kg stationary bomb is exploded in three parts having mass 1 : 1 : 3 respectively. Parts having same mass move in perpendicular directions with velocity 39 ms^-1, and then the velocity of bigger part will be

In the shown figure, find the potential drop across $4\mu F$ capacitor and $6\Omega$ resistor (at steady state)

Two bodies (which are not too near each other) are found to repel each other, what kind of fundamental force could this possibly be?

For a channel section subjected to a downward vertical shear force at its centroid, which one of the following represents the correct distribution of shear stress in flange and web ?

In the circuit shown in the figure, a voltage is applied between points $A$ and $B$ which changes with time as

$$E_0=\{\begin{array}{cc}kt& \text{for}0\le t\le t_0\\ k{t}_0& \text{for}t>t_0\end{array}$$ Plot the variation of potential difference $E$ as a function of time.

A chain of length l an mass m lies on the surface of a smooth sphere of radius R>l with one end tied to the top of the sphere. (a) Find the gravitational potential energy of the chain with reference level at the centre of the sphere. (b) Suppose the chain is released and slides down the sphere. Find the kinetic energy of the chain, when it has slid through an angle $\theta$ (c) Find the tangential acceleration $\frac{dv}{dl}$ of the chain when the chain starts sliding down.