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The ratio of radius of gyration of a solid sphere and hollow sphere of the same mass and radius about a tangential axis is

Find the derivative of the following function with respect to x. $y=\frac{sinx}{x}$

Inside an infinitely long wire, having circular cross-section, there is a hole that runs throughout the length of the wire, as shown in the diagram. A current, with current density $J$, flows through the wire over the cross-section. The magnetic field at point $P$ will be -

Find
the value of:

(i) sin
75°

(ii) tan
15°

A conductor $ABQCD$ moves along its bisector with a velocity of $v=1\text{m/s}$ through a perpendicular magnetic field of $1{\text{Wb/m}}^2$, as shown in the figure. If all the four sides are of $1\text{m}$ length each, then the induced emf between points $A$ and $D$ is

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A spring of spring constant $5\times {10}^{3}N/m$ is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm is

A parallel plate capacitor of $20\mu \text{F}$ is connected to a battery of emf $8.85\text{V}$. If the plate area of each plate is $2{\text{cm}}^2$, with what magnitude of force the positively charged plate attracts the negative one?

A body starting with an initial velocity of 3.5m/s and moving with uniform acceleration of 0.75m/s square undergoes displacement 111/8 in how much time

Positive charge Q is distributed uniformly over a circular ring of radius R . A particle having a mass m and a negative charge q, is placed on its axis at a distance x from the center. Find the force on the particle. Assume x<<<<R

Consider a simple RC circuit as shown in Figure 1





Process 1 : In the circuit the switch $S$ is closed at $t=0$ and the capacitor is fully charged to voltage $V_e$ (i.e. charging continues for time $T>>RC$ ). In the process some dissipation ($E_D$) occurs across the resistance R. The amount of energy finally stored in the fully charged capacitor is $E_c$.



Process 2 :

In a different process the voltage is first set to $\frac{V_0}{3}$ and maintained for a charging time T>>RC. Then the voltage is raised to $\frac{2V_0}{3}$ without discharging the capacitor and again maintained for a time $T>>RC$. The process is repeated one more time by raising the voltage to $V_0$ and the capacitor is charged to the same final voltage $V_0$ as in Process $1.$

These two processes are depicted in Figure 2.

In Process 1, the energy stored in the capacitor $E_c$ and heat dissipated across resistance $E_D$ are related by

Two small balls having equal positive charges $Q$ on each are suspended by two insulating strings of equal length $L$, from a hook fixed to a stand. The whole setup is taken in a satellite into space where there is no gravity (state of weightlessness), then the angle between the two strings in equilibrium is -

The work done in turning a magnet of magnetic moment $M$ by an angle of ${90}^{\circ }$ from the meridian is $n$ times the corresponding work done to turn it through an angle of ${60}^{\circ }$. The value of $n$ is-

The coefficient of static friction between the two blocks is $0.363$, what is the minimum acceleration of block $1$ so that block $2$ does not fall ?





[Take $g=9.8{\text{m/s}}^2$]

A block of mass $0.1kg$ is held against a wall by applying a horizontal force of $5N$ on the block. If the coefficient of friction between the block and the wall is 0.5, the magnitude of frictional force acting on the block is:

A standard electronic transmitter is used to monitor the water temperature in a vessel. The transmitter is calibrated such that it responds to the temperature range of ${50}^{\circ }-{100}^{\circ }C.$ what is the transmitter output when the water temperature is ${85}^{\circ }C$

In magnetic field $\overrightarrow{B}=(2\hat{i}+3\hat{j})\text{T}$ , velocity of a charged particle is given in table-1. Corresponding magnetic force is given in table-2. Match the two tables.



Table-1
Table-2

The work done per unit volume to stretch the length of area of cross-section $2{\text{mm}}^2$ by $2\%$ will be

Take the young's modulus of material as $[Y=8\times {10}^{10}{\text{N/m}}^2]$

Two rods of different materials and of equal cross-sectional areas and length $1\text{m}$ are joined face to face at one end and their free ends are fixed to rigid walls. If the temperature of the surrounding is increased by $30{}^{\circ }\text{C}$, the magnitude of the displacement of the joint of the rods is :
$[{\alpha }_1=2\times {10}^{-5}{}^{\circ }{\text{C}}^{-1},{\alpha }_2=5\times {10}^{-5}{}^{\circ }{\text{C}}^{-1},$
$Y_1=2\times {10}^{10}{\text{N/m}}^2,Y_2={10}^{10}{\text{N/m}}^2]$

A wheel of radius $2\text{m}$ is rotating at a constant angular acceleration of magnitude $10{\text{rad/s}}^2$. Its angular frequency, at $t=0$, is $\frac{60}{\pi }\text{rpm}$. What will be its angular displacement at $t=2.0\text{sec}$?

A ball is projected vertically up with a velocity of $20m{s}^{-1}$. Its velocity at the height of $15\text{m}$ is

(Take $g=10m{s}^{-2})$

In space, uniform electric field exist and electric field is parallel to xy plane. The potential of points P(1, 1) m, Q (–1, 1) m and R(1, 2) m are 2 V, 8 V and 6 V, respectively. The electric field is



त्रिविम में, एकसमान विद्युत क्षेत्र उपस्थित है तथा विद्युत क्षेत्र, xy तल के समान्तर है। बिन्दुओं P(1, 1) m, Q (–1, 1) m तथा R(1, 2) m के विभव क्रमशः 2 V, 8 V व 6 V हैं। विद्युत क्षेत्र है

A ball is thrown vertically upward with a velocity of $10\text{m/s}$ from the top of a tower of height of $50\text{m}$. Find the time taken by the ball to reach a height $10\text{m}$ from the ground. (Take $g=10{\text{m/s}}^2$)

The displacement of a body at a particular second $\left(n\right)$ is given by the expression $s_{n^{th}}=u+\frac{a}{2}(2n-1)$. The dimensional formula of $s_{n^{th}}$ in this equation is

There are two bodies of masses $100\text{kg}$ and $10000\text{kg}$ seperated by a distance $1\text{m}$. At what distance from the smaller body, the intensity of gravitational field will be zero.

What is the mean translational kinetic energy of $N_2$ at $27{}^{\circ }\text{C}$?

Find the acceleration of block of mass $m$ (in $m/s^2$) along the plane (w.r.t ground), if the acceleration of block of mass $M$ is $bm/s^2$. Assume the surface to be smooth and string inextensible.

An 8 bit successive approximation analog to digital converter has full scale reading of $2.55V$ and its conversion time for an analog input of $1V$ is 20 $\mu s.$ The conversion time for a $2V$ input will be

In figure, if the potential at point $\text{P}$ is $100\text{V}$, what is the potential at point $\text{Q}$?

Two particles of masses $m_{1}and{m}_2$ in projectile motion have velocities ${\overrightarrow{v}}_{1}and{\overrightarrow{v}}_2$ respectively at time t = 0 and that is the moment when they collide. Their velocities become ${\overrightarrow{v}}_1^{\prime }and{\overrightarrow{v}}_2^{\prime }$ at time $2t_0$ while still moving in air. The value of magnitude of change in total momentum would be