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The equation for the vibration of a string fixed at both ends vibrating in its third harmonic is given by $y=2\text{(cm)}\sin \left[\right(0.6{\text{cm}}^{-1}x\left]\cos \right[\left(500\pi {\text{s}}^{-1}\right)t].$

What is its length ?

A road is $7.5\text{m}$ wide and its outer edge is raised by $1.5\text{m}$ over the inner edge. The radius of curvature is $50\text{m}$. If $g=10{\text{m/s}}^2$, for what speed of the car is this road suited?

The refractive index of diamond with respect to glass is $1.6$ and absolute the refractive index of glass is $1.5$. Find the absolute refractive index of diamond.

A thin prism having refracting angle ${10}^o$ is made of glass of refractive index $1.42$. This prism is combined with another thin prism of glass of refractive index $1.7$. This combination produces dispersion without deviation. The refracting angle of second prism should be

A billiard ball (of radius R), initially at rest is given a harp impulse by a cue. The cue is held horizontally a distance h above the central line. The ball leaves the cue with a speed $V_0$, eventually acquires a final speed of $\frac{9}{7}{V}_0$. Find h.

The frequency (v) of charged particle, moving at right angles to the magnetic field is independent of

A tunnel is dug in the earth across one of its diameters. Two masses $m$ and $2m$ are dropped from the ends of the tunnel. If the masses collide and stick to each other and perform $\text{S.H.M.}$ then amplitude of $\text{S.H.M.}$ will be: $(R=$ radius of the earth$)$

A
solenoid 60 cm long and of radius 4.0 cm has 3 layers of windings of
300 turns each. A 2.0 cm long wire of mass 2.5 g lies inside the
solenoid (near its centre) normal to its axis; both the wire and the
axis of the solenoid are in the horizontal plane. The wire is
connected through two leads parallel to the axis of the solenoid to
an external battery which supplies a current of 6.0 A in the wire.
What value of current (with appropriate sense of circulation) in the
windings of the solenoid can support the weight of the wire? g
= 9.8 m s⁻²

A current flows in a conducting wire of length $L$. If the wire bend it in a circular form, its magnetic moment would be-

We know centre of mass of a body can lie outside of the body, like in case of a ring. Now let’s take a semicircular ring whose centre is at 0,0. Find the coordinate of it's center of mass.

A seconds pendulum clock made up of a wire shows correct time at $25{}^{\circ }\text{C}$. How much time does the clock lose or gain, in one week, when the temperature is increased to $35{}^{\circ }\text{C}$?

$[\alpha =1.2\times {10}^{-5}/{}^{\circ }\text{C}]$

Consider a non-ideal voltage source whose output voltage is measured by a non-ideal voltmeter as shown below.



Let $V_e$ be the difference between $V_s$ and the measured voltage. $V_e/V_s$ is function of

Magnetic field due to two current elements having same currents in same direction at a point $\text{P}$ in between them is zero. If $r$ is the distance of point $\text{P}$ from one of the conductors and $d$ is the distance between both the conductors, then the value of $d$ is

The lower surface of a cube is fixed. On its upper surface, force is applied at an angle of 30° from its surface. The change will be of the type

The length of the wire between two ends of a sonometer is $100\text{cm}$. What should be the positions of two bridges (from left end of sonometer) below the wire so that the three segments of the wire have their fundamental frequencies in the ratio $1:3:5$

A $750\text{Hz},20\text{V}$ (rms) source connected to a resistance of $100\Omega ,$ an inductance of $0.1803\text{H}$ and a capacitance of $10\mu \text{F}$ all in series. The time in which the resistance (heat capacity $2{\text{J}}^{\circ }{\text{C}}^{-1}$ ) will get heated by ${10}^{\circ }\text{C}$ is close to-



[Assume no loss of heat to the surroundings ]

Magnifying power of a terrestrial telescope is $50$. Find the length of the telescope if focal length of eyepiece and erecting lens is $4\text{cm}$ and $6\text{cm}$ respectively.

Two cars of unequal masses use similar tyres. If they are moving at the same initial speed, the minimum stopping distance

The figure shows three progressive waves A, B and C. What can be concluded from the figure that with respect to wave A?

A wheel has initial angular speed of $\pi \text{rad/s}$. A constant force is acting on wheel to stop it. If wheel covers $\theta =\frac{\pi }{2}\text{rad}$ before coming to rest, find the magnitude of angular acceleration of the wheel.

A ball is thrown upward from the edge of a cliff with an initial velocity of $8\text{m/s}$. How fast will it go at $0.5\text{sec}$ later.
[Take $g=10{\text{m/s}}^2$]

When a plane mirror is placed horizontally on a level ground at a distance of $60\text{m}$ from the foot of a tower, the top of the tower and its image in the mirror subtend an angle of ${90}^{\circ }$ at O. The height of the tower will be

An object is located at $2\text{km}$ beneath the surface of the water. If the fractional compression $\frac{\Delta V}{V}$ is $1.36$%, the ratio of hydraulic stress to the corresponding hydraulic strain will be:

[Given : density of water is $1000{\text{kg m}}^{-3}$ and $g=9.8{\text{m s}}^{-2}$]

On a long horizontally moving belt (Fig. 3.26), a child runs to and fro with a speed $9km{h}^{-1}$ (with respect to the belt) between his father and mother located 50 m apart on the moving belt. The belt moves with a speed of $4km{h}^{-1}$. For an observer on a stationary platform outside, what is the
a) speed of the child running in the direction of motion of the belt?
b) speed of the child running opposite to the direction of motion of the belt?
c) time taken by the child in (a) and (b) ?
Which of the answers alter if motion is viewed by one of the parents?

An anchored boat in water is hits by waves whose successive crests are $100\text{m}$ apart and velocity is $25\text{m/s}$. The boat bounces up once in every -