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A ball of mass $m=2\text{kg}$ is travelling under the influence of gravity. At height $H=5\text{m}$ speed of ball is $u=10\text{m/s}$ as shown in figure. Find the torque due to gravity about $x$ axis.

A particle of mass = 2 kg executes SHM in xy-plane between points A and B under action of force $\overrightarrow{F}=Fx\hat{i}+Fy\hat{j}$. Minimum time taken by particle to move from A to B is 1 sec. At t= 0 the particle is at x = 2 and y = 2. Then $F_x$ as function of time is

The electric field due to a short electric dipole at a distance $r$ from it, on its axial line is $E.$ The electric field on its axial line, at a distance $\frac{r}{2}$ will be equal to



​​​​​​​$[$0.77 Mark$]$

A glass flask is filled up to a mark with 50 cc of mercury at ${18}^{\circ }C$ . If the flask and contents are heated to ${38}^{\circ }C$ , how much mercury will be above the mark ? (coefficient of linear expansion for glass is 9 × ${10}^{-6}{/}^{\circ }C$ and coefficient of real expansion of mercury is 180 × ${10}^{-6}{/}^{\circ }C$)

A screw gauge with a pitch of $0.5\text{mm}$ and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the 45th division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet if the main scale reading is $0.5\text{mm}$ and the 25th division coincides with the main scale line?

If magnitude of scalar and vector products of two vectors are $48\sqrt{3}$ and $144$ respectively, then the angle between the two vectors is :

Find the resultant of the given vectors.

A cyclist riding the bicycle at a speed of $14\sqrt{3}\text{m/s}$ takes a turn around a circular road of radius $20\sqrt{3}\text{m}$ without skidding. What is his inclination to the vertical?
[Take $g=9.8{\text{m/s}}^2$]

What are the readings of the ammeters $A_1$ and $A_2$ shown in figure (45-E4). Neglect the resistances of the meters.

A message signal $m\left(t\right)=5sinc\left(500t\right)V$ is applied to a frequency modulator whose carrier frequency is $100MHz$ and frequency sensitivity is $400Hz/V$. The modulation index (or the deviation ratio) of the resultant $FM$ signal will be

$[$ Assume that, $sinc\left(t\right)=sin\left(\pi t\right)/\pi t]$

A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist ?

Car A is moving with a speed of 36 km/h on a two-lane road. Two cars B and C, each moving with a speed of 54 km/h in opposite directions on the other lane are approaching car A. At a certain instant when the distance AB = distance AC = 1 km, the driver of car B decides to overtake A before C does. What must be the minimum acceleration of car B so as to avoid an accident?

From a solid sphere of mass M and radius R, a spherical portion of radius $\frac{R}{2}$ is removed, as shown in the figure. Taking gravitational potential $V=0$ at $r=\infty$, the potential at the centre of the cavity thus formed is (G = gravitational constant)

A man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes $10$ minutes with a drift of $120m$. If he crosses the river taking shortest route, he takes $12.5$ minutes, find the velocity of the boat with respect to water.

Figure shows a $\text{DC}$ voltage regulator circuit, with a Zener diode of breakdown voltage $=6\text{V}.$ If the unregulated input voltage varies between $10\text{V}$ to $16\text{V},$ then what is the maximum Zener current ?

A ball is projected from a horizontal surface at an angle ${60}^{\circ }$ with horizontal. The maximum height of the projectile is $240\text{m}$ from the ground. Upon hitting the ground for the first time, it loses $75\%$ of its kinetic energy. Immediately after the bounce, the velocity of the ball makes angle of ${45}^{\circ }$ with the horizontal surface. The maximum height of the ball after the bounce is

Light is incident from air on oil at an angle of ${30}^{\circ }.$ After moving through oil-1 and another medium oil-2, it enters glass and then it enters water. If the refractive indices of glass and water are 1.5 and 1.3, respectively, find the angle which the ray makes with normal in water.

The dimension formula for thermal conductivity is (here $K\text{denotes the temeperature}$)

Two wires $\text{AC}$ and $\text{BC}$ are tied at $\text{C}$ to a small sphere of mass $5\text{kg}$, which revolves a constant speed $v$ in the horizontal circle of radius $1.6\text{m}$. If both wires are to remain taut and if the tension in either of the wires is not to exceed $60\text{N}$, then minimum value of $v$ is

[Take, $g=10{\text{m/s}}^2$]

A projectile starts from the origin at time $t=0$ and moves in the $\text{xy-plane}$ with a constant acceleration $\alpha$ in the $\text{y-direction}$ and constant speed along $x$ direction. If equation of motion is $y=\beta x^2$ then its velocity component in the $\text{x-direction}$ will be

The velocities of a particle in SHM at positions $x_1$ and $x_2$ are $v_1$ and $v_2$ respectively, its time period will be -

Repeated observations in an experiment gave the values 1.29, 1.33, 1.34, 1.35, 1.32,1.36, 1.30 and 1.33. Find the relative error.

An LCR circuit contains resistance of $100\Omega$ and a supply of $200\text{V}$ at $300\text{rad/s}$ angular frequency. If only capacitor is taken out from the circuit and the rest of the circuit is joined, current lags behind the voltage by ${60}^{\circ }$. If, on the other hand, only inductor is taken out, the current leads by ${60}^{\circ }$ with the applied voltage. The current flowing in the circuit is

At time $t=0$, a container has $N_0$ radioactive atoms with a decay constant $\lambda$. In addition, $C$ numbers of atoms of the same type are being added to the container per unit time. How many atoms of this type are there at $t=T$ ?

Two trains one of length $100\text{m}$ and another of length $125\text{m}$, are moving in mutually opposite directions along parallel lines, meet each other, each with speed $100\text{m/s}$. If their accelerations are $0.3{\text{m/s}}^2$ and $0.2{\text{m/s}}^2$ respectively, then the time they take to pass each other will be

Find the
derivative of x² – 2 at x = 10.

The gravitational field due to a mass distribution is given by $E=-K/x^3$ in $x-$direction. Taking the gravitational potential to be zero at infinity, find its value at a distance $x$.

The maximum shear stress induced in a solid circular shaft of diameter $15cm$ while transmitting $150kW$ power at $180rpm$ is ____ $N/m{m}^2$

The electric potential $\left(V\right)$ in $\text{volts}$ varies with $x$ $\left(\text{in meters}\right)$ according to the relation $V=5+4x^2$. The force experienced by a negative charge of $2\mu C$ located at $x=0.5\text{m}$ is

A long taut string is connected to a harmonic oscillator of frequency $f$ at one end. The oscillator oscillates with an amplitude $a_0$ and delivers power $P_0$ to the string. Due to dissipation of energy, the amplitude of wave goes on decreasing with distance $x$ from the oscillator as $a=a_0{e}^{-kx}.$ In what length of the string does ${\left(\frac{3}{4}\right)}^{th}$ of the energy supplied by the oscillator get dissipated?