A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4kg$ is at rest on this surface. An impulse of $1.0 N s$ is applied to the block at time $t=0$ so that it starts moving along the x-axis with a velocity $v\left(t\right)=v_0{e}^{-\frac{t}{\tau }}$ , where $v_0$ is a constant and $\tau =4$ . The displacement of the block, in meter, at $t=\tau$ is __________. Take $e^{-1}=0.37$

A \(0.1 \mathrm{~kg}\) mass is suspended from a wire of negligible mass. The length of the wire is \(1 \mathrm{~m}\) and its cross-sectional area is \(4.9 \times 10^{-7} \mathrm{~m}^2\). If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency \(140 \mathrm{rad} \mathrm{s}^{-1}\). If the Young's modulus of the material of the wire is \(n \times 10^9 \mathrm{Nm}^{-2}\), the value of \(n\) is

Two batteries of different emfs and different internal resistances are connected as shown. The voltage across \(A B\) in volt is

A piece of ice (heat capacity \(=2100 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}\) and latent heat \(=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}\) ) of mass \(\mathrm{m}\) gram is at \(-5^{\circ} \mathrm{C}\) at atmospheric pressure. It is given \(420 \mathrm{~J}\) of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that \(1 \mathrm{~g}\) of ice has melted. Assuming there is no other heat exchange in the process, the value of \(m\) is

A student performs an experiment for determination of \(g\left(=\frac{4 \pi^2 l}{T^2}\right), l \approx 1 \mathrm{~m}\), and he commits an error of \(\Delta l\). For \(T\) he takes the time of \(n\) oscillations with the stop watch of least count \(\Delta T\) and he commits a human error of \(0.1 \mathrm{~s}\). For which of the following data, the measurement of \(g\) will be most accurate?

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0 m s^{-1}$ and the man behind walks at a speed $2.0 m s^{-1}$ . A third man is standing at a height $12 m$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $1430 Hz$ . The speed of sound in air is $330 m s^{-1}.$ At the instant, when the moving men are $10 m$ apart, the stationary man is equidistant from them. The frequency of beats in $Hz$ , heard by the stationary man at this instant, is __________.

One mole of a monatomic ideal gas is taken through a cycle \(A B C D A\) as shown in the \(p\)-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.

Let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$, for $x\in \mathrm{ℝ}$. Then the number of real solutions of the equation $\sqrt{1+\cos \left(2x\right)}=\sqrt{2}{\tan }^{-1}\left(\tan x\right)$ in the set $\left(-\frac{3\pi }{2}, -\frac{\pi }{2}\right)\cup \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)\cup \left(\frac{\pi }{2}, \frac{3\pi }{2}\right)$ is equal to

Let $P$ be a point on the parabola $y^2=4ax$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$, where $F$ is the focus of the parabola, is $120$. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $\left(a, m\right)$ is

Match the statements in Column I with the properties Column II.

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In a thin rectangular metallic strip a constant current \(I\) flows along the positive \(x\)-direction, as shown in the figure. The length, width and thickness of the strip are \(l, w\) and \(d\), respectively.

A uniform magnetic field \(\vec{B}\) is applied on the strip along the positive \(y\)-direction. Due to this, the charge carriers experience a net deflection along the \(z\)-direction. This results in accumulation of charge carriers on the surface \(P Q R S\) and appearance of equal and opposite charges on the face opposite to \(P Q R S\). A potential difference along the \(z\)-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.






Question:

Consider two different metallic strips (\(1\) and \(2\)) of the same material. Their lengths are the same, widths are \(w_{1}\) and \(w_{2}\) and thicknesses are \(d_{1}\) and \(d_{2}\), respectively. Two points \(K\) and \(M\) are symmetrically located on the opposite faces parallel to the \(x-y\) plane (see figure). \(V_{1}\) and \(V_{2}\) are the potential differences between \(K\) and \(M\) in strips \(1\) and \(2\), respectively. Then, for a given current \(I\) flowing through them in a given magnetic field strength \(B\), the correct statement(s) is(are)

Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=\pi /(2 \omega)\), then the value of \(4 b^2 / a^2\) will be ________ .

Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction as shown in the given scheme.