Two point-like objects of masses $20\mathrm{gm}$ and $30\mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10\mathrm{cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times {10}^{-8}\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. The angular frequency of the oscillations in $n\times {10}^{-3}\mathrm{rad}{\mathrm{s}}^{-1}$. The value of $n$ is _____.

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$ .

Two infinitely long straight wires lie in the xy - plane along the lines $x=\pm R$. The wire located at $x = +R$ carries a constant current $I_1$ and the wire located at $x =-R$ carries a constant current $I_2$. A circular loop of radius R is suspended with its centre at $\left(0, 0\sqrt{3}R\right)$ and in a plane parallel to the xy - plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\widehat{j}$ direction. Which of the following statements regarding the magnetic field $\overrightarrow{B}$ is (are) true?

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.

The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
If the total energy of the particle is \(E\), it will perform periodic motion only if

A ball of mass \(0.2 \mathrm{~kg}\) rests on a vertical post of height \(5 \mathrm{~m}\). A bullet of mass \(0.01 \mathrm{~kg}\), travelling with a velocity \(v \mathrm{~m} / \mathrm{s}\) in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of \(20 \mathrm{~m}\) and the bullet at a distance of \(100 \mathrm{~m}\) from the foot of the post. The initial velocity \(v\) of the bullet is

The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1m at ${10}^{o}C.$ Now the end P is maintained at ${10}^{o}C,$ while the end S is heated and maintained at ${400}^{o}C.$ The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is $1.2\times {10}^{-5} K^{-1},$ the change in length of the wire PQ is

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is \(0.5 \mathrm{~mm}\) and there are 50 divisions on the circular scale. The reading on the main scale is \(2.5 \mathrm{~mm}\) and that on the circular scale is 20 divisions. If the measured mass of the ball has a relative error of \(2 \%\), the relative percentage error in the density is

Match the following

The packing efficiency of the two-dimensional square unit cell shown below is

A cylindrical cavity of diameter a exists inside a cylinder of diameter \(2 a\) as shown in the figure. Both the cylinder and the cavity are infinity long. A uniform current density \(J\) flows along the length. If the magnitude of the magnetic field at the point \(P\) is given by \(\frac{N}{12} \mu_{0} a J\), then the value of \(N\) is

There are two vernier calipers both of which have $1\mathrm{cm}$ divided into $10$ equal divisions on the main scale. The Vernier scale of one of the calipers $\left(C_1\right)$ has $10$ equal divisions that correspond to $9$ main scale divisions. The Vernier scale of the other caliper $\left(C_2\right)$ has $10$ equal divisions that correspond to $11$ main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in $\mathrm{cm}$) by calipers $C_1$ and $C_2$ respectively, are

Which of the following inequalities is/are TRUE?

A \(20 \mathrm{~cm}\) long string, having a mass of \(1.0 \mathrm{~g}\), is fixed at both the ends. The tension in the string is \(0.5 \mathrm{~N}\). The string is set into vibration using an external vibrator of frequency 100 \(\mathrm{Hz}\). Find the separation (in \(\mathrm{cm}\) ) between the successive nodes on the string.