If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that

In the determination of Young's modulus \(\left(Y=\frac{4 M L g}{\pi l d^{2}}\right)\) by using Searle's method, a wire of length \(L=2 \mathrm{~m}\) and diameter \(d=0.5 \mathrm{~mm}\) is used. For a load \(M=2.5 \mathrm{~kg}\), an extension \(l=0.25 \mathrm{~mm}\) in the length of the wire is observed. Quantities \(d\) and \(l\) are measured using a screw gauge and a micrometer, respectively. They have the same pitch of \(0.5 \mathrm{~mm}\). The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the \(Y\) measurement

Paragraph :
A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
The block in the above question is maintained at the position by external means and the level of liquid is lowered. The height \(h_2\) when this external force reduces to zero is

In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement(s) is(are) correct?
[Given : $e^{–1}=0.36$]

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

Paragraph :
A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
If height \(h_2\) of water level is further decreased, then

Two uniform strings of mass per unit length \(\mu\) and \(4 \mu\), and length \(L\) and \(2 L\), respectively, are joined at point \(\mathrm{O}\), and tied at two fixed ends \(\mathrm{P}\) and \(\mathrm{Q}\), as shown in the figure. The strings are under a uniform tension \(T\). If we define the frequency \(v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}\), which of the following statement(s) is(are) correct?

Consider an elliptically shaped rail PQ in the vertical plane with $\mathrm{O}\mathrm{P}\mathrm{}=\mathrm{}3\mathrm{}\mathrm{m}$ and $\mathrm{O}\mathrm{Q}\mathrm{}=\mathrm{}4\mathrm{}\mathrm{m}$ . A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is $(\mathrm{n}\times 10)$ Joules. The value of n is (take acceleration due to gravity = $10\mathrm{}\mathrm{m}{\mathrm{s}}^{–2}$ )

A circuit is connected as shown in the figure with the switch \(S\) open. When the switch is closed, the total amount of charge that flows from \(Y\) to \(X\) is

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate $\left({\mathrm{KMnO}}_4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\mathrm{\Delta }n=\left(n_1-n_2\right)\ll n_1.$ When they are connected by a tube of small length $l$ and cross-sectional area $S,{\mathrm{KMnO}}_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order ${\left(\mathrm{\Delta }n\right)}^2$. Which of the following is/are correct? ($k_B$ is the Boltzmann constant)

If $\beta =\underset{x\to 0}{\lim }\frac{e^{x^3}-{\left(1-x^3\right)}^{\frac{1}{3}}+\left({\left(1-x^2\right)}^{\frac{1}{2}}-1\right)\sin x}{x{\sin }^{2}x}$, then the value of $6\beta$ is

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

The correct option(s) about entropy (S) is(are)
[$\mathrm{R}=$ gas constant, $\mathrm{F}=$ Faraday constant, $\mathrm{T}=$ Temperature]