Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1=300\mathrm{K}\mathrm{a}\mathrm{n}\mathrm{d}{T}_2=100\mathrm{K}$as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1\mathrm{a}\mathrm{n}\mathrm{d}{K}_2$, respectively. If the temperature at the junction of the two cylinders in the steady state is $200\mathrm{}\mathrm{K}$, then $K_1/K_2$is

In a historical experiment to determine Planck's constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength \((\lambda)\) of incident light and the corresponding stopping potential (\(V_0\)) are given below:

\(\lambda(\mu m)\)	Volt
0.3	2.0
0.4	1.0
0.5	0.4

Given that \(c=3 \times 10^8 m s^{-1}\) and \(e=1.6 \times 10^{-19} C\), Planck's constant (in units of J s) found from such an experiment is :

A mixture of ideal gas containing $5$ moles of monatomic gas and $1$ mole of rigid diatomic gas is initially at pressure ${\mathrm{P}}_0,$ volume ${\mathrm{V}}_0$ and temperature ${\mathrm{T}}_0.$ If the gas mixture is adiabatically compressed to a volume $\frac{{\mathrm{V}}_0}{4}$, then the correct statement(s) is/are,
(Given $2^{1.2}=\mathrm{ }2.3\text{;} 2^{3.2}=9.2\text{;}\mathrm{ }\mathrm{R}$ is gas constant)

A geostationary satellite above the equator is orbiting around the earth at a fixed distance \(r_1\) from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance \(r_2\) from the center of the earth, such that \(r_1=1.21 r_2\). The time period of the second satellite as measured from the geostationary satellite is \(\frac{24}{p}\) hours. The value of \(p\) is

A hydrogen atom in its ground state is irradiated by light of wavelength $970 Å.$ Taking hc/e = $1.237\times {10}^{-6} eV$ m and the ground state energy of hydrogen atom as $-13.6 eV$ , the number of lines present in the emission spectrum is

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed \(\omega\), the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the \(z\)-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points \(P\) and \(Q\) ). Both these motions have the same angular speed \(\omega\) in this case


Now consider two similar systems as shown in the figure: Case
(a) the disc with its face vertical and parallel to \(x-z\) plane; Case
(b) the disc with its face making an angle of
\(45^{\circ}\) with \(x-y\) plane and its horizontal diameter parallel to \(x\)-axis. In both the cascs, the disc is welded at point \(\mathrm{P}\), and the systems are rotated with constant angular speed \(\omega\) about the \(z\) axis.

Question : Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

For any positive integer $n$, define $f_n:\left(0, \infty \right)\to R$ as $f_n\left(x\right)={\sum }_{i=1}^n{\tan }^{-1}\left(\frac{1}{1+\left(x+j\right)\left(x+j-1\right)}\right)$ for all $x\in \left(0,\infty \right)$ . $($Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$$)$ Then, which of the following statement(s) is (are) TRUE?

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors \(\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}\) such that \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}=\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}=\frac{1}{2}\). Then, the volume of the parallelopiped is

If ${\mathrm{\lambda }}_{\mathrm{c}\mathrm{u}}$ is the wavelength of ${\mathrm{K}}_{\mathrm{\alpha }}$ X-ray line of copper (atomic number 29) and ${\mathrm{\lambda }}_{\mathrm{M}\mathrm{o}}$ is the wavelength of the ${\mathrm{K}}_{\mathrm{\alpha }}$ X-ray line of molybdenum (atomic number 42), then the ratio $\frac{{\mathrm{\lambda }}_{\mathrm{c}\mathrm{u}}}{{\mathrm{\lambda }}_{\mathrm{M}\mathrm{o}}}$ is close to

Two balls, having linear momenta \(\overrightarrow{\mathbf{p}}_1=p \hat{\mathbf{i}}\) and \(\overrightarrow{\mathbf{p}}_2=-p \hat{\mathbf{i}}\), undergo a collision in free space. There is no external force acting on the balls. Let \(\overrightarrow{\mathbf{p}}_1^{\prime}\) and \(\overrightarrow{\mathbf{p}}_2^{\prime}\) be their final momenta. The following option(s) is/are NOT ALLOWED for any non-zero value of \(p, a_1, a_2, b_1, b_2, c_1\) and \(c_2\)

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct?

Paragraph:
The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
A diatomic molecule has moment of inertia I. By Bohr's quantization condition its rotational energy in the \(n\)th level ( \(n=0\) is not allowed) is