List of top Physics Questions on Mechanics asked in IIT JAM PH

In a two-level atomic system, the excited state is 0.2 eV above the ground state. Considering the Maxwell-Boltzmann distribution, the temperature at which 2% of the atoms will be in the excited state is _____ K. (up to two decimal places)
(Boltzmann constant $k_B = 8.62 \times 10^{-5}$ eV/K)

At a particular temperature T, Planck's energy density of black body radiation in terms of frequency is $\rho_T(\nu) = 8 \times 10^{-18} \text{ J/m}^3 \text{ Hz}^{-1}$ at $\nu = 3 \times 10^{14}$ Hz. Then Planck's energy density $\rho_T(\lambda)$ at the corresponding wavelength ($\lambda$) has the value \rule{1cm}{0.15mm} $\times 10^2 \text{ J/m}^4$. (in integer)
[Speed of light $c = 3 \times 10^8$ m/s]
(Note: The unit for $\rho_T(\nu)$ in the original problem was given as J/m³, which is dimensionally incorrect for a spectral density. The correct unit J/(m³·Hz) or J·s/m³ is used here for the solution.)

A planet rotates in an elliptical orbit with a star situated at one of the foci. The distance from the center of the ellipse to any foci is half of the semi-major axis. The ratio of the speed of the planet when it is nearest (perihelion) to the star to that at the farthest (aphelion) is \rule{1cm{0.15mm}. (in integer)}

A particle is moving with a constant angular velocity 2 rad/s in an orbit on a plane. The radial distance of the particle from the origin at time t is given by $r = r_0 e^{2\beta t}$ where $r_0$ and $\beta$ are positive constants. The radial component of the acceleration vanishes for $\beta = $ \rule{1cm{0.15mm} rad/s. (in integer)}

Neutrons of energy 8 MeV are incident on a potential step of height 48 MeV. As they penetrate the classically forbidden region, the distance at which the probability density of finding neutrons decreases by a factor of 100 is \rule{1cm{0.15mm} fm. (up to two decimal places)
(Take $\hbar c = 200$ MeV fm, and the rest mass energy of neutron = 1 GeV.)}

Two solid cylinders of the same density are found to have the same moment of inertia about their respective principal axes. The length of the second cylinder is 16 times that of the first cylinder. If the radius of the first cylinder is 4 cm, the radius of the second cylinder is \rule{1cm{0.15mm} cm. (in integer)}

Two particles of masses $m_1$ and $m_2$, interacting via gravity, rotate in circular orbits about their common center of mass with the same angular velocity $\omega$.
For masses $m_1$ and $m_2$, respectively, \begin{itemize
$r_1$ and $r_2$ are the constant distances from the center of mass,
$L_1$ and $L_2$ are the magnitudes of the angular momenta about the center of mass, and
$K_1$ and $K_2$ are the kinetic energies.
Which of the following is(are) correct?
(G is the universal gravitational constant)}

A particle of mass m is subjected to a potential V(x). If its wavefunction is given by
$\psi(x,t) = \alpha x^2 e^{-\beta x} e^{i\gamma t/\hbar}, x>0$
$\psi(x,t) = 0, x \le 0$,
then V(x) is ($\alpha$, $\beta$ and $\gamma$ are constants of appropriate dimensions)

Two non-relativistic particles with masses $m_1$ and $m_2$ move with momenta $\mathbf{p}_1$ and $\mathbf{p}_2$, respectively, in an inertial frame S. In another inertial frame S', moving with a constant speed with respect to S, the same particles are observed to have momenta $\mathbf{p'}_1$ and $\mathbf{p'}_2$, respectively.
Galilean invariance implies that

One end of a long chain is lifted vertically from flat ground to a height H with constant speed v by a force of magnitude F. Assume that the length of the chain is greater than H and that it has a uniform mass per unit length $\rho$. The magnitude of the force F at height H is
(g is the acceleration due to gravity)

Three particles of equal mass M, interacting via gravity, lie on the vertices of an equilateral triangle of side d, as shown in the figure below. The whole system is rotating with an angular velocity $\omega$ about an axis perpendicular to the plane of the system and passing through the center of mass. The value of $\omega$, for which the distance between the masses remains d, is
(G is the universal gravitational constant)

Two masses, $M_1$ and $M_2$, are connected through a massless spring of spring constant k, as shown in the figure below. The mass $M_1$ is at rest against a rigid wall. Both $M_1$ and $M_2$ are on a frictionless surface. The mass $M_2$ is pushed towards $M_1$ by a distance x from its equilibrium position and then released. After $M_1$ leaves the wall, the speed of the center of mass of the composite system is

Two planets $P_1$ and $P_2$ having masses $M_1$ and $M_2$ revolve around the Sun in elliptical orbits, with time periods $T_1$ and $T_2$, respectively. The minimum and maximum distances of planet $P_1$ from the Sun are $R$ and $3R$, respectively, whereas for planet $P_2$, these are $2R$ and $4R$, respectively, where $R$ is a constant. Assuming $M_1$ and $M_2$ are much smaller than the mass of the Sun, the magnitude of $\dfrac{T_2}{T_1}$ is:

For a particle moving in a central potential, which one of the following statements is correct?

Three planets orbit a star at distances $a,\;4a,\;9a$. Their orbital periods are proportional to $r^{3/2}$. If the smallest planet has period $T$, after how long will all three be aligned again?

A particle $P$ of mass $m$ is constrained to move on the surface of a cylinder under a force $-k\vec{r}$ as shown in the figure ($k$ is a positive constant). Neglect friction. Which of the following statements is correct?

A planet has the same average density as Earth but only $1/8$ the mass of Earth. If $g_p$ and $g_e$ are the surface gravities on the planet and Earth, then $\frac{g_p}{g_e} =$ ............ (Specify your answer up to one digit after decimal.)

A body of mass 1 kg moves in an elliptical orbit with semi-major axis 1000 m and semi-minor axis 100 m. The orbital angular momentum is 100 kg m$^2$ s$^{-1}$. The time period of motion is ............. hours. (Specify answer up to two digits after the decimal point.)

A particle of mass $m$ moves in a circular orbit with $x = R\cos(\omega t)$ and $y = R\sin(\omega t)$ observed in inertial frame $S_1$. Another frame $S_2$ moves with velocity $\vec{v} = \omega R \hat{i}$ with respect to $S_1$, and origins coincide at $t=0$. The angular momentum at $t = \frac{2\pi}{\omega}$ as observed in $S_2$ about its origin is $(mR^2\omega)x$. Then $x$ is ............. (Specify answer up to two digits after decimal.)

The moon moves around the earth in a circular orbit with a period of 27 days. If $R = 6.4 \times 10^6$ m is Earth's radius and $g = 9.8$ m/s$^2$, and if $D$ is the moon's orbital radius, find $D/R$. (Specify answer up to one digit after decimal point.)

A raindrop falls under gravity and captures water molecules from the atmosphere. Its mass changes at the rate $\lambda m(t)$, where $\lambda$ is a positive constant and $m(t)$ is the instantaneous mass. Assume gravity is constant and captured water is at rest before capture. Which of the following statements is correct?

At $ t = 0 $, a particle of mass $ m $ having velocity $ v_0 $ starts moving through a liquid kept in a horizontal tube and experiences a drag force $ F_d = - k \frac{dv}{dt} $. It covers a distance $ L $ before coming to rest. If the times taken to cover the distances $ L/2 $ and $ L $ are $ t_2 $ and $ t_4 $, respectively, then the ratio $ t_2/t_4 $ (ignoring gravity) is:

To demonstrate Bernoulli's principle, an instructor arranges two circular horizontal plates of radii $ b $ each with distance $ d $ ($ d \ll b $) between them (see figure). The upper plate has a hole of radius $ a $ in the middle. On blowing air at a speed $ v_0 $ through the hole so that the flow rate of air is $ \pi a^2 v_0 $, it is seen that the lower plate does not fall. If the density of air is $ \rho $, the upward force on the lower plate is well approximated by the formula (assume that the region with $ r<a $ does not contribute to the upward force and the speed of air at the edges is negligible):

An object of mass $ m $ with non-zero angular momentum $ \ell $ is moving under the influence of gravitational force of a much larger mass (ignore drag). Which of the following statement(s) is (are) correct?

The linear mass density of a rod of length $ L $ varies from one end to the other as $ \lambda_0 \left( 1 + \frac{x^2}{L^2} \right) $, where $ x $ is the distance from one end with tensions $ T_1 $ and $ T_2 $ in them (see figure), and $ \lambda_0 $ is a constant. The rod is suspended from a ceiling by two massless strings. Then, which of the following statement(s) is (are) correct?