1
JEE Advanced 2024 Paper 2 Online
+3
-1

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M(\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_{\mathrm{c}}(r)=m \alpha / r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_{\mathrm{c}}(r)$, but with a new time period $T_1$, then $\left(T_1^2-T_0^2\right) / T_1^2$ is given by

[G is the gravitational constant.]

A
$\frac{3 \alpha}{G M r_0^2}$
B
$\frac{\alpha}{2 G M r_0^2}$
C
$\frac{\alpha}{G M r_0^2}$
D
$\frac{2 \alpha}{G M r_0^2}$
2
JEE Advanced 2023 Paper 1 Online
+3
-1
Two satellites $\mathrm{P}$ and $\mathrm{Q}$ are moving in different circular orbits around the Earth (radius $R$ ). The heights of $\mathrm{P}$ and $\mathrm{Q}$ from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{P}}=R / 3$. The accelerations of $\mathrm{P}$ and $\mathrm{Q}$ due to Earth's gravity are $g_{\mathrm{P}}$ and $g_{\mathrm{Q}}$, respectively. If $g_{\mathrm{P}} / g_{\mathrm{Q}}=36 / 25$, what is the value of $h_{\mathrm{Q}}$ ?
A
$\frac{3 R}{5}$
B
$\frac{R}{6}$
C
$\frac{6 R}{5}$
D
$\frac{5 R}{5}$
3
JEE Advanced 2019 Paper 1 Offline
+3
-1
Consider a spherical gaseous cloud of mass density $$\rho$$(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If $$\rho$$(r) is constant in time, the particle number density n(r) = $$\rho$$(r)/m is [G is universal gravitational constant]
A
$${K \over {6\pi {r^2}{m^2}G}}$$
B
$${K \over {\pi {r^2}{m^2}G}}$$
C
$${3K \over {\pi {r^2}{m^2}G}}$$
D
$${K \over {2\pi {r^2}{m^2}G}}$$
4
JEE Advanced 2018 Paper 2 Offline
+3
-0.75
A planet of mass $$M,$$ has two natural satellites with masses $${m_1}$$ and $${m_2}.$$ The radii of their circular orbits are $${R_1}$$ and $${R_2}$$ respectively, Ignore the gravitational force between the satellites. Define $${v_1},{L_1},{K_1}$$ and $${T_1}$$ to be , respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $$1$$; and $${v_2},{L_2},{K_2},$$ and $${T_2}$$ to be the corresponding quantities of satellite $$2.$$ Given $${m_1}/{m_2} = 2$$ and $${R_1}/{R_2} = 1/4,$$ match the ratios in List-$${\rm I}$$ to the numbers in List-$${\rm II}.$$

LIST - I LIST - II
P. v1/v2 1. 1/8
Q. L1/L2 2. 1
R. K1/K2 3. 2
S. T1/T2 4. 8
A
$$P \to 4;Q \to 2;R \to 1;S \to 3$$
B
$$P \to 3;Q \to 2;R \to 4;S \to 1$$
C
$$P \to 2;Q \to 3;R \to 1;S \to 4$$
D
$$P \to 2;Q \to 3;R \to 4;S \to 1$$
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