JEE Advanced · Physics · 9. Gravitation

Consider a star of mass $m_2 \mathrm{~kg}$ revolving in a circular orbit around another star of mass $m_1 \mathrm{~kg}$ with $m_1 \gg m_2$. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma \mathrm{kg} / \mathrm{s}$. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is $r$, then its relative rate of change $\frac{1}{r} \frac{\mathrm{~d} r}{\mathrm{~d} t}\left(\right.$ in s $\left.^{-1}\right)$ is given by:
[Note: The original question was awarded bonus marks in the JEE Advanced 2025 exam. We have changed the options to make the question valid and solvable.]

A $\frac{3 \gamma}{2 m_2}$
B $\frac{2 \gamma}{3 m_2}$
C $\frac{2 \gamma}{m_1}$
D $\frac{3 \gamma}{2 m_1}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{2 \gamma}{3 m_2}$

Step-by-step Solution

Detailed explanation

$\begin{aligned} & v=v_0=\sqrt{\frac{G m_1}{r}} \\ & \frac{d v_0}{d t}=\frac{v_0}{2}\left[\frac{\gamma}{m_1}-\frac{1}{r} \frac{d r}{d t}\right]\end{aligned}$

$\begin{aligned} & F \alpha =m_2 \frac{d v}{d t} \\ & \frac{G m_1}{r^2} \frac{v_r}{v}=\frac{d v}{d t} \\ & \frac{v_0}{r} \cdot \frac{d r}{d t} \frac{1}{v_0}=\frac{d v}{d t}\end{aligned}$
$\frac{V_0}{\gamma} \frac{d \gamma}{d t}=\frac{X_0}{2}\left[\frac{r}{m_1}-\frac{1}{r} \frac{d r}{d t}\right]$
$\frac{3}{2} \frac{1}{\gamma} \frac{d \gamma}{d t}=\frac{r}{2 m_1} \rightarrow \frac{1}{\gamma} \frac{d r}{d t}=\frac{\gamma}{3 m_2}$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and

U_{0}

are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

	Column I		Column II
A.	$U_{1} (x) = \frac{U_{0}}{2} {[1 - {(\frac{x}{a})}^{2}]}^{2}$	P.	The force acting on the particle is zero at $x = a$
B.	$U_{2} (x) = \frac{U_{0}}{2} {(\frac{x}{a})}^{2}$	Q.	The force acting on the particle is zero at $x = 0$ .
C.	$U_{3} (x) = \frac{U_{0}}{2} {(\frac{x}{a})}^{2} \exp [- {(\frac{x}{a})}^{2}]$	R.	The force acting on the particle is zero at $x = - a$
D.	$U_{4} (x) = \frac{U_{0}}{2} [\frac{x}{a} - \frac{1}{3} {(\frac{x}{a})}^{3}]$	S.	The particle experiences an attractive force towards $x = 0$ in the region $\|x\| < a$
		T.	The particle with total energy $\frac{U_{0}}{4}$ can oscillate about the point $x = - a$

JEE Advanced 2015 Hard

Explore more questions on app

From JEE Advanced

The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.

	List I		List II
$(P)$	Etard reaction	$1$	$Acetophenone \overset{Zn - Hg, HCl}{\to}$
$(Q)$	Gattermann reaction	$2$	$Toluene \to_{(ii) {SOCl}_{2}}^{(i) {KMnO}_{4}, KOH, Δ}$
$(R)$	Gattermann-Koch reaction	$3$	$Benzene \to_{Anhyd . {AlCl}_{3}}^{{CH}_{3} Cl}$
$(S)$	Rosenmund reduction	$4$	$Aniline \frac{{NaNO}_{2} / HCl}{273 - 278 K}$
		$5$	$Phenol \overset{Zn, Δ}{\to}$

JEE Advanced 2023 Easy

Explore more questions on app