Figure shows a smooth thin non conducting rod of radius r charges with uniform linear charge density λ , fixed horizontally. A neutral and smooth ring Q of mass M can slide freely on the rod which hap

Question

Figure shows a smooth thin non conducting rod of radius r charges with uniform linear charge density λ , fixed horizontally. A neutral and smooth ring Q of mass M can slide freely on the rod which happens to just fit in it, and P is non-conducting particle having a mass m and charge q, attached to the ring Q by means of a nonconducting and incxtensible string of length R. Choose the correct options when string becomes vertical, if P is released from the position shown in figure.

TestHub · Accepted Answer

No external horizontal force acts on (Q + P), so horizontal momentum stays zero and $K E = \frac{1}{2} m v^{2} (1 + \frac{m}{M})$ .

Energy, string horizontal to vertical: P drops by R while its distance from the rod's axis changes from r to (R+r); with $Δ V = \frac{λ}{2 π ε _{0}} ln \frac{R + r}{r}$ :

$\frac{1}{2} m v^{2} (1 + \frac{m}{M}) = m g R + \frac{q λ}{2 π ε _{0}} ln (1 + \frac{R}{r})$ ⇒ $v = [(1 + \frac{m}{M})^{- 1} (2 g R + \frac{q λ}{π ε _{0} m} ln (1 + \frac{R}{r}))]^{1/2}$ - (A); (C) has $(1 - \frac{m}{M})$ instead of $(1 + \frac{m}{M})$ - wrong.

At the vertical position P moves on a circle of radius (R+r): $T = m g + \frac{m v ^{2}}{R + r}$ - (B); (D) has the wrong sign.

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Options:(select one or more)

Doubts & Discussion

JEE Advanced 2027 (Class 12+)