A particle is projected from a point P ( 3 R , 0 ) along the axis of an annular uniform disc of mass M and inner and outer radii R and 6 R respectively as shown in the figure. If the escape veloci

Question

A particle is projected from a point P ( 3 ​ R , 0 ) along the axis of an annular uniform disc of mass M and inner and outer radii R and 6 ​ R respectively as shown in the figure. If the escape velocity of the particle from this point is 2 N × R GM ​ ​ find N .

TestHub · Accepted Answer

Potential at point P due to elementary is equal to

$dv = - \frac{Gdm}{x}$

where

$dm = \frac{M}{π [ ( 6 R ) ^{2} - R ^{2} ]} 2 π rdr = \frac{2}{5} M \frac{rdr}{R ^{2}}$

Hence $dV = \frac{- G 2 Mrdr}{x 5 R ^{2}}$ as $r^{2} + (3 R)^{2} = x^{2}$ We can write, $2 rdr = 2 xdx$ Hence $dV = \frac{- G 2 M}{5 R ^{2} x} xdx = \frac{- 2}{5} \frac{GM}{R ^{2}} dx$

$V = \int_{x_{1}}^{x_{2}} \frac{- 2}{5} \frac{GM}{R ^{2}} dx = - \frac{2}{5} \frac{GM}{R ^{2}} (x_{2} - x_{1})$

Here, $x_{2} = (3 R)^{2} + (6 R)^{2} = 3 R$

$x_{1} = (3 R)^{2} + R^{2} = 2 R$

Hence $V = - \frac{2}{5} \frac{GM}{R}$ To escape a particle to infinity $\frac{1}{2} m v^{2} ⩾ \frac{2}{5} \frac{GM}{R} m$ Hence $v ⩾ \frac{4 GM}{5 R}$

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