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Physics-

General

Easy

Question

A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is

$\frac{2 G M}{7 R} (4 \sqrt{2} - 5)$
$- \frac{2 G M}{7 R} (4 \sqrt{2} - 5)$
$\frac{G M}{4 R}$
$\frac{2 G M}{5 R} (\sqrt{2} - 1)$

The correct answer is: $fraction numerator 2 G M over denominator 7 R end fraction left parenthesis 4 square root of 2 minus 5 right parenthesis$

$W equals increment U equals U subscript f end subscript minus U subscript i end subscript equals U subscript infinity end subscript minus U subscript P end subscript$
$equals negative U subscript P end subscript equals negative m V subscript P end subscript$
$equals negative V subscript P end subscript open parentheses a s blank m equals 1 close parentheses$
Potential at point $P$ will be obtained by in integration as given below. Let $d M$ be the mass of small rings as shown

$d M equals fraction numerator M over denominator pi left parenthesis 4 R right parenthesis to the power of 2 end exponent minus pi left parenthesis 3 R right parenthesis to the power of 2 end exponent end fraction open parentheses 2 pi r close parentheses d r$
$equals fraction numerator 2 M r d r over denominator 7 R to the power of 2 end exponent end fraction$
$d V subscript P end subscript equals negative fraction numerator G. d M over denominator square root of 16 R to the power of 2 end exponent plus r to the power of 2 end exponent end root end fraction$
$equals negative fraction numerator 2 G M over denominator 7 R to the power of 2 end exponent end fraction not stretchy integral subscript 3 R end subscript superscript 4 R end superscript fraction numerator r over denominator square root of 16 R to the power of 2 end exponent plus r to the power of 2 end exponent end root end fraction bullet d r$
$equals negative fraction numerator 2 G M over denominator 7 R end fraction open parentheses 4 square root of 2 minus 5 close parentheses$
$therefore W equals plus fraction numerator 2 G M over denominator 7 R end fraction left parenthesis 4 square root of 2 minus 5 right parenthesis$

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parallel

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