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Physics

Mechanics

Easy

Question

A thin rod of mass M and length L is bent into semicircle as shown is figure. What is gravitational force on a particle with mass m at the centre of curvature ?

$F_{y} = \frac{G m M}{L^{2}}$
$F_{y} = \frac{2 π G m M}{L^{2}}$
$F_{y} = \frac{2 π}{L^{2}}$
$F_{y} = \frac{L^{2}}{2 π}$

The correct answer is: $F subscript y equals fraction numerator 2 pi G m M over denominator L squared end fraction$

$d F equals fraction numerator G m left parenthesis M divided by L right parenthesis left parenthesis R d theta right parenthesis over denominator R squared end fraction text along end text O P space left square bracket text as end text d l equals R d theta right square bracket$
$table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell d F subscript x equals d F cos invisible function application theta equals fraction numerator G m M cos invisible function application theta over denominator L R end fraction end cell row cell d F subscript y equals d F sin invisible function application theta equals fraction numerator G m M sin invisible function application theta d theta over denominator L R end fraction end cell end table$

$F subscript x equals fraction numerator G m M over denominator L R end fraction integral subscript 0 superscript pi cos invisible function application theta d theta equals fraction numerator G m M over denominator L R end fraction left square bracket sin invisible function application theta right square bracket subscript 0 superscript pi equals 0 F subscript y equals fraction numerator G m M over denominator L R end fraction integral subscript 0 superscript pi sin invisible function application theta d theta equals fraction numerator G m M over denominator L R end fraction left square bracket negative cos invisible function application theta right square bracket subscript 0 superscript pi equals fraction numerator 2 pi G m M over denominator L squared end fraction F equals square root of F subscript x superscript 2 plus F subscript y superscript 2 end root equals F subscript y equals fraction numerator 2 pi G m M over denominator L squared end fraction open parentheses a s F subscript x equals z e r o close parentheses$

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