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Physics

Mechanics

Easy

Question

A particle of mass m is situated at a distance d from one end of a rod of mass M and length L as shown in fig. Find the magnitude of the gravitational force between them.

$\frac{G m M}{d (d + L)}$
$\frac{G m M}{d}$
$\frac{m M}{(d + L)}$
$\frac{G}{d (d + L)}$

The correct answer is: $fraction numerator G m M over denominator d left parenthesis d plus L right parenthesis end fraction$

$text Mass of the element end text d m equals M over L d x d F equals fraction numerator G m open parentheses M over L d x close parentheses over denominator x squared end fraction semicolon F equals integral subscript d superscript left parenthesis d plus L right parenthesis G m open parentheses M over L close parentheses d x end superscript x squared not stretchy rightwards double arrow F equals fraction numerator G m M over denominator L end fraction integral subscript d superscript left parenthesis d plus L right parenthesis end superscript x to the power of negative 2 end exponent d x equals fraction numerator G m M over denominator L end fraction open square brackets fraction numerator x to the power of negative 1 end exponent over denominator negative 1 end fraction close square brackets subscript d superscript left parenthesis d plus L right parenthesis end superscript$
$not stretchy rightwards double arrow F equals fraction numerator G m M over denominator L end fraction open square brackets fraction numerator negative 1 over denominator x end fraction close square brackets subscript d superscript negative d plus L right parenthesis end superscript equals fraction numerator G m M over denominator L end fraction open square brackets 1 over d minus fraction numerator 1 over denominator left parenthesis d plus L right parenthesis end fraction close square brackets semicolon F equals fraction numerator G m M over denominator L end fraction open square brackets fraction numerator d plus L minus d over denominator left parenthesis d plus L right parenthesis d end fraction close square brackets equals fraction numerator G m M over denominator d left parenthesis d plus L right parenthesis end fraction$

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