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physics

mechanics

Easy

Question

The change in potential energy when a body of mass m is raised to a height n R_E from earth's surface is (R_E. radius of the earth)

$\frac{m g R_{\underline{g}}}{n}$
$m g R_{E} \frac{n}{(n - 1)}$
$m g R_{E}$
$m g R_{Σ} \frac{n}{(n + 1)}$

The correct answer is: $m g R_{Σ} \frac{n}{(n + 1)}$

$U equals negative fraction numerator G m subscript capital sigma m over denominator r end fraction$
At the surface of the earth, $r equals R subscript E$
$table attributes columnalign left end attributes row cell U subscript 1 equals negative fraction numerator G M subscript E m over denominator R subscript E end fraction end cell row cell r equals R subscript E plus h equals R subscript E plus n R subscript E equals left parenthesis 1 plus n right parenthesis R subscript E end cell row cell therefore U subscript 2 equals negative fraction numerator G M subscript E m over denominator left parenthesis n plus 1 right parenthesis R subscript E end fraction end cell row cell capital delta U equals U subscript 2 minus U subscript 1 equals negative fraction numerator G M subscript E m over denominator left parenthesis n plus 1 right parenthesis R subscript E end fraction minus open parentheses negative fraction numerator G M subscript E m over denominator R subscript E end fraction close parentheses end cell row cell equals fraction numerator G M subscript E m over denominator R subscript E end fraction open parentheses 1 minus fraction numerator 1 over denominator left parenthesis n plus 1 right parenthesis end fraction close parentheses equals fraction numerator G M subscript E m m over denominator left parenthesis n plus 1 right parenthesis R subscript E end fraction end cell row cell equals m g R subscript E fraction numerator n over denominator left parenthesis n plus 1 right parenthesis end fraction open parentheses because g equals fraction numerator G M subscript E over denominator R subscript E superscript 2 end fraction close parentheses end cell end table$

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