A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02R. The percentage difference in the time periods of the two satellites is:

Option 1 -

0.7

Option 2 -

3.0

Option 3 -

2.0

Option 4 -

1.5

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1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

a month ago

Correct Option - 2

Detailed Solution:

Time period of Satellite = T = $\frac{2 π R}{v} = {(\frac{4 π^{2} R^{3}}{G M})}^{\frac{1}{2}}$

$\Rightarrow T α R^{\frac{3}{2}} \Rightarrow \frac{T_{2}}{T_{1}} = \frac{{(1.02 R)}^{\frac{3}{2}}}{R^{\frac{3}{2}}} = {(1.02)}^{\frac{3}{2}} = {(1 + 0.02)}^{\frac{3}{2}}$

The percentage difference in the time periods of the two satellites = $\frac{Δ T}{T_{1}}$ × 100

= (0.03) × 100 = 3%

Time period of Satellite = T = <math> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mrow> <mrow> <mi>v</mi> </mrow> </mfrac> <mo>=</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816089495"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mo>⇒</mo> <mi>T</mi> <mi>α</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> <mo>⇒</mo> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>.</mo> <mn>0</mn> <mn>2</mn> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </mfrac> <mo>=</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>.</mo> <mn>0</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>.</mo> <mn>0</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math>     <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816089496"> </o:OLEObject></xml><! [endif]->The percentage difference in the time periods of the two satellites = <math> <mrow> <mfrac> <mrow> <mi>Δ</mi> <mi>T</mi> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816089497"> </o:OLEObject></xml><! [endif]->× 100= (0.03) × 100 = 3%

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A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02R. The percentage difference in the time periods of the two satellites is:

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