Class 11th Gravitation Physics Physics Gravitation

Consider a binary star system of star A and star B with masses m_A and m_B revolving in a circular orbit of radii r_A and r_B, respectively. If T_A and T_B are the time period of star A and star B, respectively, then:

Option 1 - <p>T<sub>A</sub> = T<sub>B</sub></p>

Option 2 - <p>T<sub>A</sub> > T<sub>B</sub> (if m<sub>A</sub> > m<sub>B</sub>)</p>

Option 3 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </msup> </mrow> </math> </span></p>

Option 4 - <p>T<sub>A</sub> > T<sub>B</sub> (if r<sub>A</sub> > r<sub>B</sub>)</p>

6 Views|Posted 6 months ago

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Answered by

Vishal Baghel | Contributor-Level 10

6 months ago

Correct Option - 1

Detailed Solution:

Since binary mass system performs circular motion about is common centre of mass, so

$m_{A} ω_{A}^{2} r_{A} = \frac{G m_{B} m_{A}}{{(r_{A} + r_{B})}^{2}} = \frac{G m_{B} m_{A}}{r^{2}}$

$m_{A} ω_{A}^{2} * \frac{m_{B}}{(m_{A} + m_{B})} r = \frac{G m_{B} m_{A}}{r^{2}}$

$ω_{A} = \sqrt[]{\frac{G (m_{A} + m_{B})}{r^{3}}}$

Similarly we can show that

$ω_{B} = \sqrt[]{\frac{G (m_{A} + m_{B})}{r^{3}}}$

Hence their angular velocity will be same, time period will be same, i.e. TA = TB

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