A-Level AQA Physics Gravitational Potential: Derive an expression to show tha

Derive an expression to show that for satellites in a circular orbit $r^2 \propto T^2$ where $T$ is the period of orbit and $r$ is the radius of the orbit. - AQA - A-Level Physics - Question 7 - 2017 - Paper 2

Question 7

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Derive an expression to show that for satellites in a circular orbit $r^2 \propto T^2$ where $T$ is the period of orbit and $r$ is the radius of the orbit.

Worked Solution & Example Answer:Derive an expression to show that for satellites in a circular orbit $r^2 \propto T^2$ where $T$ is the period of orbit and $r$ is the radius of the orbit. - AQA - A-Level Physics - Question 7 - 2017 - Paper 2

Step 1

Derive From Centripetal Force Equation

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Answer

The centripetal force acting on a satellite in a circular orbit is given by:

$F_c = \frac{m v^2}{r}$

where $m$ is the mass of the satellite, $v$ is its orbital speed, and $r$ is the radius of the orbit. This force must equal the gravitational force acting on the satellite:

$F_g = \frac{G M m}{r^2}$

where $G$ is the gravitational constant, and $M$ is the mass of the planet around which the satellite orbits.

Step 2

Combine Equations

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Answer

Setting the centripetal force equal to the gravitational force gives:

$\frac{m v^2}{r} = \frac{G M m}{r^2}$

Cancelling $m$ and rearranging, we find:

$v^2 = \frac{G M}{r}$

The orbital period $T$ is related to the orbital speed by the circumference of the orbit:

$v = \frac{2 \pi r}{T}$

Substituting for $v$ gives:

$\left(\frac{2 \pi r}{T}\right)^2 = \frac{G M}{r}$

Rearranging leads to:

$T^2 = \frac{4 \pi^2 r^3}{G M}$

This shows that $r^2 \propto T^2$ .

Derive an expression to show that for satellites in a circular orbit $r^2 \propto T^2$ where $T$ is the period of orbit and $r$ is the radius of the orbit. - AQA - A-Level Physics - Question 7 - 2017 - Paper 2

Worked Solution & Example Answer:Derive an expression to show that for satellites in a circular orbit $r^2 \propto T^2$ where $T$ is the period of orbit and $r$ is the radius of the orbit. - AQA - A-Level Physics - Question 7 - 2017 - Paper 2

Derive From Centripetal Force Equation

Combine Equations

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