A-Level Physics October/November 2024 Q1(b)(ii): Use Newton's law of gravitation to show that h and T are related by (h + B)³ = GA T² /…

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  For a satellite in a circular orbit around a planet, the gravitational force provides the necessary centripetal force.

  Let the mass of the planet be $M$, the mass of the satellite be $m$, the radius of the planet be $R$, and the altitude of the satellite above the surface be $h$. The orbital radius is $x = R + h$.

  The gravitational force is given by Newton's law of gravitation: $F_g = \frac{GMm}{x^2}$

  The centripetal force required for an orbit with period $T$ is: $F_c = m\omega^2x = m(\frac{2\pi}{T})^2x = \frac{4\pi^2mx}{T^2}$

  Equating the two forces, $F_g = F_c$: $\frac{GMm}{x^2} = \frac{4\pi^2mx}{T^2}$

  Cancel the satellite mass $m$ from both sides: $\frac{GM}{x^2} = \frac{4\pi^2x}{T^2}$

  Now, rearrange the equation to make $x^3$ the subject: $GMT^2 = 4\pi^2x^3$ $x^3 = \frac{GMT^2}{4\pi^2}$

  Substitute $x = R + h$ into the equation: $(R+h)^3 = \frac{GMT^2}{4\pi^2}$

  Comparing this to the given expression $(h + B)^3 = \frac{GAT^2}{4\pi^2}$:

  We can identify the constants: $A = M$ (the mass of the planet) $B = R$ (the radius of the planet)

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  How the marks are awarded

  • M1 — Equating the gravitational force to the centripetal force, using the correct expressions: $GMm/x^2$ and $4\pi^2mx/T^2$.
  • A1 — Correctly completing the algebraic rearrangement after cancelling the satellite's mass, leading to the expression $x^3 = GMT^2 / 4\pi^2$.
  • B1 — Clearly substituting $x = R+h$ (or $x=B+h$) and explicitly identifying that the constant $A$ is the mass of the planet and $B$ is the radius of the planet.

  Common mistakes

  • Using an incorrect formula for centripetal force, such as $mv^2/r$, and then struggling to substitute $v=2\pi r/T$, often leading to algebraic errors.
  • Making a mistake during the algebraic rearrangement, such as incorrectly cross-multiplying to get $x^3 = 4\pi^2 / GMT^2$ or ending up with $x$ instead of $x^3$.
  • Forgetting to relate the orbital radius $x$ to the altitude $h$ and the planet's radius $R$. Some students leave the answer in terms of $x$ or incorrectly assume $x=h$.
  • Successfully deriving the final equation but failing to explicitly state what the constants A and B represent in terms of the planet's properties.

  Examiner tip: Master deriving orbital relationships by equating the gravitational force to the centripetal force, ensuring you can use the form involving the period $T$ ($F_c = 4\pi^2mr/T^2$) fluently.

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