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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