Explain why the gravitational field strength inside a uniform spherical planet decreases as one moves from the surface towards the centre.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A geophysicist models the Earth as a uniform sphere of mass M and radius R. She considers how the gravitational field strength varies at points within the planet.
Model answer (4 marks)
The field at a point inside a uniform sphere depends only on the mass that lies inside a sphere of radius r centred on the planet’s centre. The mass of the shell outside this radius exerts no net force on the point. For a uniform density ρ the enclosed mass is M_enc = ρ(4/3)πr³, so M_enc ∝ r³. The gravitational field is g = GM_enc/r². Substituting the r³ dependence gives g ∝ r³/r² = r. Thus as r decreases from the surface (r = R) towards the centre, g decreases linearly to zero at r = 0, where the enclosed mass is zero.
Examiner tips
- Show the dependence of g on the enclosed mass and radius, then substitute M_enc ∝ r³ to get g ∝ r.
- Explain that the shell theorem means only the inner mass contributes, not the total mass.
- Mention that g → 0 at the centre because M_enc → 0.
Common mistakes
- Confusing the total mass of the planet with the enclosed mass inside radius r.
- Using g = GM/r² with the total mass M instead of M_enc.
- Failing to show the r³/r² simplification that leads to g ∝ r.
Mark scheme (4 marks)
- Gravitational field strength depends on the mass that is enclosed within the sphere of radius r (the mass 'below' the point), not the total mass of the planet.
- The enclosed mass is proportional to r³ (for a uniform density planet), so it decreases as r decreases.
- Gravitational field strength g = GM_enc / r², so both the numerator (M_enc ∝ r³) and denominator (r²) change; the numerator decreases faster, giving g ∝ r.
- At the centre r = 0, the enclosed mass is zero and the gravitational field strength is zero, consistent with the linear relationship.
Key terms in this question
Related
- All IB DP Physics Standard Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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