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PhysicsDifficulty 2-3

Gravity

The attractive force between any two masses. Newton's law of universal gravitation describes it as proportional to the product of the two masses and inversely proportional to the square of the distance between them. Near Earth's surface this produces a roughly constant downward acceleration of about 9.8 m/s^2 on falling objects.

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Gravitational Potential Energy

Because the gravitational force varies with distance, the energy stored by separating two masses is not simply proportional to distance once that distance is comparable to the separation itself — the general gravitational potential energy must be found by integrating the force.

Defining U so that F = −dU/dr, and choosing U(∞) = 0 as the reference, gives the standard result for two point masses separated by r:

U(r)=−Gm1m2rU(r) = -G\dfrac{m_1 m_2}{r}

Gravitational potential energy, zero at infinite separation, found by integrating Newton's law of gravitation.

Info: U(r) is negative because gravity is attractive: work must be done on the system to increase the separation towards the zero-energy reference at infinity.

Recovering the Near-Surface Approximation

Near Earth's surface, where height h above the surface is small compared with Earth's radius R⊕, a first-order expansion of U(r) around r = R⊕ recovers the familiar linear approximation used at introductory level.

ΔU≈mgh(h≪R⊕)\Delta U \approx mgh \quad (h \ll R_{\oplus})

The near-surface linear approximation, valid only when h is small compared with Earth's radius.

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Sources and methodology

  • Newton's law of universal gravitation states that the gravitational force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between them (F = Gm₁m₂/r²).
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • The Newtonian gravitational constant G has a CODATA recommended value of 6.67430×10⁻¹¹ m³ kg⁻¹ s⁻².
  • Standard gravity (g₀), the conventional value of acceleration due to gravity at Earth's surface, is defined exactly as 9.80665 m/s² by the 3rd General Conference on Weights and Measures (1901).
    • SI Brochure: Standard Acceleration of Gravity g0 (3rd CGPM, 1901; ISO 80000-3)
  • Ignoring air resistance, gravity produces the same acceleration on every object regardless of its mass, because the extra gravitational force on a heavier object is exactly offset by its greater inertia.
    • Fundamentals of Physics (Halliday, Resnick & Walker)

Content status: published, last reviewed 30 September 2026.