Introduction
The gravitational field is a vector field describing the force per unit mass at each point in space. Field concept replaces action-at-a-distance with local interaction between mass and field. Understanding gravitational fields is essential for calculating forces on test masses and visualizing gravitational effects.
Definition of Gravitational Field
Gravitational field strength g at a point is defined as the gravitational force per unit mass experienced by a small test mass m placed at that point: g = F/m. For a point mass M: g = (GM/r2)\(\hat{r}\) (directed toward M). Units: N/kg or m/s2 (same units as acceleration). Near Earth's surface, g ≈ 9.8 m/s2 downward. The field is a property of space determined by source masses.
Field Due to Multiple Masses
By superposition principle, total gravitational field at a point is the vector sum of fields from all individual masses: g_total = Σ g_i = Σ (GM_i/r_i2)\(\hat{r}\)_i. For continuous mass distribution: g = ∫ (G dm/r2)\(\hat{r}\). This vector addition allows calculation of fields from arbitrary mass distributions - rings, disks, spheres, or complex shapes.
\nIntroduction
The gravitational field is a vector field describing the force per unit mass at each point in space. Field concept replaces action-at-a-distance with local interaction between mass and field. Understanding gravitational fields is essential for calculating forces on test masses and visualizing gravitational effects.
Definition of Gravitational Field
Gravitational field strength g at a point is defined as the gravitational force per unit mass experienced by a small test mass m placed at that point: g = F/m. For a point mass M: g = (GM/r2)\(\hat{r}\) (directed toward M). Units: N/kg or m/s2 (same units as acceleration). Near Earth's surface, g ≈ 9.8 m/s2 downward. The field is a property of space determined by source masses.
Field Due to Multiple Masses
By superposition principle, total gravitational field at a point is the vector sum of fields from all individual masses: g_total = Σ g_i = Σ (GM_i/r_i2)\(\hat{r}\)_i. For continuous mass distribution: g = ∫ (G dm/r2)\(\hat{r}\). This vector addition allows calculation of fields from arbitrary mass distributions - rings, disks, spheres, or complex shapes.
\nField Lines and Visualization
Gravitational field lines: (1) Point toward masses (field is attractive); (2) Density represents field strength (closer lines = stronger field); (3) Never intersect; (4) Form smooth curves from infinity to mass centers. Field line diagrams visualize force direction and magnitude. Uniform field (near Earth's surface) has parallel, equally spaced lines. Point mass has radial lines converging on mass.
Conservative Nature
Gravitational field is conservative: ∮g·dr = 0 around any closed path. This means work done by gravity around closed loop is zero, and potential energy can be defined. Equivalent statements: (1) Curl of g is zero (∇×g = 0); (2) g can be written as gradient of scalar potential (g = -∇φ); (3) Work between two points is path-independent.
\nCalculating Fields for Extended Objects
For a uniform spherical shell: g = 0 inside, g = GM/r2 outside (toward center). For uniform solid sphere: g = (GM/R3)r inside (linear with r), g = GM/r2 outside. These results come from integration or Gauss's law. For a thin ring at axial distance z: g = GMz/(R2+z2)^(3/2). Field calculations involve vector integration over mass elements.
\nSolved Example: Field Inside Earth
Calculate gravitational field at depth d below Earth's surface. Earth has uniform density ρ, radius R_E. Solution: By shell theorem, only mass interior to radius (R_E - d) contributes. Interior mass: M_enc = (4/3)π(R_E - d)3ρ. Total Earth mass M_E = (4/3)πR_E3ρ. So M_enc = M_E × (R_E - d)3/R_E3 = M_E × (1 - d/R_E)3. Gravitational field at depth d: g = GM_enc/(R_E - d)2 = G×M_E×(R_E - d)3/R_E3 / (R_E - d)2 = (GM_E/R_E3)×(R_E - d) = g_surface × (R_E - d)/R_E = g_surface × (1 - d/R_E). At Earth's center (d = R_E): g = 0. At half radius (d = R_E/2): g = g_surface/2. Field decreases linearly with depth, zero at center. This assumes uniform density; actual Earth has dense core so g actually increases slightly in outer core before decreasing.
\n