Interactive physics simulator
Gravitational Field Strength (g)
Explore how gravity fields curve space and attract objects. Map radial vector grids, compare uniform density gravity profiles with Earth's multi-layered PREM core curve, and locate neutral saddle points alongside co-rotating Lagrange L1–L5 coordinates.
Gravitational Field Strength Lab
Configure celestial properties, toggle interior models, and interact with test probes to inspect field telemetry.
Field Telemetry
- Planet Mass (M)
- 1.00 M_Earth
- Planet Radius (R)
- 6371 km
- Probe Distance (r)
- 12742 km (2.00 R)
- Field Strength (g)
- 2.45 N/kg (m/s²)
- Surface Gravity (g0)
- 9.81 N/kg
- Orbital Period (T)
- 4.01 hours
What is Gravitational Field Strength?
Gravitational field strength (represented by $g$) is a vector quantity that describes the intensity of gravity at any point in space. It is defined as the gravitational force ($F$) exerted per unit mass ($m$) on a small test object placed in the field:
Because $F$ is measured in Newtons (N) and $m$ is measured in kilograms (kg), the SI unit for field strength is **Newtons per kilogram** (N/kg). Since Newtons are equivalent to kg·m/s², the units are also equivalent to **meters per second squared** (m/s²), representing the acceleration that a free-falling object experiences at that position.
Planetary Gravity Fields
How the field scales outside a spherical planet:
- Point Mass Formula: For any sphere (or point mass), the field strength is calculated as: $g = G \cdot M / r^2$, where $r$ is the distance from the center.
- Inverse Square Law: Outside the planet surface ($r > R$), the field drops off inversely with the square of the distance. Going twice as far from the core reduces $g$ to 25% of its surface strength.
Gravity Profiles Inside the Planet
Decay behavior as you descend towards the center:
- Uniform Density: If density ($\rho$) is constant, the outer shells cancel out, and the field decreases linearly: $g \propto r$. At the center ($r = 0$), $g = 0$ N/kg.
- PREM Model (Earth): Earth's core is extremely dense compared to the mantle. As you descend, you get closer to this dense core, causing $g$ to increase initially and peak at $10.7$ m/s² at the core-mantle boundary before dropping to zero.
Lagrange & Saddle points
Field mechanics in multi-body systems:
- Neutral Saddle Point: In a binary system (like Earth-Moon), the point where their gravitational fields are equal and opposite, cancelling to zero (g_net = 0 N/kg).
- Lagrange Points: Five equilibrium points (L1 to L5) where the combined gravity fields of two large bodies balance the centrifugal forces of a rotating reference frame.
Solved Examples
Calculate the gravitational field strength g at the surface of Mars, given Mars has a mass of 6.42 * 10^23 kg and a radius of 3389 km.
- Identify the given values: planet mass (M) = 6.42 * 10^23 kg, planet radius (R) = 3389 km = 3.389 * 10^6 meters.
- Recall the universal gravitational constant: G = 6.674 * 10^-11 N m² / kg².
- Apply the formula for gravitational field strength: g = G * M / R².
- Substitute the values: g = (6.674 * 10^-11 * 6.42 * 10^23) / (3.389 * 10^6)².
- Calculate: g = (4.285 * 10^13) / (1.149 * 10^13) = 3.73 N/kg.
- Verify: The gravitational field strength is about 3.73 N/kg (or 3.73 m/s²), which is approximately 38% of Earth's gravity, matching observational data.
Answer: g = 3.73 N/kg (or m/s²)
Earth's surface gravitational field strength is 9.81 N/kg. A communication satellite orbits at an altitude equal to three times the Earth's radius (h = 3R). Calculate the gravitational field strength at the satellite's orbital altitude.
- Identify the total distance (r) from the center of the Earth: r = Earth Radius (R) + Altitude (h) = R + 3R = 4R.
- Recall that outside a spherical body, gravitational field strength follows the inverse square law: g is proportional to 1 / r².
- Formulate the ratio: g_orbit / g_surface = (R / r)² = (R / 4R)² = 1 / 16.
- Calculate the orbital field strength: g_orbit = g_surface / 16 = 9.81 / 16 = 0.613 N/kg.
- Verify: At four times the center distance, the field strength drops to 1/16th (6.25%) of its surface value, which is 0.613 N/kg.
Answer: g = 0.613 N/kg (or m/s²)
Consider a collinear Earth-Moon system where Earth has a mass of 5.97 * 10^24 kg, the Moon has a mass of 7.35 * 10^22 kg, and their centers are separated by d = 3.84 * 10^8 meters. Find the distance from the center of the Earth to the neutral saddle point where the net gravitational field strength is zero.
- Let x be the distance from the Earth's center to the neutral point. The distance from the Moon's center to this point is d - x.
- Set the gravitational fields of Earth and Moon equal in magnitude (since they point in opposite directions): G * M_Earth / x² = G * M_Moon / (d - x)².
- Cancel G and take the square root of both sides: sqrt(M_Earth) / x = sqrt(M_Moon) / (d - x).
- Rearrange the equation to solve for x: x = d * sqrt(M_Earth) / (sqrt(M_Earth) + sqrt(M_Moon)).
- Calculate the square roots: sqrt(5.97 * 10^24) = 2.443 * 10^12; sqrt(7.35 * 10^22) = 2.711 * 10^11.
- Substitute values: x = (3.84 * 10^8 * 2.443 * 10^12) / (2.443 * 10^12 + 2.711 * 10^11) = 3.46 * 10^8 meters.
- Verify: The saddle point lies closer to the Moon, at 3.46 * 10^8 m from Earth (or about 38,000 km from the Moon), which is where their gravitational pulls cancel.
Answer: Distance from Earth = 3.46 * 10^8 m (or 90.1% of the Earth-Moon distance)
Common Misconceptions
- "Gravity is constant inside the Earth": False. Gravity changes significantly inside the Earth. It actually rises slightly in the mantle due to density changes, before dropping to zero at the core.
- "Units of N/kg and m/s² are different things": False. They are completely equivalent. 1 N/kg is equal to 1 m/s². The former emphasizes gravity field strength (force per unit mass), while the latter emphasizes the resulting acceleration.
- "At a Lagrange point, the net gravity is always zero": False. Only at the neutral saddle point (which lies between L1 and the smaller mass) does the net gravitational field equal zero. At L1–L5, the net gravitational field is non-zero but balances the centrifugal force.
Practice Questions
1. How does the gravitational field strength of a planet change if its mass is doubled while keeping its radius constant? What if its radius is doubled instead, keeping mass constant?
Since gravitational field strength is directly proportional to mass (g = G * M / R²), doubling the mass doubles the field strength (2g). However, because it is inversely proportional to the square of the radius, doubling the radius reduces the field strength to 1/4th (g / 4 or 25%) of the original value.
2. If you descend halfway down a deep vertical tunnel into a hypothetical planet of uniform density, how does the gravitational field strength change? Explain using the shell theorem.
Under a uniform density assumption, the shell theorem states that the outer spherical shell of mass above you exerts zero net gravitational force on you. Only the sphere of mass below your current depth (with radius r) pulls you inward. Since volume scales as r³ and distance squared scales as 1/r², the internal field strength scales linearly with distance: g(r) = g_surface * (r / R). Halfway down (r = 0.5R), the field strength is exactly 50% of the surface value.
3. Explain why the actual gravitational field strength of the Earth increases slightly as you go from the surface down through the crust and mantle, reaching a maximum at the core-mantle boundary, rather than decreasing linearly.
The Earth does not have uniform density. The core (composed of dense iron and nickel) is much denser than the rocky crust and mantle. As you descend, you get closer to this extremely dense core, and the dense mass below you pulls harder than the lighter rock layer left behind. According to the Preliminary Reference Earth Model (PREM), this causes the field strength to peak at about 10.7 m/s² at the core-mantle boundary (depth of ~2900 km) before rapidly dropping through the core.
4. What is the physical significance of the L1 Lagrange point in a binary star or planet-satellite system, and how does the net gravitational field behave there?
The L1 point lies on the line connecting the two masses, between them. At this location, the gravitational pulls of the two bodies are in opposite directions, and their combined field exactly balances the centrifugal force required for an object to co-orbit with the secondary body. The net gravitational field strength (excluding rotational forces) is not zero at L1, but the combined gravitational and centrifugal forces are in perfect equilibrium.
FAQ
Frequently Asked Questions
What is gravitational field strength?
Gravitational field strength (represented by g) is the gravitational force exerted per unit mass on a small test mass placed at a point in the field. It measures the intensity of gravity at a given location and is calculated as g = F / m.
What are the SI units of gravitational field strength?
The SI units are Newtons per kilogram (N/kg) or meters per second squared (m/s²). These units are completely equivalent (1 N/kg = 1 m/s²).
How does gravitational field strength differ from gravitational force?
Gravitational force is the actual pull between two masses, measured in Newtons (N). Gravitational field strength is a property of the space surrounding a mass, indicating the acceleration a test mass would experience at that point, measured in N/kg or m/s².
What is the formula to calculate the gravitational field strength of a spherical planet?
The formula is g = G * M / r², where G is the universal gravitational constant, M is the planet's mass, and r is the distance from the planet's center to the point of measurement.
How does gravitational field strength vary with altitude?
According to the inverse square law, gravitational field strength decreases as altitude increases. Outside a planet, g is proportional to 1/r². For example, at twice the Earth's radius from the center, the field strength drops to 25% of its surface value.
What happens to the gravitational field strength at the center of the Earth?
At the exact center of the Earth, the gravitational field strength is zero (g = 0 N/kg). This is because the surrounding mass pulls equally in all directions, causing the gravitational forces to cancel out completely.
How does the field strength change inside a planet of uniform density?
Assuming uniform density, the gravitational field strength inside a planet decreases linearly with depth (g is proportional to r). As you move closer to the center, the mass of the shell above you no longer exerts any net gravitational force on you (Shell Theorem).
What is the PREM model and how does it describe Earth's internal gravity?
The Preliminary Reference Earth Model (PREM) is a realistic density profile of the Earth's layers. Because Earth's core is extremely dense, g actually increases with depth initially, peaking at about 10.7 m/s² at the core-mantle boundary before rapidly dropping to zero at the center.
What is a gravitational saddle point?
In a system with multiple masses (like the Earth and the Moon), a gravitational saddle point is a neutral point where the opposing gravitational forces cancel out, resulting in a net gravitational field strength of zero (g_net = 0).
What are Lagrange points?
Lagrange points are equilibrium positions in space near two co-orbiting bodies where their combined gravitational fields balance the centrifugal force of rotation. There are five such points, labeled L1 to L5.
Is gravitational field strength a scalar or vector quantity?
It is a vector quantity because it has both magnitude and direction. The direction of a gravitational field vector always points toward the center of the mass creating the field (inward/attractive).