13.2 Gravitation Near Earth's Surface

Gravity near Earth's surface is all about the interplay between the gravitational constant G and acceleration due to gravity g. While G is universal, g varies with location and altitude, affecting how objects move and weigh differently across Earth's surface.

Understanding gravity's effects helps us calculate celestial body masses and explains phenomena like free fall and tides. It's crucial for grasping how objects behave on Earth and in space, from weighing less at the equator to achieving escape velocity.

Gravitation Near Earth's Surface

Gravitational constant vs acceleration

• Gravitational constant $G$ measures the strength of gravitational force between two objects (Earth and an object on its surface)
• Acceleration due to gravity $g$ quantifies the acceleration experienced by objects due to Earth's gravitational pull near its surface (9.8 m/s²)
• Newton's Law of Universal Gravitation relates $G$ and $g$: $g = \frac{GM_E}{R_E^2}$
  • $M_E$ represents Earth's mass
  • $R_E$ represents Earth's radius
• $G$ is a universal constant (6.67 × 10⁻¹¹ N·m²/kg²) while $g$ varies with location and altitude
• The gravitational field describes the strength and direction of the gravitational force at any point in space

Mass calculation of celestial bodies

• Calculate the mass of celestial bodies using surface gravitational acceleration $g$ and radius $R$
• Rearrange Newton's Law of Universal Gravitation to solve for mass $M$: $M = \frac{gR^2}{G}$
• Steps to calculate mass:
  1. Measure surface gravitational acceleration $g$ on the celestial body
  2. Determine radius $R$ of the celestial body
  3. Substitute values of $g$, $R$, and $G$ into the equation $M = \frac{gR^2}{G}$
• This method allows for estimating the mass of planets, moons, and other celestial objects (Mars, Jupiter's moon Europa)

Variations in surface gravity

• Value of $g$ varies slightly based on geographical location and Earth's rotation
• Latitude dependence causes $g$ to be slightly greater at poles compared to equator (0.5% difference)
  • Due to Earth's oblate shape being slightly flattened at poles
• Altitude dependence causes $g$ to decrease with increasing altitude above Earth's surface
  • Distance from Earth's center increases, reducing gravitational acceleration
• Earth's rotation creates small centrifugal force opposing gravitational force
  • Effect is greatest at equator (reduces $g$ by 0.3%) and zero at poles
• These variations in $g$ affect the weight of objects at different locations (person weighs slightly less at equator than at poles)

Gravitational effects and motion

• Free fall occurs when an object is subject only to the force of gravity, resulting in an acceleration of $g$
• Escape velocity is the minimum speed an object needs to overcome Earth's gravitational pull and escape its orbit
• Tidal forces are caused by differences in gravitational pull on different parts of an object, leading to deformation and tidal effects on Earth