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Introduction
Hey it's a me again @drifter1!
In this article we will continue with Physics, and more specifically the branch of "Classical Mechanics".
Today's topic is Earth’s Rotation and its Effect on Gravity.
So, without further ado, let's get straight into it!
Gravity on Earth (based on simple assumptions)
Applying Newton's Law of Gravitation (otherwise known as Universal Law of Gravitation) and solving for the gravitational acceleration g, while considering that the Earth is falsly a perfect uniform sphere of mass M = 5.98 × 1024 Kg and typical radius r = 6.38 × 106 m, results in:
which is a great approximation for gravity calculations, if they don't need to be accurate.
If the distance gets doubled, d = 2 ⋅ r, g becomes:
So, can we do better?
The Motion of The Earth
The Earth doesn't move in a straight line, but is moving in a parabolic path around the Sun, which is also moving around the center of our Galaxy. The Earth also rotates around its own axis eastward or counterclockwise, when viewed in respect to the Geographic North Pole. Therefore, its easy to suspect that the actual Gravitational Force must depend on the position of the object on the Earth's surface.
Apparent Weight
The Earth rotates around its axis roughly every 24 hours. From circular motion we know that an object moving at constant speed in a circle must have a centripetal acceleration (and force) directed towards the center of that circle. In the case of the Earth, the role of the centripetal force is taken by the force and acceleration of Gravity.
If the Earth was not rotating, the total net force would be F = mg.
Considering the rotation of the Earth, from Newton's 2nd Law the total net force is:
where ac is the centripetal acceleration, and F is the centripetal force.
Because ac points in the same direction as the weight its negative, and so equal to:
where ω is the angular speed.
Substituing ac in the previous equation we get:
Supposing that the Earth's rotation is exactly 24 hours, the angular speed is equal to:
Correction on the Equator
Plugging the radius of the Earth into that equation, the "correction" is:
which is only 0.34% of g, and so not that big of a difference.
On the poles there is no centripetal acceleration, and so the gravity force is equal to F = mg.
Rotation speed for zero apparent weight
For the apparent weight to be zero, g must be equal to ac:
Which is quite fast!
The period of the Earth for this speed is:
Results away from the equator
When in a lattitude λ, the apparent weight is equal to:
where:
- λ = 90o at the poles
- λ = 0o at the equator
RESOURCES:
References
- https://geo.libretexts.org/Bookshelves/Meteorology/Book%3A_Fundamentals_of_Atmospheric_Science_(Brune)/10%3A_Dynamics_-_Forces/10.05%3A_Effects_of_Earth%E2%80%99s_Rotation-_Apparent_Forces
- https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/13-2-gravitation-near-earths-surface/
- https://www.studyadda.com/notes/jee-main-advanced/physics/gravitation/variation-in-g-due-to-rotation-of-earth/7456
- https://www.toppr.com/ask/question/the-effect-of-rotation-of-the-earth-on-the-value-of-acceleration-due-to-gravity/
- https://physics.stackexchange.com/questions/94463/how-the-apparent-weight-varies-due-to-the-rotational-motion-of-earth
Images
Mathematical equations used in this article, where made using quicklatex.
Previous articles of the series
Rectlinear motion
- Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
- Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
- Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
- Rectlinear motion exercises -> examples and tasks in rectlinear motion
Plane motion
- Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
- Projectile motion as a plane motion -> missile/bullet motion as a plane motion
- Smooth Circular motion -> smooth circular motion theory
- Plane motion exercises -> examples and tasks in plane motions
Newton's laws and Applications
- Force and Newton's first law -> force, 1st law
- Mass and Newton's second law -> mass, 2nd law
- Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
- Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
- Contact forces and friction -> contact force, friction
- Dynamics of Circular motion -> circular motion dynamics, applications
- Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
- Contact force and friction examples -> exercises in force and friction
- Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
- Advanced Newton law examples -> advanced (more difficult) exercises
Work and Energy
- Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
- Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
- Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
- Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
- Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
- Internal Energy and Work -> Internal Energy, Internal Work
Momentum and Impulse
- Conservation of Momentum -> Momentum, Conservation of Momentum
- Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
- Collision Examples -> Various Elastic and Inelastic Collision Examples
- Impulse -> Impulse with Example
- Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
- Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis
Angular Motion
- Angular motion basics -> Angular position, velocity and acceleration
- Rotation with constant angular acceleration -> Constant angular acceleration, Example
- Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
- Parallel Axis Theorem -> Parallel axis theorem with example
- Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
- Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
- Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
- Angular Momentum -> Angular Momentum and its conservation
- Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
- Exercises around Angular motion -> Angular motion examples
Equilibrium and Elasticity
- Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
- Force Couple System -> Force Couple System, Example
- Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
- Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
- Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
- Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
- Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
- Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples
Gravity
- Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
- Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
- Gravitational Fields -> Gravitational Field Mathematics and Visualization
- Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
- Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
- Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
- Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
- Kepler's Laws of Planetary Motion -> Kepler's Story, Elliptical Orbits, Kepler's Laws
- Spherical Mass Distributions -> Spherical Mass Distribution, Gravity Outside and Within a Spherical Shell, Simple Examples
Final words | Next up
And this is actually it for today's post!
Next time we will get into Black Holes..
See ya!
