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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 article will be about Exercises on Newtonian Gravity, and splitted into two parts.
So, without further ado, let's get straight into it!
Recap of Useful Formulas
The following formulas will be very useful for solving Problems around Gravity (in part 1).Universal Law of Gravitation
where:
- d is the distance of the COMs of the two masses m1 and m2
- G is the Gravitation Constant which equals 6.674 × 10-11 N
Weight or Force of Gravity
Gravitational Acceleration
where:
- r can be thought of as the median radius of a planet
- h as the height of an object in respect to the surface of the planet
Gravitational Attraction of the Earth in Various Heights (based on Ref1 Sample Problems)
Let's consider a person of mass m = 60 Kg is inside the Gravitational Field of the Earth (which has a mass of M = 5.98 x 1024 Kg and radius of r = 6.38 x 106 m).
Calculate the Force of Gravity that the Earth exerts on the person when:
- The person is standing at sea-level (distance to the center of the earth equal to the radius r)
- The person is standing at the peak of Mount Everest (highest altitude above mean sea level is h = 8,848 m
- The person is on an airplane that flies at about h = 12 km above sea-level
1. Sea-Level
2. Mount Everest
3. Airplane
The difference is minimal on so even using the value g = 9.8 m/s2 for Earth would be sufficient enough as it gives:
Calculating the Mass of the Earth
Knowing that the equatorial radius of the Earth is RE = 6.380 Km and the Gravitational acceleration is g = 9.8 m/s2, can we somehow calculate the mass of the Earth?
Solution
Let's consider a test-mass of mass m = 1 Kg.
Solving the equation of gravitational acceleration g for the mass of the Earth ME we have:
Multiple Attractors and Force of Gravity
One-Dimensional Problem
Let's consider the following problem:
[Custom Figure using draw.io]
Calculate the total force of gravity excelled on sphere m by the two spheres with equal mass M if d1 is greater than d2.
The result will be an expression of m, M, d1 and d2.
Solution
Because d1 > d2 we already know that F1 < F2.
Thus, the total force will be ΣF = F2 - F1.
Using the Universal Law of Gravitation we have:
The difference of squares a2 - b2 = (a + b)(a - b) is useful when more than 2 masses attract the object, as it simplifies the result by a lot.
Two-Dimensional Problem
Let's consider the following problem:
[Custom Figure using GeoGebra]
In this figure, A (0, 0), B (-10, -10), C (5, 8.66) and D (8, -4.618) represent the centers of mass of the point masses: mA = 4 Kg, mB = 11 Kg, mC = 9 Kg and mD = 7 Kg.
The distance of mass A to each of the other masses is dAB = 14.14 m, dAC = 10 m and dAD = 9.24 m.
The Force of Gravitational Attraction from the masses B, C and D towards A, has an angle of β = 45°, γ = 60° and δ = 30° correspondingly, always in respect to the horizontal or x-axis.
Solution
Using the Universal Law of Gravitation we can calculate the force of attraction of each mass B, C and D towards A:
In order to calculate the sum of those forces we have to split the forces into two components: one for the horizontal axis and one for the vertical axis.
The components are defined easily using Trigonometric Functions:
In the case of the force FB which attracts mass A towards mass B we have the following components:
(The results are negative because sin and cos have negative values in the 3rd Quadrant)
Similarly, for force FC from mass C we have:
Lastly, for force FD from mass D we have:
(The trigonometric function sin is negative in the 4th Quadrant, whilst cos is positive)
The signs shows us where the vectors are pointing (left-right and up-down).
Now that we have the components we can easily calculate the components of ΣF: ΣFx and ΣFy as:
Thus, the total force is:
and with an angle of:
RESOURCES:
References
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
- https://www.grc.nasa.gov/www/k-12/airplane/wteq.html
- 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
Final words | Next up
And this is actually it for today's post!
In part 2 we will get into exercises around Gravitational Fields and Gravitational Potential Energy...
See ya!
