<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Newtonian_gravity_field_%28physics%29.svg/500px-Newtonian_gravity_field_%28physics%29.svg.png">
<p>[<a href="https://commons.wikimedia.org/wiki/File:Newtonian_gravity_field_(physics).svg">Image1</a>]</p>
<h2>Introduction</h2>
<p>
Hey it's a me again <a href="https://peakd.com/@drifter1">@drifter1</a>!<br><br>
In this article we will continue with <strong>Physics</strong>, and more specifically the branch of "<strong>Classical Mechanics</strong>".
Today's article is part two of <strong>Exercises on Newtonian Gravity</strong>.
Part 1 can be found <a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-exercises-around-newtonian-gravity-part-1">here</a>.
<br><br>
So, without further ado, let's dive straight into it!
</p>
<hr>
<h2>Recap of Useful Formulas</h2>
The following formulas will be very useful for solving Problems around Gravity.
<h3>Universal Law of Gravitation</h3>
<img src="https://quicklatex.com/cache3/56/ql_514b79aa5addcaa63db269a949a0da56_l3.png"><br><br>
where:
<ul>
<li><em>d</em> is the distance of the COMs of the two masses <em>m<sub>1</sub></em> and <em>m<sub>2</sub></em></li>
<li><em>G</em> is the Gravitation Constant which equals <em>6.674 × 10<sup>-11</sup> N</em></li>
</ul>
<h3>Weight or Force of Gravity</h3>
<img src="https://quicklatex.com/cache3/04/ql_6fe3bbaf539e652ee411898ddc3f5804_l3.png">
<h3>Gravitational Acceleration</h3>
<img src="https://quicklatex.com/cache3/ab/ql_34d215fca16c6b01ba8a50062d3c9fab_l3.png"><br><br>
where:
<ul>
<li><em>r</em> can be thought of as the median radius of a planet</li>
<li><em>h</em> as the height of an object in respect to the surface of the planet</li>
</ul>
<h3>Gravitational Potential Energy</h3>
<img src="https://quicklatex.com/cache3/b6/ql_965b00a05fbbe3f743ac632c2e17e2b6_l3.png"><br><br>
or<br><br>
<img src="https://quicklatex.com/cache3/83/ql_c4c341fc2cc9da534a264db6f00c5683_l3.png"><br><br>
<em>ΔU = U<sub>2</sub> - U<sub>1</sub></em> is thus negative when the height decreases (<em>h<sub>1</sub> > h<sub>2</sub></em>).
<h3>Gravitational Potential</h3>
<img src="https://quicklatex.com/cache3/7a/ql_a4d00b5224d4f3fe4307d29000636b7a_l3.png">
<hr>
<h2>Two-Dimensional Problem's Field Acceleration</h2>
<p>
Let's start by calculating the Gravitational Acceleration <em>g</em> of the two-dimensional gravity problem that we solved during Part 1.
</p>
<p>
The total force aplied on mass <em>m<sub>A</sub> = 4 Kg</em> was calculated to be <em>ΣF = 2.059 x 10<sup>-11</sup></em>.
</p>
<p>
Thus, the Gravitational acceleration <em>g</em> at the point where <em>m<sub>A</sub></em> is in the field of the three other masses is:<br><br>
<img src="https://quicklatex.com/cache3/d1/ql_fb58714e3fcf976ede77f05087fdf2d1_l3.png">
</p>
<hr>
<h2>Gravitational Field on the Axis of a Ring (based on Ref1)</h2>
<p>
Let's consider the following gravitational field along the axis of a uniform ring:<br><br>
<img src="https://i.ibb.co/m9npsPf/ring-field.jpg"><br><br>
[Custom Figure using <a href="https://app.diagrams.net/">draw.io</a>]
</p>
<p>
The ring has a mass of <em>M</em> and a radius of <em>a</em>, and the points <em>P</em> are taken at a distance <em>b</em> along the axis of the ring.
At the center of the ring the Gravitational field of course has a strengh of zero.
</p>
<p>
Let's find the maximum strength of the Gravitational Field along the axis of the ring in respect to the distance from the center <em>O</em>.
</p>
<h3>Solution</h3>
<p>
In order to calculate the field of a ring, the ring has to be split into small masses <em>dM</em>, whose fields can then be summed up together.
</p>
<p>
The distance towards each of those small masses is equal to <em>c</em>, which can be easily calculated using Pythagoras's Theorem:<br><br>
<img src="https://quicklatex.com/cache3/d2/ql_228a6ad947bbc6b08d5b672786bc50d2_l3.png">
</p>
<p>
The total gravitational field strengh <em>dg</em> of each of those masses <em>dM</em> is:<br><br>
<img src="https://quicklatex.com/cache3/80/ql_f31ecff12f2a1efdf2bfb9b68fbea580_l3.png">
</p>
<p>
Because each field <em>dg</em> points towards another direction a split of each vector into two components is necessary.<br><br>
Doing that its easy to notice that there is another mass <em>dM'</em> that cancels out one of the components completely, which is something that happens for all masses <em>dM</em>:<br><br>
<img src="https://i.ibb.co/SfxFD86/field-strengths.jpg"><br><br>
[Custom Figure using <a href="https://app.diagrams.net/">draw.io</a>]
</p>
<p>
Thus, only the component parallel to the axis of the ring will contribute towards the field strength.<br><br>
Using the trigonometric function of <em>cosinus</em>, this component is defined as:<br><br>
<img src="https://quicklatex.com/cache3/15/ql_5119de4b409fc91aaf7da9377ec98015_l3.png">
</p>
<p>
Summing up the contributions of all those small masses <em>dM</em>, the total gravitational field <em>g</em> along the axis of the ring is calculated to be:<br><br>
<img src="https://quicklatex.com/cache3/6a/ql_70db17b018d490702343d1a41345af6a_l3.png">
</p>
<p>
The maximum value is reached at the point(s) where the derivative of <em>g</em>, <em>g'</em>, is zero.<br><br>
The derivative is equal to:<br><br>
<img src="https://quicklatex.com/cache3/ab/ql_e0c14d69a88e9b4c216d0e3b03e4d9ab_l3.png">
</p>
<p>
The nominator is zero when:<br><br>
<img src="https://quicklatex.com/cache3/f3/ql_46e7ec687fac1495622340b819552df3_l3.png"><br><br>
Therefore, the maximum value is at a distance of <em>b = a / √2</em>, which gives us a field strength of:<br><br>
<img src="https://quicklatex.com/cache3/46/ql_92be5b89bb811b66faabd206ce232a46_l3.png">
</p>
<p>
The plot of the gravity field strength <em>g</em> in respect to the distance <em>b</em>, supposing that <em>G = M = a = 1</em>, looks as following:<br><br>
<img src="https://i.ibb.co/WxdCGYX/graph.jpg"><br>
[Custom Figure using <a href="https://www.geogebra.org/">GeoGebra</a>]
</p>
<hr>
<h2>Speed of a Rollercoaster (Inversed Example 2 of Reference 2)</h2>
<p>
Consider a rollercoaster starts going down from an unknown height <em>h</em> and reaches a final speed of <em>v = 25 m / s</em>.<br><br>
Find the value of <em>h</em> if friction is negligible and <em>g = 9.8 m/s<sup>2</sup></em>.
</p>
<h3>Solution</h3>
<p>
The initial velocity of the rollercoaster is zero (<em>K<sub>initial</sub> = 0</em>) and thus the rollercoaster initially has only gravitational potential energy.<br><br>
The final potential energy is zero (<em>U<sub>final</sub> = 0</em>), and thus, from Energy conservation, the initial potential energy equals the final kinetic energy, or mathematically:<br><br>
<img src="https://quicklatex.com/cache3/dc/ql_8f6914ebb5c03cc985d02800ebbfaedc_l3.png"><br><br>
The mass is cancelled out and the rest is known and thus the initial height <em>h</em> is:<br><br>
<img src="https://quicklatex.com/cache3/a6/ql_d63cb1ee9355c28f8c98a990d41c3aa6_l3.png">
</p>
<hr>
<h2>RESOURCES:</h2>
<h3>References</h3>
<ol>
<li><a href="https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Book%3A_Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.04%3A_The_Gravitational_Fields_of_Various_Bodies/5.4.02%3A_Field_on_the_Axis_of_a_Ring">https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Book%3A_Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.04%3A_The_Gravitational_Fields_of_Various_Bodies/5.4.02%3A_Field_on_the_Axis_of_a_Ring</a></li>
<li><a href="https://courses.lumenlearning.com/physics/chapter/7-3-gravitational-potential-energy/">https://courses.lumenlearning.com/physics/chapter/7-3-gravitational-potential-energy/</a></li>
</ol>
<h3>Images</h3>
<ol>
<li><a href="https://commons.wikimedia.org/wiki/File:Newtonian_gravity_field_(physics).svg">https://commons.wikimedia.org/wiki/File:Newtonian_gravity_field_(physics).svg</a></li>
</ol>
<p>Mathematical equations used in this article, where made using <a href="http://quicklatex.com/">quicklatex</a>.</p>
<hr>
<h2>Previous articles of the series</h2>
<h3>Rectlinear motion</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-velocity-and-acceleration-in-a-rectlinear-motion">Velocity and acceleration in a rectlinear motion</a> -> velocity, acceleration and averages of those</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rectlinear-motion-with-constant-accelaration-and-free-falling">Rectlinear motion with constant acceleration and free falling</a> -> const acceleration motion and free fall</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rectlinear-motion-with-variable-acceleration-and-velocity-relativity">Rectlinear motion with variable acceleration and velocity relativity</a> -> integrations to calculate pos and velocity, relative velocity</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rectlinear-motion-exercises">Rectlinear motion exercises</a> -> examples and tasks in rectlinear motion</li>
</ul>
<h3>Plane motion</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-position-velocity-and-acceleration-vectors-in-a-plane-motion">Position, velocity and acceleration vectors in a plane motion</a> -> position, velocity and acceleration in plane motion</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-projectile-motion-as-a-plane-motion">Projectile motion as a plane motion</a> -> missile/bullet motion as a plane motion</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-smooth-circular-motion">Smooth Circular motion</a> -> smooth circular motion theory</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-plane-motion-exercises">Plane motion exercises</a> -> examples and tasks in plane motions</li>
</ul>
<h3>Newton's laws and Applications</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-force-and-newton-s-first-law">Force and Newton's first law </a>-> force, 1st law</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-mass-and-newton-s-second-law">Mass and Newton's second law</a> -> mass, 2nd law</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-newton-s-3rd-law-and-mass-vs-weight">Newton's 3rd law and mass vs weight</a> -> mass vs weight, 3rd law, friction</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-applying-newton-s-laws">Applying Newton's Laws</a> -> free-body diagram, point equilibrium and 2nd law applications</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-contact-forces-and-friction">Contact forces and friction</a> -> contact force, friction</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-dynamics-of-circular-motion">Dynamics of Circular motion</a> -> circular motion dynamics, applications</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-object-equilibrium-and-2nd-law-application-examples">Object equilibrium and 2nd law application examples</a> -> examples of object equilibrium and 2nd law applications</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-contact-force-and-friction-examples">Contact force and friction examples</a> -> exercises in force and friction</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-circular-dynamic-and-vertical-circle-motion-examples">Circular dynamic and vertical circle motion examples </a>-> exercises in circular dynamics</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-advanced-newton-law-examples">Advanced Newton law examples</a> -> advanced (more difficult) exercises</li>
</ul>
<h3>Work and Energy</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-work-and-kinetic-energy">Work and Kinetic Energy</a> -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises</li>
<li><a href="https://steemit.com/busy/@drifter1/physics-classical-mechanics-conservative-and-non-conservative-forces">Conservative and Non-Conservative Forces</a> -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises</li>
<li><a href="https://steemit.com/busy/@drifter1/physics-classical-mechanics-potential-and-mechanical-energy">Potential and Mechanical Energy</a> -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips</li>
<li><a href="https://steemit.com/busy/@drifter1/physics-classical-mechanics-force-and-potential-energy">Force and Potential Energy</a> -> Force as Energy Derivative (1-dim) and Gradient (3-dim)</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-potential-energy-diagrams">Potential Energy Diagrams</a> -> Energy Diagram Interpretation, Steps and Example </li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-internal-energy-and-work">Internal Energy and Work</a> -> Internal Energy, Internal Work</li>
</ul>
<h3>Momentum and Impulse</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-conservation-of-momentum">Conservation of Momentum</a> -> Momentum, Conservation of Momentum</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-elastic-and-inelastic-collisions">Elastic and Inelastic Collisions</a> -> Collision, Elastic Collision, Inelastic Collision</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-collision-examples">Collision Examples</a> -> Various Elastic and Inelastic Collision Examples</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-impulse">Impulse</a> -> Impulse with Example</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-motion-of-the-center-of-mass">Motion of the Center of Mass</a> -> Center of Mass, Motion analysis with examples</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-explaining-the-physics-behind-rocket-propulsion">Explaining the Physics behind Rocket Propulsion</a> -> Required Background, Rocket Propulsion Analysis </li>
</ul>
<h3>Angular Motion</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-angular-motion-basics">Angular motion basics</a> -> Angular position, velocity and acceleration </li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rotation-with-constant-angular-acceleration">Rotation with constant angular acceleration</a> -> Constant angular acceleration, Example</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rotational-kinetic-energy-and-moment-of-inertia">Rotational Kinetic Energy & Moment of Inertia </a> -> Rotational kinetic energy, Moment of Inertia</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-parallel-axis-theorem">Parallel Axis Theorem</a> -> Parallel axis theorem with example</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-torque-and-angular-acceleration">Torque and Angular Acceleration</a> -> Torque, Relation to Angular Acceleration, Example</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rotation-about-a-moving-axis-rolling-motion">Rotation about a moving axis (Rolling motion)</a> -> Fixed and moving axis rotation</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-work-and-power-in-angular-motion">Work and Power in Angular Motion</a> -> Work, Work-Energy Theorem, Power</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-angular-momentum">Angular Momentum</a> -> Angular Momentum and its conservation</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-explaining-the-physics-behind-mechanical-gyroscopes">Explaining the Physics behind Mechanical Gyroscopes</a> -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-exercises-around-angular-motion">Exercises around Angular motion</a> -> Angular motion examples</li>
</ul>
<h3>Equilibrium and Elasticity</h3>
<ul>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-rigid-body-equilibrium">Rigid Body Equilibrium</a> -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-force-couple-system">Force Couple System</a> -> Force Couple System, Example</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-tensile-stress-and-strain">Tensile Stress and Strain</a> -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-volumetric-stress-and-strain">Volumetric Stress and Strain</a> -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility</li>
<li><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-cross-sectional-stress-and-strain">Cross-Sectional Stress and Strain</a> -> Shear Stress, Shear Strain, Shear Modulus</li>
<li><a href="https://peakd.com/steemstem/@drifter1/physics-classical-mechanics-elasticity-and-plasticity-of-common-materials">Elasticity and Plasticity of Common Materials</a> -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials</li>
<li><a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-rigid-body-equilibrium-exercises">Rigid Body Equilibrium Exercises</a> -> Center of Gravity Calculation, Equilibrium Problems</li>
<li><a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-exercises-on-elasticity-and-plasticity">Exercises on Elasticity and Plasticity</a> -> Young Modulus, Bulk Modulus and Shear Modulus Examples </li>
</ul>
<h3>Gravity</h3>
<ul>
<li><a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-newton-s-law-of-gravitation">Newton's Law of Gravitation</a> -> Newton's Law of Gravity, Gravitational Constant G</li>
<li><a href="https://peakd.com/hive-163521/@drifter1/physics-classical-mechanics-weight-the-force-of-gravity">Weight: The Force of Gravity</a> -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System</li>
<li><a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-gravitational-fields">Gravitational Fields</a> -> Gravitational Field Mathematics and Visualization</li>
<li><a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-gravitational-potential-energy">Gravitational Potential Energy</a> -> Gravitational Potential Energy, Potential and Escape Velocity</li>
<li><a href="https://peakd.com/hive-196387/@drifter1/physics-classical-mechanics-exercises-around-newtonian-gravity-part-1">Exercises around Newtonian Gravity (part 1)</a> -> Examples on the Universal Law of Gravitation</li>
</ul>
<hr>
<h2>Final words | Next up</h2>
<p>
And this is actually it for today's post!<br><br>
Next time we will get into a Physics explanation of the circular motion of Satellites...<br><br>
See ya!
</p>
<p><img src="https://media.giphy.com/media/ybITzMzIyabIs/giphy.gif" width="500" height="333"/></p>
Keep on drifting!
hiveblocks