Math Easy Solutions

Math Easy Solutions

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In this video I go over a very surprising connection between the centroid of a region and the resulting volume of revolution obtained from it. This connection states that the volume of a shape formed by rotating a region about the line is equal to the area of the region multiplied by distance the centroid of the region travels around the line of revolution. This connection is called the Theorem of Pappus, named after the founder Pappus of Alexandria in the 4th century A.D. This is a remarkable yet surprising connection between volumes and centers of mass because these two topics seem on the surface very different from each other.

As well as the proof of the Pappus Theorem, I also go over a brief math (mainstream) history lesson on Pappus, who was a great Greek mathematician far ahead of his time, so make sure to watch this video!

Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104685&authkey=!AByGvDBNcGgx40k&ithint=file%2cpdf

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-moments-and-centers-of-mass-theorem-of-pappus

Related Videos:

Moments and Centers of Mass: Example 4: x and x^2: https://youtu.be/NSE92mZgz7A
Moments and Centers of Mass: Region Bounded by 2 Curves: https://youtu.be/D-9tS_3XvLM
Moments and Centers of Mass: Example 3: cos(x): https://youtu.be/fvvVqhzmT4s
Moments and Centers of Mass: Example 2: Semi-Circle: https://youtu.be/vfCMuvqOCFM
Moments and Centers of Mass: Example 1: https://youtu.be/55jrfODcbWY
Moments and Centers of Mass: Introduction: https://youtu.be/lLSo5Hck6FM
Hydrostatic Pressure and Force: Example 2: Force on a Drum: https://youtu.be/8tZ86Iw68l8
Hydrostatic Pressure and Force: Example 1: Force on a Dam: https://youtu.be/Gr5H4icS4CQ
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Simplified Filter Criteria: A Dam Filter Example: http://youtu.be/yGEeYAb4Olw
Soil Mechanics 101 - Phase Relations: http://youtu.be/DtKheQcL2BU
Types of Tailings Embankments: Upstream, Downstream and Centerline Construction Methods: http://youtu.be/1wm1XR6z-QE
Buoyancy - What is Archimedes' Principle and it's Proof: http://youtu.be/mXzccaH2KN
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E .

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In this video I go over an example on applying the Theorem of Pappus, which I covered in my last video, in determining the volume of a torus which is formed by rotating a circle about a line. This shape is simply a round ring or a donut. Applying Pappus's Theorem allows us to easily solve for the volume of the torus, which is simply the area of the circle multiplied by the distance the centroid of the circle travels around the line, which is the same as the circumference of the circle with radius being equal to the distance from the line to the centroid of the circle. This is a remarkably simple way of determining the volume and as I will cover in my next video, it is far easier than having to solve for the volume using basic integration techniques. So stay tuned for that!

Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104709&authkey=!AI8NZsC6OOdsJD4&ithint=file%2cpdf

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-moments-and-centers-of-mass-example-5-theorem-of-pappus

Related Videos:

Moments and Centers of Mass: Theorem of Pappus: https://youtu.be/TLxmMLFC8tc
Moments and Centers of Mass: Example 4: x and x^2: https://youtu.be/NSE92mZgz7A
Moments and Centers of Mass: Region Bounded by 2 Curves: https://youtu.be/D-9tS_3XvLM
Moments and Centers of Mass: Example 3: cos(x): https://youtu.be/fvvVqhzmT4s
Moments and Centers of Mass: Example 2: Semi-Circle: https://youtu.be/vfCMuvqOCFM
Moments and Centers of Mass: Example 1: https://youtu.be/55jrfODcbWY
Moments and Centers of Mass: Introduction: https://youtu.be/lLSo5Hck6FM
Hydrostatic Pressure and Force: Example 2: Force on a Drum: https://youtu.be/8tZ86Iw68l8
Hydrostatic Pressure and Force: Example 1: Force on a Dam: https://youtu.be/Gr5H4icS4CQ
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Simplified Filter Criteria: A Dam Filter Example: http://youtu.be/yGEeYAb4Olw
Soil Mechanics 101 - Phase Relations: http://youtu.be/DtKheQcL2BU
Types of Tailings Embankments: Upstream, Downstream and Centerline Construction Methods: http://youtu.be/1wm1XR6z-QE
Buoyancy - What is Archimedes' Principle and it's Proof: http://youtu.be/mXzccaH2KN .

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In this video I go over the same example as in example 5 but this time determine the volume of a torus using basic integration techniques as opposed to simply using the Theorem of Pappus. This is to show how applying theorems such as the Pappus's Theorem can save you a lot of time and effort in calculations which is very useful when developing computer algorithms that run millions of computations. In this example I show that using the Theorem of Pappus to calculate the volume of a torus, otherwise known as a donut shape, takes just several minutes as opposed to almost half an hour when actually integrating to calculate the volume.

Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104725&authkey=!ADb8BTmRaSNQIzA&ithint=file%2cpdf

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-moments-and-centers-of-mass-example-6-theorem-of-pappus

Related Videos:

Moments and Centers of Mass: Example 5: Theorem of Pappus: https://youtu.be/ClsK9TODEGk
Moments and Centers of Mass: Theorem of Pappus: https://youtu.be/TLxmMLFC8tc
Moments and Centers of Mass: Example 4: x and x^2: https://youtu.be/NSE92mZgz7A
Moments and Centers of Mass: Region Bounded by 2 Curves: https://youtu.be/D-9tS_3XvLM
Moments and Centers of Mass: Example 3: cos(x): https://youtu.be/fvvVqhzmT4s
Moments and Centers of Mass: Example 2: Semi-Circle: https://youtu.be/vfCMuvqOCFM
Moments and Centers of Mass: Example 1: https://youtu.be/55jrfODcbWY
Moments and Centers of Mass: Introduction: https://youtu.be/lLSo5Hck6FM
Hydrostatic Pressure and Force: Example 2: Force on a Drum: https://youtu.be/8tZ86Iw68l8
Hydrostatic Pressure and Force: Example 1: Force on a Dam: https://youtu.be/Gr5H4icS4CQ
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Simplified Filter Criteria: A Dam Filter Example: http://youtu.be/yGEeYAb4Olw
Soil Mechanics 101 - Phase Relations: http://youtu.be/DtKheQcL2BU
Types of Tailings Embankments: Upstream, Downstream and Centerline Construction Methods: http://youtu.be/1wm1XR6z-QE
Buoyancy - What is Archimedes' Principle and it's Proof: http://youtu.be/mXzccaH2KN
Equation of a Circle and it's proof: http://youtu.be/xMXYJ9UeF4I .

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In this video I show how we can apply the concept of integrals and integrations to economics by going over the concept of Consumer Surplus. In economics the consumer surplus is the amount of money that consumers would save in purchasing a product or commodity that is currently priced below what each person individually values the product. For example if you value a TV at $1000 but the price is $600 then you would have a savings of $400. Summing these savings up for all the different type of consumers that value the TV higher than it is worth gives the total Consumer Surplus. Visually the consumer surplus is represented by the area under the demand curve graph and above the current selling price of the product. Note that the demand curve is the price per unit vs. number of units expected to sell.

Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104746&authkey=!ANkQYGynpe5iCtI&ithint=file%2cpdf
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-economics-consumer-surplus

Related Videos:

Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
https://youtu.be/TLxmMLFC8tc
https://youtu.be/55jrfODcbWY
Moments and Centers of Mass: Introduction: https://youtu.be/lLSo5Hck6FM
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg .

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In this video I go over a simple example on illustrating how to determine the consumer surplus when we are given the demand curve function and the amount of products we are looking to sell. The demand curve in this example is given by the function p(x) = 1200 - 0.2x - 0.0001x^2 and the number of sales we are assuming is 500. Once again, the consumer surplus is the amount the consumers overall save when the price is below their perceived value or price willing to pay for the product.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhrI-2D4QjTMRpQ_SVQ
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-economics-consumer-surplus-example-1

Related Videos:

Economics: Consumer Surplus: https://youtu.be/rcJCm1wi4lo
Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
https://youtu.be/TLxmMLFC8tc
https://youtu.be/55jrfODcbWY
Moments and Centers of Mass: Introduction: https://youtu.be/lLSo5Hck6FM
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg .

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In this video I go over deriving an integral formulation of the Cardiac Output, which is the amount of blood that the heart pumps into the aorta. The aorta is the main artery that sends out oxygenated blood through the rest of the arteries and to all parts of the human body.

The method that is usually used to measure the cardiac output is known as the Dye Dilution Method and involves injected an known amount of dye into the right atrium of the heart and then measure the concentration of dye through time in the aorta. Oxygen-depleted blood from the body travel through the veins into the right atrium then get pumped to the lungs through the pulmonary arteries, where they get oxygenated and then get send through the pulmonary veins into the left atrium which pumps the blood into the aorta, thus completing the cycle. The concentration of the dye is measured with a probe in the aorta at specific time intervals until there is no more dye left. From this, we can formulate the amount of blood pumped by the heart per unit time by summing up the concentration measurements per time and thus obtaining an integral formula.

This is a very interesting video on applying integral calculus principles to the cardiovascular system so make sure to watch this video because you may learn a few things about the human body and heart!

Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!106285&authkey=!ANBvZD5_mei3jJQ&ithint=file%2cpdf
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-cardiac-output-rate-of-blood-pumped-by-the-heart

Related Videos:

Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Derivatives Application: Blood Flow: http://youtu.be/nTFJ57uDwtw
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg .

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In this video I go over an introduction into probability and how we can model probability through looking at the proportion of specific ranges of any continuous random variable over many experimental trial tests. From the experimental data, I show how we can create a model of the data, called the probability density function. This function is unique in that the area under the curve represents the probability that the random variable lies in that interval. Since we are dealing with areas under a curve, this naturally ties well into integrals. This is a very interesting and in-depth video that goes over the very core and basis of probability so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhr5nCXj3h1h7Dt3L3w
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-probability-introduction

Related Videos:

Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Approximate Integration: Simpson's Rule: Proof: https://youtu.be/aDvSpOHQoLU .

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In this video I go over an example on determining whether a function is a Probability Density Function (PDF) as well as determine the probability that the random variable lies within a given range. As explained in my last video, the probability density function is modeled from real world experimental data so to ensure that the function given is actually a PDF, we need to ensure it meets the following two conditions:

1) The function is always positive.
2) The area under the function is equal to 1.

This is a useful video for understanding PDFs and proving if a given function is one. I also go over some very interesting algebra techniques in solving the example without needing to use a calculator, so make sure to watch this video!

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/6jzzjp-conics-in-polar-coordinates-unified-theorem-parabola-proof
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhr5womcvwVHbh8-CAA

Related Videos:

Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Approximate Integration: Simpson's Rule: Proof: https://youtu.be/aDvSpOHQoLU .

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In this video I go over another example on probability and this time determine the exact form of a probability density function that is modeled by a decreasing exponential function. Such density functions model phenomena such as call waiting times, or equipment failure times, very well. In this example, I look at the case of call waiting when calling a company, and getting forwarded to their customer service representative. Such a case would realistically have a higher probability of lower wait times and very low probability that you have to wait very long. This type of probability density function is in contrast to the usual bell curve.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhr8KdhcvxQbiT2B-wg

Related Videos:

Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Power Functions Part 2: What is x^0 and 0^0???: http://youtu.be/VzFUDiLzRiE
Limits at Infinity: Horizontal Asymptotes: http://youtu.be/6pdgb09wRvI .

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In this video I go over deriving the average value when given the probability density function of a random continuous variable. In this case, I use the random variable being the call waiting time whenever you goal a company, i.e. calling their customer support phone-line. Using the long-run proportion as the basis of the probability we can thus approximate the total wait time by looking at the area under the graph multiplied by the number of experimental data we have, in this case the number of callers. The average wait time would thus just be the total time waited by all callers divided by the number of callers. This result, which is written as an integral, is also the x-coordinate of the centroid, or center of mass, of the region underneath the probability density function.

The average value, is also called the mean value and is traditionally written by the greek letter mu, μ.

This is a very in-depth video on determining the average value of any given probability density function so make sure to watch it!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhr8xfltWIER_ncymZA

Related Videos:

Probability: Example 2: Exponentially Decreasing Probability Density Function: https://youtu.be/QeU_9NDCXoA
Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Moments and Centers of Mass: Constant Density: https://youtu.be/3bglr1sRWUc .

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In this video I go over another example on probability and this time look at determining the average or mean value of the exponential decreasing probability density function, which was determined in my earlier "Probability Example 2" video. This example involves using integration by parts as well as using L'Hospital's Rule to determine an indeterminate limit when solving for the average value. This is a very interesting video on probability and integration in general so make sure to watch it!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhr9JQcs4TGUEDtGZfw

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/probability-average-value-example-1

Related Videos:

Probability: Average Value (or Mean Value): https://youtu.be/gF-zRmdTUWw
Probability: Example 2: Exponentially Decreasing Probability Density Function: https://youtu.be/QeU_9NDCXoA
Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Moments and Centers of Mass: Constant Density: https://youtu.be/3bglr1sRWUc
L'Hospital's Rule and Indeterminate Forms - Intro: http://youtu.be/x-djc6wrY5w
Integration by Parts: Proof: http://youtu.be/TZhEOct5u_M .

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In this video I go over another example on probability and average values and this time look at the probability of waiting longer than the average time we expect to wait when calling a company representative. What's interesting is that the average wait time does not necessarily mean that 50% of people wait at least this long. This is rather the median wait time and I will look at this further in my next video so stay tuned for that. The reason why the average wait time differs from the median wait time is because people with very long wait times skew the average value to be higher than when we base it strictly on finding the median value. This is a good example illustrating this concept so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhr9tBUO-GsP8-NHkSQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/probability-average-value-example-2

Related Videos:

Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value (or Mean Value): https://youtu.be/gF-zRmdTUWw
Probability: Example 2: Exponentially Decreasing Probability Density Function: https://youtu.be/QeU_9NDCXoA
Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Moments and Centers of Mass: Constant Density: https://youtu.be/3bglr1sRWUc
Limits at Infinity: Horizontal Asymptotes: http://youtu.be/6pdgb09wRvI .

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In #MESExperiments Part 8, I have uploaded the late Francis J. McCabe’s amazing Gyro Wheel Torque “Over-Unity” demonstration, which he had up on his YouTube Channel in June 27, 2012. Francis showed that a large 50 pound, 32 inch diameter gyro, precessing from about 25 inches from the pivot point, can still be “torqued-up” by a very small and visibly weak 6 volt battery with a “Torque Rating” of 2.5 inch-pounds at 12 volts.

Crunching in the numbers, we get roughly:

- Output Torque = 50 pounds * 25 inches = 50 * 25 = 1,250 inch-pounds
- Input Torque = 2.5 inch-pounds * (6 volts input) / (12 volts rating) = 1.25 inch-pounds.
- Output Torque / Input Torque = 1250/1.25 = 1000 TIMES OVER-UNITY!

Recall that the mainstream definition of “Torque” is essentially force multiplied by the distance perpendicular from the fulcrum or point of rotation. For a fixed horizontal wheel, the torque would be the “Moment of Inertia” multiplied by the Acceleration. But while precessing, this torque is also equal to the weight multiplied by the distance to the pivot. More on this *mainstream* torque and precession can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/rotv2.html

Essentially what this means is that although a very weak motor won’t be able to spin a fixed horizontal heavy wheel, but if spun initially through the vertical near zero torque position and then allowed to precess, the motor magically is able to increase spin speed and maintain precession!

This is literally as if the very magical act of precession is causing the gyroscope to be “lifted” in a horizontal spiral plane. Later Francis turns off the power and the gyro-wheel begins precessing in a downwards spiral.

This is an absolutely mind-boggling demonstration by Francis J. McCabe, and just adds to amazing properties of gyroscopes shown in my earlier experiments.

To see the original video on Francis’ YouTube Channel, you can watch here:

https://youtu.be/QmqIg00xSFo

To learn more about Francis J. McCabe, such as his previous work for both NASA and Boeing as a Rocket Scientist before realizing that Gyro Science is where the real science is at, then make sure to read up his life on the following links:
- https://en.wikipedia.org/wiki/Francis_J._McCabe
- https://www.chestnuthilllocal.com/2013/01/08/mt-airy-inventor-finds-new-uses-for-the-gyroscope/
- https://www.chestnuthilllocal.com/2014/03/26/francis-j-mccabe-inventor-business-owner/
- http://www.franmccabe.com/
- https://www.youtube.com/user/Mccabefj/videos

Of note is that Francis viewed the Earth and in fact everything as a gyroscope; which I tend to agree with him on this! And it is no surprise that in his lecture, he mentioned he worked with THE Eric Laithwaite:

https://youtu.be/TVoXdgkSOfY?t=54

The world is definitely a small place, and nothing smaller than the number of people awakened to the magical and game-changing secrets (in plain sight) of gyroscopes….

Stay Tuned for #MESExperiments Number 9!

Related Videos:

#MESExperiments Video Series: https://mes.fm/experiments-playlist
#AntiGravity Video Series: https://mes.fm/antigravity-playlist
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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In this video I go over an example on determining the median of a probability density function, which in this example is from my earlier video on Probability: Average Value: Example 2. The median is shown to be 3.5 minutes which is less than the average or mean value of 5 minutes solved previously. This is a reasonable result, which as explained in my last video on Median, the large values can skew the average more so than the median.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsAhO0k15l5pvAWuNA

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/probability-median-example-1

Related Videos:

Probability: Median: https://youtu.be/C8ZU8BjHIqQ
Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value (or Mean Value): https://youtu.be/gF-zRmdTUWw
Probability: Example 2: Exponentially Decreasing Probability Density Function: https://youtu.be/QeU_9NDCXoA
Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Moments and Centers of Mass: Constant Density: https://youtu.be/3bglr1sRWUc
Limits at Infinity: Horizontal Asymptotes: http://youtu.be/6pdgb09wRvI .

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In this video, I go over an extremely extensive tutorial on Sequences as part of my new video series on Sequences and Series. Sequences are ordered lists of numbers where each term is categorized by its value and the integer corresponding to its place or order in the list. Series are summations of sequences; which I have shown in my earlier videos in finding areas and hence integrals of functions.

The topics I cover in this video include: Sequences and Notation, Definition and Precise Definition, Infinite Limits, Monotonic and Bounded Sequences, as well as the famous Fibonacci Sequence and its relationship with the also very famous Golden Ratio. And at the end of this video I go over a brief mainstream history overview of Leonardo Fibonacci. The Fibonacci sequence is the sequence in which the initial 2 terms are 1 and 1 (or 0 and 1 in modern computing) and then the terms that follow are just the summation of the previous two terms. The limit of the ratio of consecutive Fibonacci terms approach the Golden Ratio (φ) (which is the ratio of any two numbers such that it is the same as the ratio of their summation to the larger number). Fibonacci and Golden numbers/sequences/ratios/spirals/angles/etc. appear very often in all aspect of life from art, nature, design, finance, and even advertising! #AmazingStuff

Here is a list of the topics covered and their start times in the video:

1. @ 55:54 - Introduction to Sequences and Series
2. @ 9:25 – Sequences and Notation
3. @ 11:06 - Examples 1 to 3
3. @ 27:30 – Limit of a Sequence
4. @ 33:59 – Definition 1: Limit of a Sequence
5. @ 37:38 – Definition 2: Precise Definition of a Sequence
6. @ 47:33 – Theorem 1
7. @ 52:26 – Definition 3: Infinite Limits of Sequences at Infinity
8. @ 58:17 – Limit Laws for Sequences
9. @ 1:01:39 – Squeeze Theorem for Sequences
10. @ 1:08:10 – Theorem 2
11. @ 1:08:51 - Examples 4 to 7
12. @ 1:18:51 – Theorem 3
13. @ 1:19:48 – Examples 8 to 9
14. @ 1:26:46 – Creating Graphs of Sequences
15. @ 1:29:33 – Example 10
16. @ 1:41:46 – Definition 4: Monotonic Sequences
17. @ 1:43:42 – Examples 11 and 12
18. @ 1:54:08 – Definition 5: Bounded Sequences
19. @ 1:59:31 – Completeness Axiom
20. @ 2:03:43 - Monotonic Sequence Theorem
21. @ 2:09:27 – Example 13: Recursion and Mathematical Induction
22. @ 2:22:27 – Exercise 1: Convergence
23. @ 2:33:20 - Exercise 2: Proof of Theorem 2
24. @ 2:39:41 - Exercise 3: Proof of Theorem 3
25. @ 2:48:23 - Exercise 4: Fibonacci Sequence
26. @ 3:04:17 – Mainstream History of Fibonacci
27. @ 3:12:04 – Overview of Fibonacci Numbers
28. @ 3:17:33 – Overview of the Golden Ratio
29. @ 3:21:53 – Relationship Between Fibonacci Sequence and Golden Ratio
30. @ 3:27:06 – Golden Angle/Spiral and Approximations with Fibonacci Spiral
31. @ 3:31:30 – Fibonacci Numbers and Golden Ratio in Nature, Art, Design, & MORE! #Amazing

Stay tuned for my next super long math video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh68vMw9e1A75d63O3w

Related Videos:

Limits: https://www.youtube.com/playlist?list=PLFE39AF0D7E0651AC
Areas and Summation: https://www.youtube.com/playlist?list=PLai3U8-WIK0E6biFIz0glhAnUwE1qaviI
Parametric Equations and Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0H0AMIZV8HU3LVxSSf_Y9pi .

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In this video I go over a brief introduction on the most commonly used probability density function, the normal distribution. This function is represented by a bell-shaped curve, often referred to as the Bell Curve, and models many natural phenomena well, such as test scores, rainfall, heights, weights, etc. Although I go over the basics of the normal distribution function and its properties in terms of the standard deviation, I leave the actual derivation of the formula to later videos, as it is more advanced. So stay tuned for those videos!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsBSj6EePdgyR-ZAAg

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/probability-normal-distribution

Related Videos:

Probability: Median: Example 1: https://youtu.be/nDAfJfCa4Fc
Probability: Median: https://youtu.be/C8ZU8BjHIqQ
Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value (or Mean Value): https://youtu.be/gF-zRmdTUWw
Probability: Example 2: Exponentially Decreasing Probability Density Function: https://youtu.be/QeU_9NDCXoA
Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Moments and Centers of Mass: Constant Density: https://youtu.be/3bglr1sRWUc .

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In this video I go over an example on normal distributions and this time look at the very well known application that follows a normal distribution probability density function, the Intelligence Quotient (IQ) scores. IQ tests have been widely used for about a century now and they, like many phenomena, are normally distributed. In the example I show that about 2/3rd of people (or 68%) have an IQ between 85 and 115 while less than 0.4% have an IQ higher than 140.

I also go over a brief history of the IQ tests as well its uses and drawbacks. IQ tests focus mainly on cognitive abilities such as memory, attention, speed, and pattern recognition. This is a very useful video if you are interested in learning more about the popular IQ testing concept so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsBg8srx4qaRfYXtnQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/probability-normal-distribution-example-1-iq-scores

Related Videos:

Probability: Normal Distribution: https://youtu.be/b17gJ4F_v54
Probability: Median: Example 1: https://youtu.be/nDAfJfCa4Fc
Probability: Median: https://youtu.be/C8ZU8BjHIqQ
Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value: Example 2: https://youtu.be/1q_yYIoQ7no
Probability: Average Value (or Mean Value): https://youtu.be/gF-zRmdTUWw
Probability: Example 2: Exponentially Decreasing Probability Density Function: https://youtu.be/QeU_9NDCXoA
Probability: Example 1: https://youtu.be/SDJPja8GJ1Q
Probability: Introduction: https://youtu.be/H_sfVMH0VpQ
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Three Prisoners Problem: http://youtu.be/8vY66MD7nsM
Odds of Having a Perfect NCAA March Madness Bracket: http://youtu.be/It1sCq9cAFM
Odds of Winning the Lottery: http://youtu.be/dVNFhu6tMQc
Blood Flow: Poiseuille's Law: https://youtu.be/X6aU0p7wJzI
Applications of Integrals: Hydrostatic Pressure and Force: https://youtu.be/fesMt6vmXIo
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Moments and Centers of Mass: Constant Density: https://youtu.be/3bglr1sRWUc
Can We Integrate All Continuous Functions?: http://youtu.be/OFEDLJYqYps .

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In this video I go over an introduction to differential equations and explain a bit about how they are the most important applications of calculus to the real-world. Differential equations are simply equations that consist of a function and some of its derivatives. These types of equations model real-world applications very well because often times physical concepts change at a rate that is proportional to its current state.

One such example is in population growth in which the rate of growth depends on what the current population is. In this video I use differential equations to model population growth. The first differential equation for population growth that I go over is for ideal conditions and is simply stated as the rate of growth is proportional to the current population. But a more accurate model assumes that there is a maximum carrying capacity in which the population levels off. This latter model is known as the Logistic Differential Equation and was first proposed by the Dutch mathematical biologist Pierre-François Verhulst in the 1840s. I also go over a brief history lesson on Verhulst.

This is an extremely important video as it lays the foundation for my later videos on one of the most powerful mathematical concepts, differential equations, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsEMZvMcUkjvxzId8g

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-population-growth

Related Videos:

Natural Exponential Function: y = e^x: http://youtu.be/vGsOA2eqkig .

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In this video I go over further into differential equations and this time use Hooke's Law to establish a relationship between the resistance force of a spring as being proportional to the length it stregths (or compresses). Since we are dealing with a force, we can apply Newton's Second Law of motion, which states that a force is equal to its mass times its acceleration. The acceleration of an object can also be considered as the second derivative of position of the spring. Thus we can model the motion of a spring through a second-order differential equation. It is called second-order because of it involves a function and its second derivatives.

Functions that fit this description are the sine and cosine functions, since the second derivatives both are themselves but with a negative sign. And similar to trigonometric functions, the motion of a spring oscillates which makes trig functions reasonable solutions to the motion of a spring differential equation.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsEgtmClosN58R6igg

Related Videos:

Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA
Integrals and Work: Example 2 - Hooke's Law: http://youtu.be/h8lwvB9rOng
Higher Derivatives Example on Acceleration: http://youtu.be/PVxUQjJ1vUA .

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In #911Truth Part 12 I have uploaded the infamous March 2, 2007 interview between Democracy Now’s Amy Goodman and Retired US General Wesley Clark regarding the United States PRE-PLANNED “7 countries in 5 years” invasion of the Middle East. Clark details how a mere 10 DAYS after 9/11, a general at the Pentagon told him that the decision to go war with Iraq was already made! And the reason given was “I don’t know”…. In other words, 10 days after 9/11, and before the media driven lie about “Iraq WMDs” the decision had already been made. Furthermore, Clark reveals that several weeks later, upon which the United States were already bombing Afghanistan, he went to see the unnamed general again to see if the Iraq War plan was still going to happen. But the General revealed that it was much worse and gave him a classified memo that listed “7 Countries in 5 Years” to be “take(n) out”: Iraq, Syria, Lebanon, Libya, Somalia, Sudan, and finally IRAN!

Now while the United States war machine has taken longer than “5 years” to carry out their plan, the first 6 countries have already been decimated resulting in mass genocide, a global immigration crisis, and the ballooning of fiat debt money across the globe that further increases the taxation upon ordinary hard working people. And even though Clark said he heard of the plan for the Middle East “War” aka Mass Ritual Sacrifice and Genocide "10 days after 9/11", it is clear the plan was made well in advance; and thus so too was the planned media brainwashing on a global scale planned as well.

There is now one country left on the list: IRAN. And from the current Puppet President, Donald Trump, and his “demented” tweets to the blatant government run Psychological Operation known as “Q Anon”, the level of outright threats towards Iran make it clear that the agenda for War will continue; but this time towards a more global war that includes Russia and China.

Since the war agenda continues regardless of who is president of ANY country, the inevitable realization comes to mind: we (as in the average ordinary citizen) are under global control. In fact, I would go so far as saying that the global power structure is not made up of “countries” but rather a two tier class: those that rule, and those that are ruled. But on an even deeper philosophical level, it is the very POSITION of power over others that will always lead to war; in other words, it doesn’t matter who is the president, even if it was yours truly, the result would be the same. Those that rule do so simply because they believe the global population are not fit to rule themselves, and everyday we prove them right by our very lifestyles, and ultimately by our very willingness to kill when asked to. Afterall, the “global elite” can only do so much in making us think that killing our fellow human being is EVER a viable decision. And ultimately the decision for peace, for self-governance, and saying no to war of ANY kind rests with us. #BoycottWar

Stay Tuned for #911Truth Part 13…

Full Amy Goodman and Wesley Clark Interview: [https://youtu.be/JOtbNC4oJ54](https://youtu.be/JOtbNC4oJ54)
“7 Countries in 5 Years” Transcript: https://genius.com/General-wesley-clark-seven-countries-in-five-years-annotated

View Video Notes on Steemit: https://steemit.com/war/@mes/911truth-part-12-gen-wesley-clark-reveals-middle-east-invasion-was-pre-planned-and-iran-is-next

Related Videos:

#911Truth Video Series: https://mes.fm/911truth-playlist
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist
#AntiGravity Video Series: https://mes.fm/antigravity-playlist
#MESExperiments Video Series: https://mes.fm/experiments-playlist
#Occult Video Series: https://mes.fm/occult-playlist
#PizzaGate Video Series: https://mes.fm/pizzagate-playlist .

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In this video I go over an example on differential equations and this time show that the solutions to the second-order differential equation y" + 9y = 0 are in the form of a combination of sine and cosine functions. This differential equation can be re-arranged to y" = -9y which is the same form as that in my last video on the motion of a spring. And thus as expected the solutions should be involving trigonometric functions because they model the oscillations of a spring very well.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsE1Ms_uLy2M-31Bnw

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-spring-motion-example-1

Related Videos:

Differential Equations: Motion of a Spring: https://youtu.be/mk2TiR5dwVs
Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA
Integrals and Work: Example 2 - Hooke's Law: http://youtu.be/h8lwvB9rOng
Derivative of Trigonometry Functions: Derivative of sin(x): http://youtu.be/elEvQ4Wu7Pk
Derivative of Trigonometry Functions: Derivative of cos(x): http://youtu.be/LMjVp-GsrCw .

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In this video I go over a general overview on differential equations as well as a few of the definitions and terms associated with them. I discuss briefly about the order of a differential equation as well as what the solution to a differential equation is. I also allude to how most real-world models of differential equations are complex and thus don't usually have explicit solutions to them. Instead we often have to approximate the solutions and I will show how in later videos so stay tuned!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsFhf69mv--wxTqDtQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-general-overview

Related Videos:

Differential Equations: Spring Motion: Example 1: https://youtu.be/Twu30EJ93Wg
Differential Equations: Motion of a Spring: https://youtu.be/mk2TiR5dwVs
Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA .

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In this video I go over an example on differential equations and show that the family of functions y = (1 + ce^t) / (1 - ce^t), where c is a constant, is a solution to the differential equation y' = 1/2(y^2 - 1). The process of proving that it is indeed a solution is to simply take the derivative of the family of functions and ensure that it satisfies the differential equation, which I show that it clearly does.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsFwy6vbLODQ0spUnA

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-example-1

Related Videos:

Differential Equations: General Overview: https://youtu.be/jit59tIY4UI
Differential Equations: Spring Motion: Example 1: https://youtu.be/Twu30EJ93Wg
Differential Equations: Motion of a Spring: https://youtu.be/mk2TiR5dwVs
Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA
Derivative Rules - Proof of the Quotient Rule: http://youtu.be/fJcgnLKkISE
Foil Method - Simple Proof and Quick Alternative Method: http://youtu.be/tmj_r94D6wQ .

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In #MESExperiments number 7, I have repeated the same experiment as in the 5th experiment but this time removed the plastic and metal bases so that the gyroscope is directly in contact with the ice block. This means that the contact surface between the ice and the stem of the gyroscope has very little rotational friction and which can be see by the gyroscope outer casing spinning very rapidly; even though the gyroscope’s center disk is the only thing initially spun.

I positioned the gyroscope at about 35 degrees from the vertical and yet again the gyroscope magically rises upwards and with ZERO “centripetal force” as can be seen by the lack of horizontal movement as the gyroscope rises upwards. This experiment demonstrates that even if the gyroscope’s outer casing is free to rotate, the gyroscope still rises upwards. Thus the ability of the gyroscope to rise upwards is not due to horizontal friction (no centripetal force) and not due to rotational friction either. Then the question arises: how is the gyroscope precessing upwards? Stay tuned for my later experiments and my epic #AntiGravity Part 6 video which shows that “gravity” itself is being interacted with through the simple rotation of matter… #StayTuned

The gyroscope takes about 3.5 minutes to fully rise upwards and after which maintains its vertical position for about 1 minute. Then, as also shown in experiment number 5, the gyroscope magically “regains” its “centripetal force” as it precesses downwards and with the ice block moving in large horizontal circular movements. This upwards rising is even more impressive given the fact that the gyroscope stem has literally dug directly into the ice block. Thus the gyroscope rises upwards not only against gravity but against the surrounding ice wall. #AbsolutelyFascinating

Also, later in the video I show that as the gyroscope loses spin speed, it still has the ability to precess in a circular downwards spiral; albeit more chaotic as the spin speed lowers. But even at extremely low spin speeds, the gyroscope still has the magical ability to avoid falling directly downwards but instead still spiral downwards which also “cushions” the fall of the gyro. #AmazingStuff

Stay Tuned for #MESExperiments Number 8!

View Video on Steemit: https://steemit.com/mesexperiments/@mes/mesexperiments-7-gyroscopes-precess-upwards-on-ice-even-while-outer-casing-spins-magic

Related Videos:

#MESExperiments Video Series: https://mes.fm/experiments-playlist
#AntiGravity Video Series: https://mes.fm/antigravity-playlist
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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In this video I go over another example on differential equations and this time discuss the more practical need for finding a particular solution to a differential equation as opposed to a general solution. One such solution involves satisfying the additional requirement that the solution has a specific initial value. Thus in solving such a problem, known as an initial value problem, we get one particular solution. In this example I look back at the same differential equation from Example 1 but this time apply the initial condition y(0) = 2.

Thus in the physical sense, the particular solution to an initial value problem means that we take our initial known or measured condition and then predict the future values from this starting point. This is a pretty interesting video on how to go about solving an initial value problem so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhsIsflB-HCEIQkOQrQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-example-2-initial-value-problem

Related Videos:

Differential Equations: Example 1: https://youtu.be/n575RKO48Ro
Differential Equations: General Overview: https://youtu.be/jit59tIY4UI
Differential Equations: Spring Motion: Example 1: https://youtu.be/Twu30EJ93Wg
Differential Equations: Motion of a Spring: https://youtu.be/mk2TiR5dwVs
Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA .

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Created 1 year, 6 months ago.

347 videos

CategoryEducation

I mainly teach math, but also do controversial videos that are censored on YouTube, as well as creating #FreeEnergy technology!