Q: A pendulum of length \(L\) is oscillating with a small angle \(\phi\). At its end hangs a small spherical ball of diameter \(d\) and \(d \ll L\). Another ball of the same diameter is set in motion at a random time such that it crosses the lowest point of the pendulum (see fig). What is the probability of a collision.
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (And Everone Else!)
A: As the second ball rolls towards the pendulum at random, the probability of a collision is simply the fraction of time the pendulum ball spends at the lower most phase of oscillatory motion. The period of a pendulum, assuming a small angle of swing is given as
$$ T = 2\pi \sqrt{\frac{L}{g}} $$
Note, that during the period of the pendulum it crosses the lowest point twice.
The Pendulum: A Case Study in Physics
The maximum velocity of the pendulum occurs at the point where it is the lowest from the top, where its kinetic energy is the highest and potential energy the lowest. The equation for the maximum velocity is given as
$$v_{max} = \sqrt{2gL(1 - cos(\phi))}$$
Therefore the time spent at the bottom phase of the pendulum in a manner that can cause a collision is
$$t_{collision} = \frac{2d}{v_{max}}$$
The sought probability is the ratio of \(t_{collision}\) and \(T\)
$$P(\text{collision}) = \frac{d}{\pi L \sqrt{2(1-cos(\phi))}}$$
Note, it is independent of \(g\), unsurprising but interesting.
Some good books to learn the art of probability
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
Introduction to Probability, 2nd Edition
The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.
Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
A: As the second ball rolls towards the pendulum at random, the probability of a collision is simply the fraction of time the pendulum ball spends at the lower most phase of oscillatory motion. The period of a pendulum, assuming a small angle of swing is given as
$$ T = 2\pi \sqrt{\frac{L}{g}} $$
Note, that during the period of the pendulum it crosses the lowest point twice.
The Pendulum: A Case Study in Physics
The maximum velocity of the pendulum occurs at the point where it is the lowest from the top, where its kinetic energy is the highest and potential energy the lowest. The equation for the maximum velocity is given as
$$v_{max} = \sqrt{2gL(1 - cos(\phi))}$$
Therefore the time spent at the bottom phase of the pendulum in a manner that can cause a collision is
$$t_{collision} = \frac{2d}{v_{max}}$$
The sought probability is the ratio of \(t_{collision}\) and \(T\)
$$P(\text{collision}) = \frac{d}{\pi L \sqrt{2(1-cos(\phi))}}$$
Note, it is independent of \(g\), unsurprising but interesting.
Some good books to learn the art of probability
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
Introduction to Probability, 2nd Edition
The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.
Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
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