The document contains 14 problems related to classical mechanics. Problem 1 describes two ladders leaning against two walls, with their intersection point 3 m above the ground, and asks for the distance between the walls. Problem 3 involves a pendulum whose length below a board can be varied, and asks the reader to show various relationships involving the semi-vertical angle, angular speed, and pendulum length are constant. Problem 4 describes a rod initially vertical on a table that is given an angular displacement and falls over, and asks for expressions involving the rod's angular speed, center speed, lower end speed, and the table's normal reaction.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
Neil Lambert - From D-branes to M-branesSEENET-MTP
Lecture by prof. dr Neil Lambert (Department of Mathematics, King’s College, London, UK & CERN, Geneva, Switzerland) on October 22, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
Solution manual for introduction to finite element analysis and design nam ...Salehkhanovic
Solution Manual for Introduction to Finite Element Analysis and Design
Author(s) : Nam-Ho Kim and Bhavani V. Sankar
This solution manual include all problems (Chapters 0 to 8) of textbook.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
Neil Lambert - From D-branes to M-branesSEENET-MTP
Lecture by prof. dr Neil Lambert (Department of Mathematics, King’s College, London, UK & CERN, Geneva, Switzerland) on October 22, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
Solution manual for introduction to finite element analysis and design nam ...Salehkhanovic
Solution Manual for Introduction to Finite Element Analysis and Design
Author(s) : Nam-Ho Kim and Bhavani V. Sankar
This solution manual include all problems (Chapters 0 to 8) of textbook.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Background MaterialAnswer the following questions after revi.docxwilcockiris
Background Material
Answer the following questions after reviewing the “Kepler's Laws and Planetary Motion” and “Newton and Planetary Motion” background pages.
Question 1: Draw a line connecting each law on the left with a description of it on the right.
planets move faster when close to the sun
planets orbit the sun in elliptical paths
planets with large orbits take a long time to complete an orbit
Question 2: When written as P2 = a3 Kepler's 3rd Law (with P in years and a in AU) is applicable to …
a) any object orbiting our sun.
b) any object orbiting any star.
c)
any object orbiting any other object.
Question 3: The ellipse to the right has an eccentricity of about … a) 0.25
b) 0.5
c) 0.75
d) 0.9
Question 4: For a planet in an elliptical orbit to “sweep out equal areas in equal amounts of time” it must …
a) move slowest when near the sun.
b) move fastest when near the sun.
c) move at the same speed at all times.
d) have a perfectly circular orbit.
Question 5: If a planet is twice as far from the sun at aphelion than at perihelion, then the strength of the gravitational force at aphelion will be as it is at perihelion.
a) four times as much
b) twice as much
c) the same
d) one half as much
e) one quarter as much
Kepler’s 1st Law
If you have not already done so, launch the NAAP
Planetary Orbit Simulator
.
·
Tip:
You can change the value of a slider by clicking on the slider bar or by entering a number in the value box.
Open the Kepler’s 1st Law tab if it is not already (it’s open by default).
· Enable all 5 check boxes.
· The white dot is the “simulated planet”. One can click on it and drag it around.
· Change the size of the orbit with the semimajor axis slider. Note how the background grid indicates change in scale while the displayed orbit size remains the same.
· Change the eccentricity and note how it affects the shape of the orbit.
Be aware that the ranges of several parameters are limited by practical issues that occur when creating a simulator rather than any true physical limitations. We have limited the semi-major axis to 50 AU since that covers most of the objects in which we are interested in our solar system and have limited eccentricity to 0.7 since the ellipses would be hard to fit on the screen for larger values. Note that the semi-major axis is aligned horizontally for all elliptical orbits created in this simulator, where they are randomly aligned in our solar system.
· Animate the simulated planet. You may need to increase the animation rate for very large orbits or decrease it for small ones.
· The planetary presets set the simulated planet’s parameters to those like our solar system’s planets. Explore these options.
Question 6: For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit?
Question 7: Create an orbit with a = 20 AU and e = 0. Drag the planet first to the far left of the ellip.
Phase, Pendulums, & Damping An investigation into swingers .docxmattjtoni51554
Phase,
Pendulums, & Damping
An investigation into swingers
4/11/2108
1
Last Time…
Our model:
2
frictionless surface
(spring equilibrium position)
The Phase Constant
If the block is pulled to A and released from rest, the phase constant is zero.
3
frictionless surface
(spring equilibrium position)
A
phase
phase constant
The Phase Constant
What if the block isn’t released from rest at A?
There is still an oscillation between ±A,
but now we need a phase constant
The phase constant exists to provide that shift away from A when t=0
4
Note: This thing moves when you run the power point, the x location at t = 0 is no longer A
Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is the phase constant ?
5
What does your
calculator say?
Which one is it? We
need a way to figure
this out.
T
T = 4 s
Finding Φ
To determine the sign of the phase constant, we’ll turn to uniform circular motion
Now what if the object doesn’t start on the x-axis?
6
The x-component of the object’s position vector is:
Now, the x-component of the object’s position vector is:
Great, but what do I need to memorize here?
Nothing… but…
We know initial conditions x(0) and v(0)
So we can determine the correct quadrant for Φ
Finding Φ
7
Consider that you can draw a circle and determine these things without memorizing them. The motion starts at the x axis, and the x velocity starts out negative and becomes positive once you start to come back around the circle. Furthermore, you can tell when the x velocity is increasing or decreasing in magnitude.
7
Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is the phase constant ?
8
T
For this problem:
slope < 0
Whiteboard Problem 15-5
Below is a velocity vs. time graph of a particle in simple harmonic motion.
What is the amplitude of the oscillation?
What is the phase constant? (LC)
What is the position at t = 0?
9
Note this is a plot of velocity, it’s a little different than the one that we just did.
Tick… Tock…
Swing on over to Mastering Physics and start the PhET Lab: Pendulums
Work as a group, but submit individually.
Sometime before you finish the PhET, calculate the length of the pendulum in the lobby of Kreger (LC) using the equation below:
You’ll have about 20 minutes to finish the PhET now, and whatever isn’t finished will be due at 6 today
10
A simple pendulum is a point mass on a massless string that can swing around a pivot point
The motion of the pendulum can be
described with torque
The Simple Pendulum
11
+
This is not SHM, but if we consider only small angles:
Whiteboard Problem 15-6
What is the period of a 1.0m long pendulum on:
Earth?
Venus? (LC)
12
Pivot
point
Physical Pendula
A physical pendulum is a rea.
11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
Aquí se estudian y deducen las relaciones entre el momento flexionante y los esfuerzos normales por flexión que se producen, y entre fuerzas cortantes verticales y los esfuerzos cortantes, y asimismo, diversos temas de importancia práctica en el diseño de vigas.
I am Craig D. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Stochastic Processes.
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You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
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Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
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Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Mission to Decommission: Importance of Decommissioning Products to Increase E...
Soal latihan1mekanika
1. 1
APPENDIX A
Miscellaneous Problems
In this Appendix I offer a number of random problems in classical mechanics. They are
not in any particular order – they come just as I happen to think of them, and they are not
necessarily related to any of the topics discussed in any of the chapters. They are
intended just to occupy you on those dull, rainy days when you have nothing better to do.
Solutions will be in Appendix B – except that whenever I add any new problems to
Appendix A, which I shall from time to time, I shall wait a few days before posting the
solutions in Appendix B.
_________________________
1. No book on classical mechanics is complete without a problem of a ladder leaning
against a wall. Here, then, is a ladder problem – except that it is nothing whatever to do
with mechanics, and it is put here just for fun. It is a problem only in geometry, yet it is
one which some people at first find difficult. It even seems difficult to try to find an
approximate solution by trying to draw it accurately to scale, and I have deliberately not
drawn it to scale, so you can’t find the answer merely by taking a ruler and measuring it!
10 m
8m
3m
Two ladders, of lengths 8 m and 10 m, are leaning against two walls as shown. Their
point of intersection is 3 m above the ground. What is the distance between the walls?
2. 2
2.
α
l0
*
*
Ω
A pendulum of length l0, is set into motion so that it describes a cone as shown of semivertical angle α, the bob describing a horizontal circle at angular speed Ω
Show that
cos α =
g .
l0 Ω 2
This, of course, is a very trivial problem not worthy of your mettle. It is given only as an
introduction to the next problem.
3. (a) A string of a pendulum passes through a board as shown in the figure below, in
such a manner that, by lowering or raising the board, the length of the string below the
board can be varied. The part below the board is initially of length l0, and it is set into
motion as a conical pendulum so that the angular speed and the semi vertical angle are
related by
cos α =
g .
l0 Ω 2
As the board is raised or lowered (or alternatively the pendulum is lowered or raised) and
consequently the length l below the board is varied, the semivertical angle θ will change
and so will the angular speed ω. (The symbols l0, α and Ω represent the initial values of
these quantities.)
3. 3
α
l0
*
*
Ω
Show that
i.
ii.
iii.
(b)
l 3 sin 3 θ tan θ is constant,
ω3 cot 2 θ is constant,
ω3 (ωl 2 − Ωl02 sin 2 α ) is constant.
Start with the following initial conditions:
l0 = 50 cm
Ω = 5 rad s−1
and assume that g = 9.8 m s−2, so that α = 38o 22'.
i. Plot a graph of θ (vertically) versus l (horizontally), for l = 0 to 1 m. When l =
40 cm, what is θ correct to one arcmin?
ii. Plot a graph of ω (vertically) versus θ (horizontally), for θ = 0 to 70o.
iii. Plot a graph of ω (vertically) versus l (horizontally) for l = 16 cm to 1 m. When l
= 60 cm, what is ω correct to four significant figures?
________________
4. 4
The next few problems involve a rod with its lower end in contact with a horizontal table
and the rod falling over from an initial vertical (or inclined) position. There are several
versions of this problem. The table could be smooth, so that the rod freely slips over the
table. Or the lower end could be freely hinged at the table, so that the lower end does not
move as the rod falls over. Or the table might be rough, so that the rod might or might
not slip.
4. A uniform rod of mass m and length 2l is initially vertical with its lower end in contact
with a smooth horizontal table. It is given an infinitesimal angular displacement from its
initial position, so that it falls over. When the rod makes an angle θ with the vertical,
find:
The angular speed of the rod;
The speed at which the centre of the rod is falling;
The speed at which the lower end of the rod is moving;
The normal reaction of the table on the rod.
Show that the speed of the lower end is greatest when θ = 37o 50'.
If the length of the rod is 1 metre, and g = 9.8 m s−2, what is the angle θ when the speed
of the lower end is 1 m s−1?
5. A uniform rod is initially vertical with its lower end smoothly hinged to a horizontal
table. Show that, when the rod falls over, the reaction of the hinge upon the rod is
vertical when the rod makes an angle 48o 11' with the vertical, and is horizontal when the
rod makes an angle 70o 31' with the vertical.
6. A uniform rod of length 1 metre, with its lower end smoothly hinged to a horizontal
table, is initially held at rest making an angle of 40o with the vertical. It is then released.
If g = 9.8 m s−1, calculate its angular speed when it hits the table in a horizontal position
(easy) and how long it takes to get there (not so easy).
7.
A uniform rod is initially vertical with its lower end in contact with a rough
horizontal table, the coefficient of friction being µ.
Show that:
a. If µ < 0.3706, the lower end of the rod must slip before the rod makes an
angle θ with the vertical of 35o 05'.
b. If µ > 0.3706, the rod will not slip before θ = 51o 15', but it will certainly
slip before θ = 70o 31' .
If µ = 0.25, at what angle θ will the lower end slip? If µ = 0.75?
5. 5
8. It is time for another ladder problem. Most ladders in elementary mechanics problems
rest on a rough horizontal floor and lean against a smooth vertical wall. In this problem,
both floor and wall are smooth. The ladder starts making an angle of α with the vertical,
and then it is released. It immediately starts to slip, of course. After a while it will cease
contact with the smooth vertical wall. Show that, at the moment when the upper end of
the ladder loses contact with the wall, the angle θ that the ladder makes with the vertical
is given by cos θ = 2 cos α .
3
9. If you managed that one all right, this one, which is somewhat similar, should be
easy. Maybe.
A uniform solid semicylinder of radius a and mass m is placed with its curved surface
against a smooth vertical wall and a smooth horizontal floor, its base initially being
vertical.
It is then released. Find the reaction N1 of the floor on the semicylinder and the reaction
N2 of the wall on the semicylinder when its base makes an angle θ with the vertical.
Show that the semicylinder loses contact with the wall when θ = 90o, and that it then
continues to rotate until its base makes an angle of 39o 46' with the vertical before it
starts to fall back.
_________________________
6. 6
Many problems in elementary mechanics involve a body resting upon or sliding upon an
inclined plane. It is time to try a few of these. The first one is very easy, just to get us
started. The two following that might be more interesting.
10. A particle of mass m is placed on a plane which is inclined to the horizontal at an
angle α that is greater than tan −1 µ , where µ is the coefficient of limiting static friction.
What is the least force required to prevent the particle from sliding down the plane?
11. A cylinder or mass m, radius a, and rotational inertia ka2 rolls without slipping down
the rough hypotenuse of a wedge on mass M, the smooth base of which is in contact with
a smooth horizontal table. The hypotenuse makes an angle α with the horizontal, and the
gravitational acceleration is g. Find the linear acceleration of the wedge as it slips along
the surface of the table, in terms of m, M, g, a, k and α .
[Note that by saying that the rotational inertia is ka2, I am letting the question apply to a
hollow cylinder, or a solid cylinder, or even a hollow or solid sphere.]
12.
x
V0
α
y
7. 7
A particle is placed on a rough plane inclined at an angle α to the horizontal. It is
initially in limiting static equilibrium. It is given an initial velocity V0 along the x-axis.
Ignoring the small difference between the coefficients of moving and limiting static
friction, show that at a point on the subsequent trajectory where the tangent to the
trajectory makes an angle ψ with the x-axis, the speed V is given by
V =
V0
.
1 + cos ψ
What is the limiting speed reached by the particle after a long time?
13.
xa
a
Calculate the moment of inertia of a hollow sphere, mass M, outer radius a, inner radius
xa. Express your answer in the form
2
I = 5 Ma 2 × f ( x) .
What does your expression become if x = 0? And if x → 1?
14.
xa
a
8. 8
Calculate the moment of inertia of a spherical planet of outer radius a, consisting of a
dense core of radius xa surrounded by a mantle of density s times the density of the core.
Express your answer in the form
2
I = 5 Ma 2 × f ( x , s ) .
Make sure that, if the density of the core is zero, your expression reduces to the answer
you got for problem 13.
2
Draw graphs of I / (5 Ma 2 ) versus x (x going from 0 to 1), for s = 0.2, 0.4, 0.6 and 0.8.
Show that, for a given mass M and density ratio s, the moment of inertia is least for a
core size give by the solution of
2(1 − s ) x 5 + 15 x 2 − 9 = 0 .
For a mantle-to-density ratio of 0.6, calculate the core size for which the moment of
inertia is least and calculate (in units of 2 Ma 2 ) the moment of inertia for that core.
5
Now let’s see if we can determine the core size from a knowledge of the moment of
inertia. It is sometimes asserted that one can determine the moment of inertia (and hence
the core size) of a planet from the rate of precession of the orbit of a satellite. I am not
sure how this would work with a planet such as Mercury, which has never had a satellite
in orbit around it. (Mariner 10, while in orbit around the Sun, made three fly-bys past
Mercury). Unless a planet departs from spherical symmetry, the orbit of a satellite will
not precess, since the gravitational planet is then identical with that from a point mass.
And, even if a planet were dynamically oblate, the rate of precession allows us to
determine the dynamical ellipticity (C − A)/C, but not either moment of inertia separately.
Nevertheless, let’s suppose that the moment of inertia of a planet is (0.92 ± 1%) %
2
2
5 Ma ; specifically, let’s suppose that the moment of inertia has been determined to be
between 0.911 and 0.929 % 2 Ma 2 , and that the mantle-to-core density ratio is known
5
(how?) to be 0.6. Calculate the possible range in the value of the core radius x.
15. A rectangular brick of length 2l rests (with the sides of length 2l vertically) on a
rough semicylindrical log of radius R. The drawing below shows three such bricks. In
the first one, 2l is quite short, and it looks is if it is stable. In the second one, 2l is rather
long, and the equilibrium looks decidedly wobbly. In the third one, we’re not quite sure
whether the equilibrium is stable or not. What is the longest brick that is stable against
small angular displacements from the vertical?
10. 10
16. A Thing with a semicylindrical (or hemispherical) base of radius a is balanced on top
of a rough semicylinder (or hemisphere) of radius b as shown. The distance of the centre
of mass of the Thing from the line (or point) of contact is l. Show that the equilibrium is
stable if
1
1
1
>
+ .
l
a
b
*
l
a
b
If a = b, is the equilibrium stable if the Thing is
(1)
(2)
(3)
(4)
A hollow semicylinder?
A hollow hemisphere?
A uniform solid semicylinder?
A uniform solid hemisphere?
11. 11
17.
*
*
A log of square cross-section, sides 2a, rests on two smooth pegs a distance 2ac apart,
one of the diagonals making an angle θ with the vertical.
Show that, if c < 1 / 8 = 0.354 the only equilibrium position possible is θ = 90o, but
that this position is unstable; consequently, following a small displacement, the logs will
fall out of the pegs. Show that if the pegs are farther apart, with c > 0.354, three
equilibrium positions are possible. Which of them are stable, and which are unstable? If
c = 1 , what are the possible equilibrium values of θ?
2
18.
θ
12. 12
A uniform solid hemisphere of radius a rests in limiting static equilibrium with its curved
surface in contact with a smooth vertical wall and a rough horizontal floor (coefficient of
limiting static friction µ). Show that the base of the hemisphere makes an angle θ with
the floor, where
sin θ =
Calculate the value of θ if (a) µ =
1
4
8µ .
3
3
and µ = 8 .
3
What happens if µ > 8 ?
19.
A uniform rod of length 2l rocks to and fro on the top of a rough semicircular
cylinder of radius a. Calculate the period of small oscillations.
13. 13
20.
A uniform solid hemisphere of radius a with its curved surface in contact with a rough
horizontal table rocks through a small angle. Show that the period of small oscillations is
P = 2π
1.04a .
g
21. The density ρ of a solid sphere of mass M and radius a varies with distance r from
the centre as
r
ρ = ρ 0 1 − .
a
Calculate the (second) moment of inertia about an axis through the centre of the sphere.
Express your answer in the form of constant % Ma2.
22.
*
α−θ
α+θ
*
14. 14
Two identical particles are connected by a light string of length 2aα. The system is
draped over a cylinder of radius a as shown, the coefficient of limiting static friction
being µ. Determine the angle θ when the system is in limiting equilibrium and just about
to slide.
F
23.
α
M
A mass M hangs from a light rope which passes over a rough cylinder, the coefficient of
friction being µ and the angle of lap being α. What is the least value of F, the tension in
the upper part of the rope, required to prevent the mass from falling?
24.
x
*
y
15. 15
A wooden cube floats on water. One of its faces is freely hinged to an axis fixed in the
surface of the water. The hinge is fixed at a distance from the top of the face equal to x
times the length of a side. The opposite face is submerged to a distance y times the
length of a side. Find the relative density s (that is, relative to the density of the water) of
the wood in terms of x and y.
25. A uniform solid sphere sits on top of a rough semicircular cylinder. It is given a
small displacement so that it rolls down the side of the cylinder. Show that the sphere
and cylinder part company when the line joining their centres makes an angle 53o 58'
with the vertical.
26.
12 cm
6 cm
9 cm
16. 16
A student has a triangular sandwich of sides 9 cm, 12 cm, 15 cm. She takes a
semicircular bite of radius 3 cm out of the middle of the hypotenuse. Where is the centre
of mass of the remainder? Is it inside or outside the bite?
27. An rubber elastic band is of length 2πa and mass m; the force constant of the rubber
is k. The band is thrown in the air, spinning, so that it takes the form of a circle, stretched
by the centrifugal force. (This takes much practice, skill and manual dexterity.) Find a
relation between its radius and angular speed, in terms of a, m and k.
28. Most of us have done simple problems on friction at high school or in first year at
college or university. You know the sort – a body lies on a rough horizontal table. A
force is applied to it. What happens? Try this one.
Find a uniform rod AB. A ruler will do as long as it is straight and not warped. Or a
pencil of hexagonal (not circular) cross-section, provided that it is uniform and doesn’t
have an eraser at the end. Place it on a rough horizontal table. Gradually apply a
horizontal force perpendicular to the rod at the end A until the rod starts to move. The
end A will, of course, move forward. Look at the end B – it moves backward. There is a
point C somewhere along the rod that is stationary. I.e., the initial motion of the rod is a
rotation about the point C. Calculate – and measure – the ratio AC/AB. What is the
force you are exerting on A when the rod is just about to move, in terms of its weight and
the coefficient of friction?
17. 17
29. Some of the more dreaded friction problems are of the “Does it tip or does it slip?”
type. This and the following four are examples of this type.
θ
A uniform solid right circular cone of height h and basal radius a is placed on an inclined
plane whose inclination to the horizontal is gradually increased. The coefficient of
limiting static friction is µ. Does the cone slip, or does it tip?
30.
2a
x
A cubical block of side 2a rests on a rough horizontal table, the coefficient of limiting
static friction being µ. A gradually increasing horizontal force is applied as shown at a
distance x above the table. Will the block slip or will it tip? Show that, if µ < ½, the
block will slip whatever the value of x.
18. 18
31.
τ
2ka
A cylindrical log of diameter 2a and mass m rests on two rough pegs (coefficient of
limiting static friction µ) a distance 2ka apart. A gradually increasing torque τ is applied
as shown. Does the log slip (i.e. rotate about its axis) or does it tip (about the right hand
peg)?
When you’ve done that one, you can try a variant (which I haven’t worked out and
haven’t posted a solution) in which a cylinder of radius a is resting against a kerb (or
curb, if you prefer that spelling) of height h, and a torque is applied. Will it tip or will is
slip?
h
19. 19
32.
•
h
2d
This problem reveals the severe limitations of my artistic abilities, but the drawing above,
believe it or not, represents a motor car seen from behind. You can see the driver and
passenger. The height of the centre of mass is h and the distance between the wheels is
2d. The car is travelling on a horizontal road surface, coefficient of friction µ, and is
steering to the left in a circle of radius R, the centre of curvature being way off to the left
of the drawing. As they gradually increase their speed, will the car slip to the right, or
will it tip over the right hand wheel, and at what speed will this disaster take place?
Fortunately, driver and passenger were both wearing their seat belts and neither of them
was badly hurt, and never again did they drive too fast round a corner.
33.
l+a
A
l−a
20. 20
A uniform rod of length 2l rests on a table, with a length l − a in contact with the table,
and the remainder, l + a sticking over the edge – that is, a is the distance from the edge of
the table to the middle of the rod. It is initially prevented from falling by a force as
shown. When the force is removed, the rod turns about A. Show that the rod slips when
it makes an angle θ with the horizontal, where
tan θ =
2µ
.
2 + 9(a/l ) 2
Here µ is the coefficient of limiting static friction at A.