This document discusses kinematics of rigid bodies, including:
- Types of rigid body motion such as translation, rotation about a fixed axis, and general plane motion.
- Translation motion is further divided into rectilinear and curvilinear types.
- Key terms related to rotation about a fixed axis like angular position, displacement, velocity, and acceleration.
- Relations between linear and angular velocity and acceleration.
- Two special cases involving rotation of pulleys - a pulley connected to a rotating block, and two coupled pulleys rotating without slip.
- Five sample problems calculating values like angular velocity and acceleration, revolutions, linear velocity and acceleration for rotating bodies.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
After reading this module, you should be able to . . .
10.01 Identify that if all parts of a body rotate around a fixed
axis locked together, the body is a rigid body. (This chapter
is about the motion of such bodies.)
10.02 Identify that the angular position of a rotating rigid body
is the angle that an internal reference line makes with a
fixed, external reference line.
10.03 Apply the relationship between angular displacement
and the initial and final angular positions.
10.04 Apply the relationship between average angular velocity, angular displacement, and the time interval for that displacement.
10.05 Apply the relationship between average angular acceleration, change in angular velocity, and the time interval for
that change.
10.06 Identify that counterclockwise motion is in the positive
direction and clockwise motion is in the negative direction.
10.07 Given angular position as a function of time, calculate the
instantaneous angular velocity at any particular time and the
average angular velocity between any two particular times.
10.08 Given a graph of angular position versus time, determine the instantaneous angular velocity at a particular time
and the average angular velocity between any two particular times.
10.09 Identify instantaneous angular speed as the magnitude
of the instantaneous angular velocity.
10.10 Given angular velocity as a function of time, calculate
the instantaneous angular acceleration at any particular
time and the average angular acceleration between any
two particular times.
10.11 Given a graph of angular velocity versus time, determine the instantaneous angular acceleration at any particular time and the average angular acceleration between
any two particular times.
10.12 Calculate a body’s change in angular velocity by
integrating its angular acceleration function with respect
to time.
10.13 Calculate a body’s change in angular position by integrating its angular velocity function with respect to time.
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After reading this module, you should be able to . . .
10.01 Identify that if all parts of a body rotate around a fixed
axis locked together, the body is a rigid body. (This chapter
is about the motion of such bodies.)
10.02 Identify that the angular position of a rotating rigid body
is the angle that an internal reference line makes with a
fixed, external reference line.
10.03 Apply the relationship between angular displacement
and the initial and final angular positions.
10.04 Apply the relationship between average angular velocity, angular displacement, and the time interval for that displacement.
10.05 Apply the relationship between average angular acceleration, change in angular velocity, and the time interval for
that change.
10.06 Identify that counterclockwise motion is in the positive
direction and clockwise motion is in the negative direction.
10.07 Given angular position as a function of time, calculate the
instantaneous angular velocity at any particular time and the
average angular velocity between any two particular times.
10.08 Given a graph of angular position versus time, determine the instantaneous angular velocity at a particular time
and the average angular velocity between any two particular times.
10.09 Identify instantaneous angular speed as the magnitude
of the instantaneous angular velocity.
10.10 Given angular velocity as a function of time, calculate
the instantaneous angular acceleration at any particular
time and the average angular acceleration between any
two particular times.
10.11 Given a graph of angular velocity versus time, determine the instantaneous angular acceleration at any particular time and the average angular acceleration between
any two particular times.
10.12 Calculate a body’s change in angular velocity by
integrating its angular acceleration function with respect
to time.
10.13 Calculate a body’s change in angular position by integrating its angular velocity function with respect to time.
Week 3 OverviewLast week, we covered multiple forces acting on.docxmelbruce90096
Week 3 Overview
Last week, we covered multiple forces acting on an object. This week we will cover motion in two dimensions, inclined planes, circular motion, and rotation.
Forces in Two Dimensions (1 of 2)
So far you have dealt with single forces acting on a body or more than two forces that act parallel to each other. But in real life situations more than one force may act on a body. How are Newton's laws applied to such cases? We will restrict the forces to two dimensions.
Since force and acceleration are vectors, Newton's law can be applied independently to the X and Y-axes of a coordinate system. For a given problem you can choose a suitable coordinate system. But once a coordinate system is chosen, we have to stick with it for that problem. The example that follows shows how to find the acceleration of a body when two forces act on it at right angles to each other.
Forces in Two Dimensions (2 of 2)
To find the resultant acceleration we draw an arrow OA of length 3 units along the X-axis and then an arrow AB of length 4 units along the Y-axis. The resultant acceleration is the arrow OB with the length of 5 units. Therefore, the acceleration is 5 m/s2 in the direction of OB. Also when you measure the angle AOB with a protractor, we find it to be 53°.
The acceleration caused by the two forces is 5 m/s2 at an angle of 53°.
Uniform Circular Motion
When an object travels in a circular path at a constant speed, its motion is referred to as uniform circular motion, and the object is accelerated towards the center of the circle. If the radius of the circular path is r, the magnitude of this acceleration is ac = v2 / r, where v is its speed and ac is called the centripetal acceleration. A centripetal force is responsible for the centripetal acceleration, which constantly pulls the object towards the center of the circular path. There cannot be any circular motion without a centripetal force.
Banking
When there is a sharp turn in the road or when a turn has to be taken at a high speed as in a racetrack, the outer part of the road or the track is raised from the inner part of the track. This is called banking. It provides additional centripetal force to a turning vehicle so that it doesn't skid.
The angle of banking is kept just right so that it provides all the centripetal force required and a motorist does not have to depend on the friction force at all.
Inclined Planes
Forces on an Inclined Plane
The inclined plane is a device that reduces the force needed to lift objects. Consider the forces acting on a block on an inclined surface. The inclined surface exerts a normal force FN on the block that is perpendicular to the incline. The force of gravity, FG, points downward. If there is no friction, the net force, Fnet, acting on the block is the resultant of FN and FG. By Newton's second law the net force must point down the incline because the block moves only along the incline and not perpendicular to it.
The vector triangle shows .
Mechanical Engineering is the Branch of Engineering.The mechanical engineering field requires an understanding of core areas including mechanics, dynamics, thermodynamics, materials science and structural analysis,Fluid Mechanics, Metrology and Instrumentation, Dynamics of Machinery- II, Manufacturing Processes II, Industrial Drafting and Machine Design, Engineering Graphics, Power Plant Engineering. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus. Visit us: https://ekeeda.com/streamdetails/stream/mechanical-engineering
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Civil Engineering is the Branch of Engineering.The Civil engineering field requires an understanding of core areas including Mechanics of Solids, Structural Mechanics - I, Building Construction Materials, Surveying - I, Geology and Geotechnical Engineering, Structural Mechanics, Building Construction, Water Resources and Irrigation, Environmental Engineering, Transportation Engineering, Construction and Project Management. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Civil Engineering is the Branch of Engineering.The Civil engineering field requires an understanding of core areas including Mechanics of Solids, Structural Mechanics - I, Building Construction Materials, Surveying - I, Geology and Geotechnical Engineering, Structural Mechanics, Building Construction, Water Resources and Irrigation, Environmental Engineering, Transportation Engineering, Construction and Project Management. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
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Kinetics of particles impulse momentum methodEkeeda
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Coplanar forces res & comp of forces - for mergeEkeeda
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Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us/l http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Ekeeda Provides Online Engineering Subjects Video Lectures and Tutorials of Mumbai University (MU) Courses. Visit us: https://ekeeda.com/streamdetails/University/Mumbai-University
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
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P
INTRODUCTION
In this Chapter we will do motion analysis of rigid bodies without involving
the forces responsible for the motion. We will learn to find the position,
velocity and acceleration of the different particles which together form a rigid
body.
In kinematics of particles the size or dimensions of the body were not taken
into consideration during motion analysis, since we treated the entire body
irrespective of its size (as large as a car, lift, train or airplane) as one single
particle. In rigid body kinematics, we shall involve the size and dimensions of
the body also. Therefore, now the body would be considered as made up of
several particles, connected to each other, such that their relative positions
do not change, as the body performs its motion.
There are various types of rigid body motion. We will begin with first
classifying them and then we will study each type in detail and thereby
develop our base for study of kinetics of rigid bodies.
Types of Rigid Body Motion
The motion of the rigid body can be classified under the following types,
1) Translation
2) Rotation about fixed axis
3) General plane motion
4) Motion about a fixed point
5) General motion
TRANSLATION MOTION
In this type of motion all the particles forming the body travel along parallel
paths. Also the orientation of the body does not
change during motion. Translation motion is
further classified as:
a) Rectilinear Translation
b) Curvilinear Translation
Kinematics of Rigid Bodies
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Rectilinear Translation
In this type of translation motion, all the particles travel along straight
parallel paths. Consider the motion of a body from position (1) to position (2).
Here we note two things, firstly a straight line joining particles P and Q in
position (1) maintains the same direction in position (2) This indicates that
the orientation of the body remains the same throughout the motion. The
second thing which is special to rectilinear translation motion, is that the
path described by the various particles, such as P and Q in our example, are
parallel and they are straight.
Examples of rectilinear translation motion are,
1) A piston which travels in the fixed slot.
2) Motion of train on a straight track.
3) Motion of lift etc.
Curvilinear Translation
In this type of translation motion, all the particles travel along curved parallel
paths. Consider the motion of a body from position (1) to position (2)
Here we note that the line joining particles P and Q in position (1) has the
same orientation in position (2) This is indicative of translation motion of the
body. Further the path described by all the particles, such as P and Q in our
example, are parallel and they are curved. This indicates curvilinear
translation motion.
At a given instant, for a translation motion, both rectilinear and curvilinear,
all the particles of the body have same velocity and same acceleration.
P Qi.e. v = v …… [13.1 (a)]
P Qand a = a …… [13.1 (b)]
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ROTATION ABOUT FIXED AXIS
In this motion all the particles forming the body
travel along circular paths of different radii,
around a common centre. This common point is
known as centre of rotation. An axis
perpendicular to the plane of the body and
passing through the centre of rotation is known
as the axis of rotation. Since the axis of rotation
remains stationary or fixed, we call such motion
as Rotation about Fixed Axis.
Important Terms
1) Angular Position
Let O, be the centre of rotation and z axis be
the fixed axis of rotation. Let the x axis be the
reference axis with respect to which we
measure the position of the rotating body. Let
us mark an arbitrary point P on the body.
The angle measured in the anticlockwise
direction which the line OP makes with the x
axis would then be the angular position of the
particle.
As the body rotates and occupies position (2), the angular position of the
body also changes and would have a new value of . Angular position is
measured in units of radians (rad). It may also be measured in units of
revolution or degree. These are related as
1 revolution = 2 rad = 360
For example, if the body’s initial angular position is 30 0.5236 rad and it
completes two revolutions, its new angular position would be,
750 13.09 rad .
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2) Angular Displacement
The change in angular position of the body during its motion is known as
the angular displacement of the body.
If 1 , is the angular position of the body in position (1) and if this changes
to 2 at position (2), the angular displacement of the body is,
2 1=
3) Angular Velocity
The rate of change of angular position with respect to time is the angular
velocity of the rotating body.
d
dt
……. [13.2]
The magnitude of angular velocity is denoted by notation (omega) and its
direction acts along the axis of rotation, the sense being defined by right
hand rule. i.e. for a body in the x-y plane rotating about a fixed axis
parallel to the z axis in the anticlockwise direction, will have positive
angular velocity. The same body, if its rotates in the clockwise direction
will have negative angular velocity. Angular velocity is represented by
curved arrow representing clockwise or anticlockwise sense.
Units of angular velocity are rad/s, through other unit like revolutions per
minute (rpm) is also commonly used. They are related as
2
1 rpm = rad/s
60
4) Angular Acceleration:
The rate of change of angular velocity with respect to time is the angular
acceleration of the rotating body.
d
dt
……. [13.3]
The magnitude of angular acceleration is denoted by notation (alpha)
and its direction acts along the axis of rotation. The sense of angular
acceleration is same as the sense of angular velocity, if the angular velocity
increases with time, and is opposite to the sense of angular velocity, if the
angular velocity decreases with time. Angular acceleration is represented
by curved arrow, representing clockwise or anticlockwise sense.
Units of angular acceleration are rad/s2
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Types of Rotation Motion about Fixed Axis
Rotation about fixed axis can be classified under three categories,
1) Uniform Angular Velocity Motion:
In this case of rotation motion the angular velocity of the rotating body
remains constant during motion. For such motions, we use a simple
relation relating , and t as
.........[13.4]
t
2) Uniform Angular Acceleration Motion:
In this case of rotation motion, the angular velocity of the rotating body is
not constant, but increases or decreases at a constant rate.
0
0
If initial angular velocity
final angular velocity
angular acceleration
angular displacement of the body
t time interval, then
we relate , , ,
and t as
0= + a t …… [13.5 (a)]
2
0= t + ½ a t …… [13.5 (b)]
2 2
0= + 2 a …… [13.5 (c)]
3) Variable Angular Acceleration Motion:
In this case of rotation motion the angular velocity changes during motion
and the rate of change of angular velocity is variable.
To solve problem on variable angular acceleration motion we make use of
the basic differential equations discussed earlier.
d
dt
.........[13.2]
d
dt
…… [13.2]
d
dt
…… [13.3]
From equations 13.2 and 13.3, we have
d
d
…… [13.6]
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Relation between Linear Velocity and Angular Velocity
Consider a body rotating about a fixed axis passing through O.
Let at the given instant, the angular velocity of the body be rad/s
anticlockwise.
All the particles on the rotating body will have the same angular velocity
but different linear velocity. If vP is the linear velocity of a particle P and
also rPO is the radial distance from P to O, then
P POv r
The sense of vP would be consistent with the direction of .
In general linear velocity v of any particle located at a radial distance r
from the axis of rotation O is related to the angular velocity of the body
by a relation.
v r ………. [13.7]
Relation between Linear Acceleration and Angular Acceleration
Consider a body rotating about a fixed axis passing through O, having an
angular velocity and angular acceleration , at the given instant as
shown.
We have studied in Chapter 9 that a particle
in curvilinear motion, has acceleration 'a'
which can be resolved into normal component
an and tangential component at. Similarly if
ap is the acceleration of particle P and if (an)p
and (at)p are its components, then
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2
2
po
n p
po
r ×ωv
a
r
2
n pop
a r
also
po
t po t pop p
d rdv dω
a r a r
dt dt dt
In general the linear acceleration a of any particle located at a radial
distance r from the axis of rotation O, has its components related to
angular velocity and angular acceleration by the relation
2
.na r ………. [13.8 (a)]
.ta r a ………. [13.8 (b)]
Also total linear acceleration a = 2 2
n ta a
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Special case 1 of rotation motion: A block connected to s rotating
pulley
Consider a pulley of radius r, hinged at the centre and supporting a block by
a string wound over it. The rotational motion of the pulley is related to
translation motion of the block. This relation can be worked out.
Let at a given instant, the pulley have an angular position of rad, angular
velocity of rad/s and angular acceleration of
rad/s2.
Let xA, vA and aA be the corresponding position, velocity
and acceleration respectively of the block A at this
instant.
Since the block translates, the string connecting it also
translates. We find point P on the pu1ley is common to
the rotating pulley and the translating block.
Refer figure. If point P belongs to the pulley, its
position, linear velocity and tangential acceleration is
given by
ps = r and pv = r and t p
a r
Relating these parameters to the motion of the connected block, we have
A p Ax s x r .……… [13.9 (a)]
A p Av v v r ………. [13.9 (b)]
and A t Ap
a a a r ………. [13.9 (c)]
Equations 13.9 (a), (b) and (c) relate the motion of a block hanging from a
rotating pulley.
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Special case 2 of rotation motion: A pulley coupled to another pulley
and they rotate without slip
Consider a pulley A of radius rA, be
coupled to another pulley B of radius rB,
and the two rotate without slip. The
rotational motion of pulley A is related to
the rotational motion of pulley B. This
relation can be worked out. Let at a given
instant, pulley A have an angular
position of A rad, angular velocity of A
rad/s and angular acceleration of A rad / s2.
Let B , B and B be the corresponding angular position, angular velocity
and angular acceleration of pulley B at this instant.
Since the pulley A rotates, it causes pulley B to also rotate. We find point P is
a common point to the two pulleys.
If point P belongs to pulley A, its position, linear velocity and tangential
acceleration is given by
P A AS = r θ and P A Av = r ω and t P A A(a ) = r α
Similarly point P also belongs to pulley B, therefore if the above parameters
are related to the pulley B, we have
P A A B Bs = r θ = r θ ………. (a)
P A A B Bv = r ω = r ω ………. (b)
t P A A B B(a ) = r α = r α ………. (c)
Equation (a), (b) and (c) relate the motions of two pulleys or gears engaged
with one another.
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EXERCISE 1
1. The tub of a washing machine is rotating at 60 rad/sec when the power is
switched off. The tub makes 49 revolutions before coming to rest.
Determine the constant angular deceleration of the tub and the time it
takes to come to a halt.
2. P2. A wheel rotating about a fixed axis at 15 rpm is uniformly accelerated
for 60 sec during which it makes 30 revolutions. Find
a) angular velocity in rpm at the end of the interval.
b) time required to attain a speed of 30 rpm. .
3. A point on the rim of a flywheel has a peripheral speed of 6 m/s at an
instant which is decreasing at a rate of 30 m/s2. If the magnitude of the
total acceleration of the point at this instant is 50 m/s2, find the diameter
of the flywheel.
4. A 1 m diameter flywheel has an initial clockwise angular velocity of 5 rad/s
and a constant angular acceleration of 1.5 rad/s2. Determine the number
of revolutions it must make and the time required to acquire a clockwise
angular velocity of 30 rad/s. Also find the magnitude of linear velocity and
linear acceleration of a point on the rim of the flywheel at t = 0.
5. A windmill fan during a certain interval of time has an angular acceleration
defined by a relation α = 18 e - 0-3 t rad/s2. The blades of the fan
describes a circle of radius 2.5 m. If at t = 0, ω = 0, determine at t = 5 sec
a) angular velocity of the fan
b) revolutions undergone by the fan.
c) Speed of the tip of fan blade.
6. A concrete mixer drum A is being rotated by two gears B and C. If the
concrete mixer is designed to attain a speed of 6 rad/s uniformly in 30 sec,
starting from rest and then maintain this speed, determine
d) the number of revolutions undergone
by the drum at t = 300 sec
e) the angular acceleration to be
developed in the driving gears.
f) velocity of concrete particles touching
the inner wall of the drum during uniform rotation motion of the
drum
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7. The variation of angular speed with time of a fan is shown. Find number of
revolutions undergone by the fan during a 10 minutes interval.
a) the angular acceleration and angular deceleration during this time
interval.
b) the magnitude of velocity and acceleration of a point on the tip of the
fan at t = 9 minutes, knowing that the fan described a circle of 1200
mm diameter.
8. The angular displacement of the rotating wheel is defined by the relation θ
= 1/4 t3 + 2 t2 + 18 rad. Determine the angular velocity and angular
acceleration of the wheel at t = 5 sec.
9. The angular acceleration of a rotating rod is given by the relation α = 9.81
cos θ -2.22 rad/s2. The rod starts from rest at θ = 0. Find
g) the angular velocity and angular acceleration of the rod at 0 = 30°.
h) the maximum angular velocity and the corresponding angle 0.
10. A belt is wrapped over two pulleys transmitting the motion without
slipping. If the angular velocity of the driver pulley A is increased uniformly
from 2 rad/s to 16 rad/s in 4 sec, determine
i) the acceleration of the straight position of the belt
j) the magnitude of total acceleration of a point on the rim of pulley B at
t = 4 sec.
k) the number of revolutions turned by the two pulleys at t = 4 sec.
11. Find the angular velocity in rad/s for
l) the second hand, the minute hand and hour hand of a watch,
m) the earth about its own axis.
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GENERAL PLANE MOTION
Translation motion and rotation about fixed axis motion are plane motions
since the motion of the body can be analysed by taking a representative slab
or a plane of the body. They may also be referred to as a Two Dimensional
Motion.
Any plane motion which does not fall under
the category of rotation about fixed axis or
translation motion can be put under the
category of general plane motion. In fact a
general plane motion is a combination of
translation motion and rotation motion. Three
examples of body performing general plane
motion are shown below.
In Fig. (a), Two blocks A and B travel is fixed slot performing translation
motion. The blocks are pin-connected by a link AB. The link AB moves from
position (1) to position (2) performing general plane motion. We observe that
the link AB rotates, but not about a fixed axis. Thus the centre of rotation of
the link AB performing general plane motion keeps on moving at every
instant.
In Fig. (b), rod AB, rod BC and block C, form a pin-connected mechanism.
The rod AB performs rotation motion about fixed axis at A. Piston C is free to
perform translation motion in the fixed slot. It is the rod BC which neither
performs pure translation or pure rotation motion. It is therefore said to
perform general plane motion. The centre of rotation of rod BC keeps on
changing as it performs general plane motion.
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In Fig. (c), a wheel rolls without slip on the
ground. The rolling wheel rotates as well as
translates. It therefore performs general plane
motion. In general any-body which rolls
without slip performs general plane motion.
Though G.P. bodies don’t actually translate
and then rotate in succession, but the motion can be duplicated by first
translating the body and then rotating it.
General Plane Motion = Translation Motion + Rotation Motion
Consider a body which has moved from position (1) to position (2) performing
G.P. motion. Refer Fig. Let P and Q be two arbitrary points chosen on it. We
may duplicate the motion by first translating the body to its position (2), i.e.
segment PQ maintains its orientation and remains parallel. Now we can
rotate the body about P to get the true orientation of the body. Hence G.P.
Motion is said to be a sum of Translation Motion and Rotation Motion.
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RELATIVE VELOCITY METHOD
We know that a G.P. Motion is a sum of translation and rotation motion.
To find the angular velocity of a body performing G.P. Motion, we may use
the relative Velocity Method, which is one of the methods for analysing
G.P. motion. To understand this method we note down the required steps
and simultaneously take up an example and apply the procedure.
Consider a rod AB pin-connected to pistons A and B which move in fixed
slots as shown. Let piston B have a known velocity vB to the right. Let us
learn to find angular velocity AB of the G.P. body AB.
Step 1. Locate a point on the G.P. body whose magnitude and direction of
velocity is known. Such a point is referred to as the reference point
and the known velocity is referred to as the translating velocity.
In our example, the velocity of point B i.e. vB is known. Hence B is
the reference point and vB is the translating velocity.
Step 2. Locate another point on the G.P body whose direction of velocity is
known. Such a point is referred to as translation point.
In our example, the direction of velocity of point A is known, since it
is constrained to move in the vertical slot. Point A is therefore the
translation point.
Step 3. Translate the body with the translating velocity and then rotate it
about the reference point with the angular velocity of the G.P body.
In our example, the bar is translated with translating velocity vB and
then rotated about reference point B with angular velocity AB .
Step 4. Write the relation for the relative linear velocity of translation point.
In our example, we write the relative linear velocity of translation
point A as
A/B AB ABV = r + …… (1)
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Step 5. Write the relation for the absolute velocity of the
translation point. Simultaneously draw the vector
diagram for the same. Obtain the relative velocity of
the translation point and hence obtain the angular
velocity of the G.P body.
In our example,
A B A/BV = V + V ….. (2)
From relation (2), A
B/
V can be found out, which in turn when
substituted in relation (1) gives the angular velocity AB .
Instantaneous Centre Method
Instantaneous Centre is defined as the
point about which the G.P. body rotates
at the given instant. This point keeps on
changing as the G.P. body performs its
motion. The locus of the instantaneous
centres during the motion is known as
centroid. Instantaneous Centre may be
denoted by letter I.
Let us understand the Instantaneous Centre Method to find the angular
velocity of a G.P body. Let us work with the earlier example we took in
article 13.5.1
Given- vB i.e. velocity of block B
To find - Angular velocity of rod AB at given instant
Step 1. Locate a point on the G.P body whose magnitude, direction and
sense of velocity is known and another point whose direction of
velocity is known. Mark the direction of velocity (Dov) of these two
points.
In our example, the magnitude and Dov of point B (DovB) is known
and also Dov of point A is known (DovA)
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Step 2. Draw perpendiculars to the direction of velocities (Dov) and extend
them to intersect at a point. Call this point as I.
Step 3. Point I i.e. the instantaneous centre, is the centre of rotation of the
G.P body at the given instant. Now treating the G.P body as a
rotating body about I, and using v = r ω relation, the angular
velocity of the G.P body can be found out.
In our example, the radial length rBI can be found out by geometry of
ABI.
Next using vB = rBI × ωAB
The angular velocity ωAB can be found out
Also now knowing ωAB , velocity of block A can be found out, using
vA - rAI × ωAB
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ROTATION ABOUT FIXED POINT
In this type of rigid body motion, the body rotates about a fixed point, but the
axis of rotation passing through the fixed point is not stationary as its
direction keeps on changing. Such motion is a three dimensional motion.
Example of this type of motion is of a top rotating about the pivot at O. The
axis of rotation is not fixed but changes its direction as the top rotates about
fixed pivot point O. Refer Fig.
Another example of rotation about a fixed point is of the motion of a boom
which is ball and socket supported at a point O on the crane. Fig. 13.15
shows the boom of a crane being raised up by rotation about the z axis, and
at the same time the crane itself rotates about the y axis to position the boom
over its target.
GENERAL MOTION
Any other motion of the rigid bodies which do not fit in any of the above
four types of rigid body motion, may be classified as a General Motion.
Motion analysis of a rigid body having Rotation Motion about a Fixed Point
of General Motion of a rigid body are beyond the scope of this book.
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EXERCISE 2
1. The rod ABC is guided by two blocks A and B which move in channels as
shown. At the given instant, velocity of block A is 5 m/s downwards.
Determine
a) the angular velocity of rod ABC
b) velocities of block B and end C of rod.
2. A rod AB 26 m long leans against a vertical wall. The end ‘A’ on the floor is
drawn away from the wall at a rate of 24 m/s. When the end ‘A’ of the rod
is 10 m from the wall, determine the velocity of the end ‘B’ sliding down
vertically and the angular velocity of the rod AB.
3. A slider crank mechanism is shown
in figure. The crank OA rotates
anticlockwise at 100 rad/s. Find
the angular velocity of rod AB and
the velocity of slider at B.
4. In a slider crank mechanism as
shown in figure the crank is
rotating at as constant speed of
120 rev/min. The connecting
rod is 600 mm long and the
crank is 100 mm long. For an angle of 30°, determine the absolute velocity
of the crosshead P.
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5. Block ‘D’ shown in figure moves with a speed of 3 m/s. Determine the
angular velocities of links BD and AB and the velocity of point B at the
instant shown. Use method of instantaneous centre of zero velocity.
6. For the position shown, the angular velocity
of bar AB is 10 rad/s anticlockwise.
Determine the angular velocities of bars BC
and CD.
7. Rod BCD is pinned to rod AB at B and
has a slider at C which slides freely in
the vertical slot. At the instant shown,
the angular velocity of rod AB is 4 rad/s
clockwise. Determine
a) angular velocity of rod BD
b) velocity of slider C
c) velocity of end D of the rod BD
8. A rod AB 1.8 m long, slides against an inclined plane and a horizontal floor.
The end A has a velocity of 5 m/s to the right. Determine the angular
velocity of the rod and the magnitude of velocity of end B and the midpoint
M of the rod for the instant shown.
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9. A slender beam AB of length 3 m which
remains always in a same vertical plane as its
ends A and B are constrained to remain in
contact with a horizontal floor and a vertical
wall as shown. Determine the velocity at point
B using instantaneous centre method.
10. A wheel of radius 0.75 m rolls without slipping on a horizontal surface to
the right. Determine the velocities of the points P and Q shown in figure
when the velocity of centre of the wheel is 10 m/s towards right.
11. One end of rod AB is pinned to the cylinder of
diameter 0.5 m while the other end slides
vertically up the wall with a uniform speed of 2
m/s. For the instant, when the end A is
vertically over the centre of the cylinder, find
the angular velocity of the cylinder, assuming it
to roll without slip.
12. A flanged wheel rolls to the left on a horizontal rail as shown. The velocity of
the wheel’s centre is 4 m/s. Find velocities of points D, E, F and H on the
wheel.
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13. Rod AB has a constant angular velocity of 50
rpm clockwise. For the given position of the
mechanism, find the angular velocity of rods BC
and CD.
14. If the link CD is rotating at 5 rad/sec anticlockwise, determine the angular
velocity of link AB at the instant shown.
15. Blocks A, B and C slide in fixed slots as shown.
The blocks form a mechanism, being
interconnected by pin-connected links AB and
BC. L (AB) = 400 mm and L (BC) = 600 mm. At
the given instant, block A has a velocity of 0.15
m/s downwards. Determine the velocities of
blocks B and C for the given instant.
16. In the mechanism shown the angular
velocity of link AB is 5 rad/s anticlockwise.
At the instant shown, determine the
angular velocity of link BC, velocity of
piston C and velocity of midpoint M of link
BC.
17. In the engine system shown, the crank AB has a constant clockwise
angular velocity of 2000
rpm. For the crank
position shown determine
the angular velocity of
connecting rod BD and
velocity of the piston D. Use ICR method.
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18. Locate the instantaneous center of rotation
for the link ABC and determine velocity of
points B & C. Angular velocity of rod OA is
15 rad/sec counter clock wise. Length of OA
is 200 mm, AB is 400 mm and BC is 150
mm.
19. The bar BC of the linkage shown has an angular velocity of 3 rad /sec
clockwise at the instant shown. Determine the angular velocity of bar AB
and linear velocity of point P on the bar BC.
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EXERCISE 3
Theory Questions
Q.1. Classify types of motion for rigid body with suitable examples.
Q.2. Explain the characteristics of a Translating Body.
Q.3. For a rotating body explain in brief the terms
1) Angular Position,
2) Angular Displacement
3) Angular Velocity
4) Angular Acceleration.
Q.4. What sire the different types of Rotation Motion about fixed axis and list
the corresponding equations applicable?
Q.5. Explain Instantaneous Centre of Rotation.
Q.6. Write a short note on General Plane Motion.
UNIVERSITY QUESTIONS
1. For the link and slider mechanism shown in figure, locate the
instantaneous centre of rotation of link AB. Also find the angular velocity of
link ‘OA’. Take velocity of slider at B = 2500mm/sec. (10 Marks)
2. ‘C’ is a uniform cylinder to which a rod ‘AB’ is
pinned at ‘A’ and the other end of the rod ‘B’ is
moving along a vertical wall as shown in fig. If the
end ‘B’ of the rod is moving upward along the wall
at a speed of 3.3 m/s find the angular velocity of
the cylinder assuming that it is rolling without
slipping. (6 Marks)
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3. Find the forces in members BD, BE and CE by
method of section only for the truss shown in
the fig. Also find the forces in other members
by method of joints. (8 Marks)
4. Collar B moves up with constant velocity
BV =2m/s. Rod AB is pinned at B.
Find out angular velocity of AB and velocity of A. (6 Marks)
5. Fig. shows the crank and connecting rod mechanism. The crank AB rotates
with an angular velocity of 2 rad/sec in clockwise direction. Determine the
angular velocity of Connecting Rod BC and the velocity of piston C using
ICR method. AB = 0.3 m and CD = 0.8 m (6 Marks)
6. In the mechanism shown the angular velocity of link AB is 5 rad/sec
anticlockwise. At the instant shown, determine the angular velocity of link
BC and velocity of piston C. (6 Marks)
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7. Due to slipping, points A and B on the rim of the disk have the velocities as
shown in figure. Determine the velocities of the centre point C and point D
on the rim at this instant. Take radius of disc 0.24 m (6 Marks)