Mechanics of Rigid Body Ia. Physics08

8/2/2019 Mechanics of Rigid Body Ia. Physics08

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Mechanics of Rigid BodyKinematics, Kinetics and Static

1.- Introduction2.- Kinematics. Types of Rigid Body Motion:

Translation,

RotationGeneral Plane Motion

3.- Kinetics. Forces and Accelerations. Energy and

Momentum Methods.

Angular Momentum and Moment of Inertia

Fundamental Equations of Dynamics

4.- Statics. Equilibrium.


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Mechanics of Rigid Body

System of Particles. Internal and External forces

Rigid Body is defined as a particular system

of particles which does not deform. Let A and Bbe any two of its particles; then, in a rigid body the

distance between A y B will remain without changes.Limits of this assumption: elasticity and break

Introduction. Rigid Body as a particular system of particles

AB

AB rrr =

CrAB AB == /


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1.- Introduction: Forces acting on a rigid body Forces acting of rigid bodies can bealso separated in two groups: (a) The external forces, represent the action of other bodies on

the rigid body under consideration; (b) The internal forces are the forces which hold together

the particles forming the rigid body. Only external forces can impart to the rigid body a motion

of translation or rotation or both

Transmissibility principle: The effect of an external force on a rigid body remainsunchanged if that force is moved along its line of action. We will need the math concept of moment

of a force (torque) to describe its principle.

Connected Rigid Bodies :Mechanisms. i.e. slider-crank, gear

box


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Free Body Diagrams:

Showing all the forces acting on the body

Reactions at supports and connections


Keeping Constant Speed over a

incline with angle

Exercise: Free-Body Diagram

on the rod (beam) embedded

in the wall

Exercise: Free-Body Diagram on the

front wheel and on the rear wheel (a)

constant speed (b) accelerating (c)

Acting brakes

Exercise: Free-Body Diagram

on the sliding stair with

friction in supporting points

Restaurant


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Introduction. Mass Center (Center of Gravity) of a System of Particles:Concept. Static and Dynamic properties

The center of mass of a system of particles (Rigid Body is a particular case) is the point of the space

where the system of gravitational forces formed by all elementary gravitational forces acting on each

elementary particle (dm g), is equivalent to one force (mg) placed there. The potential energy of a

system of particles is simply mgy, where yis the height of center of mass. This concept provides a

method to find the center of mass of a body.

The center of mass moves like a particle of mass m = mi under the influence of the external forcesacting on the system,

==i

iiCMCMext vmvmoramF

To compute the point where is placed the center of mass

===

==

i

iiCM

i

iiCM

i

iiCM

CM

i

iiCM

zmzmymymxmxm

dmrrmorrmrm

;;

Find the center of mass (with math and without math)

m 2m


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Fig. 15.1 a 15.4Mechanics of Rigid Body. Kinematics

1.- Kinematics. Types of motion: TRANSLATION, ROTATION about a fixed axis

, GENERAL PLANE MOTION, MOTION about a fixed pint, GENERAL MOTION

TRANSLATION.A motion is said to be atranslation if any straight line inside the

body keeps the same direction during the

movement.

All the particles forming the body move

along parallel paths. If these paths are

straight lines, the motion is said a rectilinear

translation; if the paths are curved lines, the

motion is a curvilinear motion

ROTATION about a fixed axis. Theparticles forming the rigid body move in

parallel planes along circles centered onthe same fixed axis. If this axis, called the

axis of rotation intersects the rigid body, the

particles located on the axis have zero

velocity and zero acceleration

Exercise: Distinguish between curvilinear translation and rotation about a fixed axis


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Mechanics of Rigid Body. Kinematics

GENERAL PLANE MOTION.

Any plane motion which is neither atranslation or a rotation is referred

as a general plane motion. Plan

motion is that in which all the

particles of the body move in parallel

planes. Translation and rotation areplane motions.

MOTION about a fixed point. Thethree-dimensional motion of a rigid

body attached at a fixed point, for

example, the motion of a top on a rough

floor, is known as motion about a fixed

point.

GENERAL MOTIONAny motion ofa rigid body which does not fall in any

of the cathegories above described.

Fi 15 1 15 4


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Fig. 15.1 a 15.4Mechanics of Rigid Body. Kinematics

Exercise: Identifydifferent types of

motion


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Mechanics of Rigid Body. Kinematics

TRANSLATION. Motion equations Fig. 15.1 Fig. 15.7 pag 918

Conclusion: A rigid body in

translation can be considered as a

particle

ABAB rrr =/

will be constant in magnitude(rigid body) and in direction

(translation motion), then

zeroisrofderivativethe AB/

AB vv =

AB

aa =

When a rigid body is in translation all the

points of the body have the same

velocity and the same acceleration.

M h i f Ri id B d Ki ti R t ti b t fi d i


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Mechanics of Rigid Body. Kinematics. Rotation about a fixed axis

Angular velocity,

Angular acceleration,

dt

d=

dt

d =

Representative slab

PR

v

ROTATION about a fixed axis.Motion equations. Velocity

Rv

Rdt

d

dt

ds

Rddsradiusxanglearc

=

=

=

=

Where angle is in radians!!!

Basic relationships

curvilinear motion

Angular velocity and angular acceleration are invariants.

They are the same for all points of the solid. They are a

characteristic of the rotating motion of the solid

Exercise: A compact disk rotating at 500 rev/min is scanned by a laser that begins at the inner radiusof about 2.4 cm and moves out the edge at 6.0 cm. Which is the linear (tangential) velocity of the disk

where the laser beam strikes: (a) at the beginning of scanning and (b) at the end?. The same for

acceleration

M h i f Rigid B d Ki ti Rotation about a fixed axis


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ROTATION about a fixed axis. Motion equations. Acceleration

RRdt

d

dt

Rd

dt

dvaT

====

)(

dt

d =

aT

aNP

RR

R

R

v

aN2

22)(

===

R

Representative slab

Mechanics of Rigid Body Kinematics Rotation about a fixed axis


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ROTATION about a fixed axis. The vector motion equations

ABB rv /=

)(// ABABB rra +=

Vector expressions for velocity andacceleration in rotation about a fixed axis

Mechanics of Rigid Body Kinematics Rotation about a fixed axis


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ROTATION about a fixed axis. MechanismRigid Body connected

The red arrow shows the angular velocityof the horizontal gear 1. Draw the angular

velocity for the other gear, 2 and 3. Solve

the problem with quantitative values:

1 = 500 rev/min; R1 = 2 cm

2 = ? rev/min; R2 = 5 cm; R2=10 cm

3 = ? rev/min; R4 = 10 cm;

The bucket falls from the

rest with a constant linear

acceleration of 0.3 g. (a)

Estimate the speed of the

bucket after 5 secondsand the fallen distance.

(b) Compute the angular

acceleration of the pulley

How fast will it rotate

after 5 s.

Gear 1 rotates

clockwise at

angular velocity of

12 rad/s. How fast

will gear 2 and 3rotate. Data: R1:5

cm; R2:10 cm;

R3:20 cm.

1

3

2

1

23

Mechanics of Rigid Body Kinematics General Plane Motion


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Mechanics of Rigid Body. Kinematics. General Plane Motion

GENERAL PLANE MOTION.Any general plane motion can be considered as atranslation plus a rotation Eulers

Theorem



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g y

GENERAL PLANE MOTION.Any general plane motion can be considered as atranslation plus a rotation

Angular velocity

and angular

acceleration of rod

are independent ofthe selected point

to rotate

Sliding rod



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Rolling without slipping.

Rolling with slipping.An object slides and rolls

.C

RaRv

Rs

C

C

===

Rolling without

slipping.

As the wheel of radius Rrotates through

angle , the point of the contact betweenthe wheel and the plane moves a

distance s that is related with by s= R.

If there is no sliding, the distance traveled

by point C is exactly the same s.

.C.

Ra Rv

Rs

C

C

Rolling with

slipping.



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g y

A bicycle travels with a speed of 40 km/h. How

fast the cycle rider pedals in rev/min?. Data:Sprocket radius: 2.5 cm; Front gear radius: 10

cm; rear wheel radius: 40 cm

Find the angular velocity of

sliding stair of length 3 m,

when the velocity of contactpoint with the soil is 3 m/s.

The angles between the stair

and the floor is 45

The slider-crank mechanism

converts the rotational motion of

crank in linear motion of slider.Find the relationship between the

angular velocity of crank and the

linear velocity of slider piston

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Mechanics of Rigid Body Ia. Physics08