12€¦ · Web viewn-axis - perpendicular to the t- axis, directed from P towards the center of curvature. Positive direction , will be designated by the unit vector, u n ( normal

ENT252 - DYNAMICS

CHAPTER 12

INTRODUCTION TO DYNAMICS

Chapter ObjectivesTo introduce the concepts of position, displacement, velocity and accelerationTo study particle motion along a straight line and represent this motion graphicallyTo investigate particle motion along a curved path using different coordinate

systemsTo present an analysis of dependent motion of two particlesTo examine the principles of relative motion of two particles using translating axes

12.1 Introductions

Mechanics – branch of the physical science that is concerned with the state of

rest or motion of bodies subjected to the action of the forces

Mechanics of rigid body - divided into statics and dynamics

Statics - concerned with the equilibrium of the body that is either at the rest or

moves with constant velocity

Dynamics - concerned with the accelerated motion of a body. Presented in 2 parts:

a) Kinematics – geometric aspect of motion

b) Kinetics – analysis of the force causing the motion

- 1 - ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS

12.2 Rectilinear Kinematics: Continuous Motion

Rectilinear Kinematics – at any given instant, the particles position, velocity and acceleration.

Position – the straight line path of a particle. From the origin (o), position vector r specify the location of the particle (p).

Convenient (r) represent by (s)

Displacement – the change in its positionEg : If the particle moves from P to P’, the displacement is Δr = r’- rΔs = s’ – s

Δs is positive – particles final position is to the right of its initial position, ie : s’>s.

Displacement of a particle – vector quantity Distance traveled is a positive vector.

{Velocity}


r

O s

r’

r r

s s

s’

Displacement

P P’

Os

s

Position

P

ENT252 - DYNAMICS

If the particle moves through a displacement Δr, from P-P¹ during the time interval Δt, the average velocity

Vavg = Δr , V = dr : instantaneous velocityΔt dt

V as an algebraic scalar, V = dsdt

Δt or dt always positive:1. particle moving the right, velocity is positive 2. particle moving to the left velocity is negative.

The magnitude of the velocity is known as the speed.( units : m/s )

vavg = st

Δt


v

P P’ O s

s

ENT252 - DYNAMICS

{ Acceleration }

Provided the velocity of the particle is known at two point P² P¹, the average acceleration

aavg =

– difference in the velocity during the time interval = ¹-

Acceleration: a = acceleration

a = deceleration

deceleration – when the particle is slowing down - speed decreasing

- is negative

acceleration is zero – when velocity is constant. -

( unit = m/s )

a ds = v dv a = & v =

constant acceleration – each of three kinematics equations a =


a

P P’ O s

v v’

ENT252 - DYNAMICS

a = a

v = , a ds = vdv

maybe integrated to obtained formula that related a,v,s,t.

The three formulas of constant acceleration :

1) Velocity as a function of time

v = v + a t

2) Position as a function of time

+ a t) dt

s = s + v t + a t

3) Velocity as a function of position

v.dv = a .ds

= v = v + 2a ( s-s )

This formula only useful when the acceleration is constant and when t = o , s = s , v = v

e.g.– a body fall freely toward the earth.

See Example:


+

ENT252 - DYNAMICS

- 12.1

- 12.2

- 12.3

- 12.4

- 12.5

Exercise : 12.1

- 12.2

- 12.3

- 12.4

- 12.5

0194579207


ENT252 - DYNAMICS

12.3 Rectilinear Kinematics: Erratic Motion

When particles motion during a time is erratic, may best be describes graphically using a series of curves.

Using the kinematics equations:

a) Given s-t graph, construct the v-t equations

s

t

b) graph graph


=

By experimentally, if the position can be determined during the time of period, graph s-t can be plotted.

By = the graph v-t can be plotted. ( “ slope of s-t graph = velocity” ).

(“slope of graph = acceleration”)

See example 12.6, page 19

ENT252 - DYNAMICS

c) graph , tv graph


using ,

( change in velocity = area under )

ENT252 - DYNAMICS

d) graph graph


(displacement = area under graph)

See example 12.7, page 21

ENT252 - DYNAMICS

e) graph graph.

between the limits at

( =

at = area under graph

f) graph graph

acceleration = velocity times slope of graph.

See Example 12.8.Exercise:

- 12.42- 12.43


ENT252 - DYNAMICS

- 12.44- 12.45- 12.46


ENT252 - DYNAMICS

12.4 General Curvilinear Motion

- curvilinear motion occurs when the particle moves along a curved path.

- position- considered a particle located at point p on a space curve defined by the path function s.

position vector r = r ( t ) magnitude and direction change as the

particle moves along the curve.

- displacement- during small line t, the particle moves a distance s along the curves.

r’ = r + r

the displacement r represent the change in the particle’s position.

r = r’ - r

- velocity – during the time , the average velocity.

V = , V =

will be tangent to the curve at p, the direction of V is also tangent to the curve.


PathP

r s O

s Position

P’s

r P r’

r O

s

Displacement

v

P

r

O

s

Velocity

ENT252 - DYNAMICS

The magnitude of v, called ‘speed’. V = .

V , where v = v - v

Instant a new acceleration, 0

Velocity vector is always directed tangent to the path.

a tangent to the hodograph, not tangent to the path of motion.


ENT252 - DYNAMICS

12.5 Curvilinear Motion: Rectangular Components

Displacement

r = x + y + zk

magnitude of r always positive

r = (x + y + z )

unit vector ur = (1/r)r

Velocity

v = v v v

v = =

( x ) + ( y

) + ( z )

v = = v + v + v ,

where : v =

v =

v =

The velocity has a magnitude defined as the positive value of

v = v 2 + v 2 + v 2

Acceleration :


ENT252 - DYNAMICS

a = = a + a + a

where : a = =

a = =

a = =

The acceleration has a magnitude defined by the positive value of

a = a + a + a

See Example 12.9 and 12.10.


ENT252 - DYNAMICS

12.6 Motion of a Projectile

The free-flight motion of a projectile – studied in terms of its rectangular components. The projectiles acceleration always act in the vertical direction.

Projectile launched at point ( , ) , initial velocity is V , having two components ( V )x and ( V )y . The projectile has a constant downward acceleration, a c = g = 9.81 .

Horizontal motion : -

Since a x = 0 ; v = vo + ac t ; vx = ( vo ) x

x = xo+ vo t + at ; x = xo + ( vo ) t

v = vo + 2ac ( s-so ) ; vx = ( vo )

First and last equation indicated that the horizontal component of velocity always remains constant during the motion.


ENT252 - DYNAMICS

Vertical motion : -

Since ay = -g. + ↑ v = vo + ac t ; vy = ( vo )y – gt.

y = yo + vo t + ac t ; y = yo + (vo) yt - g

v = vo + 2 ac ( y- y ) ; vy = (vo ) -2g (y)

Only two of the above three equations are independent of one another.

Problems involving the motion of projectile can have at most three unknowns since only three independent equations can be written.

- one equations in the horizontal direction. - two equations in the vertical direction.

Once vx and vy are obtained, the resultant velocity v which is always tangent to the path.

See Example:

- 12.11

- 12.12

- 12.13

Exercise:- 12.71

- 12.72

- 12.73

- 12.74

- 12.75


ENT252 - DYNAMICS

12.7 Curvilinear motion: Normal and tangential components

When the path along which a particle is moving is known, it is convenient to describe the motion using n and t components (normal and tangent) to the path, and at the instant considered here their origin located at the particle.

Planer motion :-

( at instant considered )o’ - center of curvature.s - radius of curvature.

t–axis - tangent to the curve at P.

n-axis - perpendicular to the t- axis, directed from P towards the center of curvature.

Positive direction , will be designated by the unit vector, u n ( normal ) and u t ( tangent ).

Velocity :-

Since the particle moving , s is a function of time. The particle velocity v has a direction that is always tangent to the path, and the magnitude that is determined by taking the time derivative of the path function s = s(t) .

v =

v = vu t

where v =


ENT252 - DYNAMICS

Acceleration :-

The acceleration of the particle is the time rate of the change of the velocity. = = u t + v t

by formulation ,

t = u n = u n = u n

substitute to the above equation

= t u t + n u n

where t =

or t ds = v.dv

and n =

magnitude of acceleration is the positive value of =

Two special cases of motion :

1) The particle moves along a straight line , s . , =

The tangential components of acceleration represents the time rate of change in the magnitude of the velocity.

2) The particle moves along a curve with a constant speed then


ENT252 - DYNAMICS

12.8 Relative – motion analysis of two particles using Translating Axes

Position :

The axes of this frame are only persuitted to translate relative to the fixed frame . The

relative position of “B with respect to A”

is designate by a relative position vector r .

r = r + r

Velocity : An equation that related the velocities of the particle can be determined by taking the time derivative.

v = v + v

where v =

v =

v =

v and v - refer to absolute velocities - observed from the fixed frame.

v - relative velocity- observed from the translating frame.


vB/A

vB

vA

ENT252 - DYNAMICS

Acceleration :

is the acceleration of as seen by the observer located at and translating with the x’,y’,z’ reference frame.


aB/A

aA aB

Documents

12€¦ · Web viewn-axis - perpendicular to the t- axis, directed from P towards the center of curvature. Positive direction , will be designated by the unit vector, u n ( normal