Engineering Dynamics Linear Motion

8/9/2019 Engineering Dynamics Linear Motion

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Kinematics in One DimensionThe Objective:The Objective:

Determine motion by using calculation or graphicalDetermine motion by using calculation or graphical

method.method.

At the end of the lesson, student should be able to:At the end of the lesson, student should be able to:

Differentiate between displacement, velocity, andDifferentiate between displacement, velocity, and

accelerationacceleration

Predict the graph of the motion of an objectPredict the graph of the motion of an object

Explain the use of a negative sign to indicate direction inExplain the use of a negative sign to indicate direction invector quantitiesvector quantities

Use a motion graph to describe the motion of an objectUse a motion graph to describe the motion of an object

Determine the slope of a graph and use that informationDetermine the slope of a graph and use that information

to determine the velocity or acceleration of an object.to determine the velocity or acceleration of an object.

Engineering Mechanics (BPB 11303)


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Kinematics in One Dimension

Distance and DisplacementDistance and Displacement

Average VelocityAverage Velocity

Instantaneous VelocityInstantaneous Velocity AccelerationAcceleration

Graphical Analysis of Linear MotionGraphical Analysis of Linear Motion


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MechanicThe study of motion of objects, and related concepts of force and energy.

KinematicsKinematicsHow objects moveHow objects move

DynamicsDynamicsDeal with force and why objects moveDeal with force and why objects move


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Quantities Two types

ScalarsScalars

Common numbers we use everydayCommon numbers we use everyday

Scalars give us anScalars give us an amountamount

Distance, speed, mass, volumeDistance, speed, mass, volume

VectorsVectors

Like scalars they show anLike scalars they show an amountamount

Unlike scalars they showUnlike scalars they show directiondirection

Displacement, Velocity, accelerationDisplacement, Velocity, acceleration


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Consider the Following

Right now, this very instance, are you moving?Right now, this very instance, are you moving?

Distance, Displacement, Speed , Velocity,

Acceleration


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The Earth in Space

Earth rotates around its axis at: 1,043 mphEarth rotates around its axis at: 1,043 mph

Earth revolves around the sun at: 66,660 mphEarth revolves around the sun at: 66,660 mph

Solar system moves toward Vega at: 43,200 mphSolar system moves toward Vega at: 43,200 mph

Solar system revolves around the Milky Way Galaxy at:Solar system revolves around the Milky Way Galaxy at:

489,600 mph489,600 mph


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Whats it Mean?

Relative to some point in space you are movingRelative to some point in space you are moving

approximately 600,503 mph or 166.81 miles everyapproximately 600,503 mph or 166.81 miles every

second!second!

But, are you moving relative to the classroom?But, are you moving relative to the classroom?


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Motion - Definitions

Motion isMotion is relativerelative

Motion: Occurs when an objectMotion: Occurs when an object

changes its position relative to achanges its position relative to areference pointreference point


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Distance vs. Displacement

Distance how far an object hasDistance how far an object has

moved ( scalar)moved ( scalar)

Displacement distance andDisplacement distance anddirection from a starting pointdirection from a starting point

(vector)(vector)


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Distance Distance how far an object has moved (Distance how far an object has moved (magnitude onlymagnitude only).).

EastWest

South

North

70 km

30 km

Total Distance = 70 km + 30 km = 100 km

Find total Distance if :

20km to the East 50 km to the West again 10 km to the West.


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Displacement Displacement - a quantity that has bothDisplacement - a quantity that has both magnitudemagnitude andand directiondirection. Such. Such

quantities are called Vector.quantities are called Vector. Displacement is how far the object is from itsDisplacement is how far the object is from its starting pointstarting point (Change in(Change in

position of the object fromposition of the object from reference pointreference point).).

EastWest

North

70 Km

30 Km

Reference point

Displacement

Displacement =70 Km30 Km = 40 Km to the East ( Right )

*Direction: Right = +ve, Left = -ve


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Example:Example:

Displacement

EastWest 10 20 30 40

x1 x2

Displacement is x2 x1 x = x2 x1 = 40 km -10 km = 30 km to the East

Delta () means change in x.Distance = 40 km 10 km = 30km

(Km)


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Question:Question:

Displacement

EastWest 10 20 30 40

x1 x2

Displacement is x2 - x1

x = x2 x1 = 10 km -30 km = -20 km to the West

Distance = 30 km -10 km = 20 km

(km)


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Speed vs. Velocity

Linear MotionLinear Motion--

Motion Along a LineMotion Along a Line


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Speed:Speed:

Speed the distance an object travels per unit ofSpeed the distance an object travels per unit of

time (scalar)time (scalar)

Speed a change in distance over time also calledSpeed a change in distance over time also calledaa raterate

Rate any change over timeRate any change over time

Speed = distance / timeSpeed = distance / time Speed = x / t (m/s)Speed = x / t (m/s)

Speed vs. Velocity


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Types of Speed:Types of Speed: Speed that doesnt change over time isSpeed that doesnt change over time is

calledcalled constant speedconstant speed

Speed is usually not constant in our day-to-Speed is usually not constant in our day-to-day lives most objects have a changingday lives most objects have a changingspeed because of other forces acting onspeed because of other forces acting onthemthem

Average speed = total distance / total timeAverage speed = total distance / total timeelapsedelapsed

Instantaneous speed =speed at a givenInstantaneous speed =speed at a givenpoint in time (measured)point in time (measured)

Speed vs. Velocity


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Speed vs. Velocity

What is the difference betweenWhat is the difference between

speed and velocity?speed and velocity?


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Velocity:Velocity:

Is used to signify bothIs used to signify both magnitudemagnitude of howof how

fast an object is moving and thefast an object is moving and the directiondirection inin

which it is moving.which it is moving.

Therefore velocity is aTherefore velocity is a vectorvector..

Speed vs. Velocity


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Average Velocity ( v ):Average Velocity ( v ):

Is defined in terms of Displacement divide byIs defined in terms of Displacement divide by

time it takes to travel.time it takes to travel.

Average Velocity = Displacement /timeAverage Velocity = Displacement /time

Displacement =Displacement =xx

.: Average Velocity =.: Average Velocity =x /x /tt

Speed vs. Velocity


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Questions:Questions:

Can you have a negative speed?Can you have a negative speed?

Can you have a negative velocity?Can you have a negative velocity? Is distance a vector or a scalar?Is distance a vector or a scalar?

How about displacement?How about displacement?

Speed vs. Velocity


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Speed vs. Velocity

EastWest

North70 m

30 m

Reference point

Displacement

Displacement =70 m30 m = 40 m to the East ( Right )

*Direction: Right = +ve, Left = -ve

Distance = 70 m + 30 m = 100 m Average Speed = Total Distance / time elapsed = 100 m / 70s = 1.4 m /s

t = 70s

Average Velocity = x / t = 40 m / 70s = 0.57 m /sAverage velocity is +ve for an object moving to the right along x axis and ve

when the object move to the left. Direction of Velocity is always same as the direction of the Displacement.


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Speed vs. Velocity

The runner as a function of time is plotted as moving along the x axis of

coordinate system. During a 3.00s time interval, the runners position changes

from x1 = 50.0m to x2 = 30.5m, as shown below. What is runners average

velocity?

10 20 30 40

x1x2

50 60

x

Distance (m)

Solution:

Displacement = x = x2 x1 = 30.5m 50m = -19.5m

Time interval = t = 3.00s

Average Velocity = v = x / t = -19.5m / 3.00s = - 6.50m/s


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Speed vs. Velocity

How far can a cyclist travel in 2.5h along a straight road if her average

speed is 18 km/h?

Solution:

From equation v = x / t ,

.: x = v t = (18 km/h) (2.5h) = 45km


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Speed vs. Velocity

Instantaneous Velocity:Instantaneous Velocity:

Average velocity over an infinitesimally short time interval.Average velocity over an infinitesimally short time interval.

t2 t5 t10 t12

Average Velocity ( v ) unable to display the whole even happen for everyAverage Velocity ( v ) unable to display the whole even happen for every

seconds in figure above.seconds in figure above.

Instantaneous velocity is velocity that happen for particular t above.Instantaneous velocity is velocity that happen for particular t above.

m/s

Average Velocity ( v )

t1 t3 t4 t6 t7 t8 t9 t11s


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Speed vs. Velocity

From equation Average VelocityFrom equation Average Velocity = x / t ifif t ( different in t) becoming extremely small (t

0 ). We can write the definition of instantaneous

velocity (v) as: v = lim t0 (x / t) = x / t

The notation lim t0 means the ratio x / t is to

be evaluated in the limit of t approaching zero.

Instantaneous velocity always equals to

instantaneous speed when they becomeinfinitesimally small.


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Acceleration

When an objectWhen an object changes speedchanges speed orordirectiondirection, it is, it is

accelerationacceleration

Acceleration tells us how fast the velocity changes,Acceleration tells us how fast the velocity changes,

whereas velocity tells us how fast the positionwhereas velocity tells us how fast the positionchanges.changes.


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Acceleration

Average Acceleration ( a ):Average Acceleration ( a ):

Average acceleration is defined as a changeAverage acceleration is defined as a change

velocity divided by time taken to make this change:velocity divided by time taken to make this change:

a =a =v /v /t = (vt = (v22 vv11 ) / (t) / (t22 tt11))

Average acceleration is vector quantity.Average acceleration is vector quantity.


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Acceleration

Instantaneous Acceleration ( a ):Instantaneous Acceleration ( a ):

Instantaneous acceleration can be defined inInstantaneous acceleration can be defined in

analogy to instantaneous velocity, for any specificanalogy to instantaneous velocity, for any specific

instant:instant: a = lima = lim

tt 00((v /v /t) =t) = v / tv / t

Instantaneous acceleration always equals toInstantaneous acceleration always equals to

instantaneous acceleration when they becomeinstantaneous acceleration when they becomeinfinitesimally small.infinitesimally small.

Average acceleration is vector quantity.Average acceleration is vector quantity.


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Acceleration

Questions:Questions:A car accelerates along a straight road from rest to 75km/h in 5.0s.

What is the magnitude of its acceleration?

Solution:

The car starts from rest, so v1 = 0. The final velocity is v2 = 75km/h. From

equation of average acceleration, the average acceleration is

a = (v2 v1) / (t2 t1) = (75km/h 0km/h )/ (5.0s 0s) = 15(km/h)/s


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Acceleration

Questions:Questions:An automobile is moving to the right along a straight highway, which we

choose to be positive x axis , and then the driver puts on the brakes. If

the initial velocity is v1= 15.0m/s and it takes 5.0s to slow down to v2 =

5.0m/s, what was the cars average acceleration?

Solution:

a = ( v2- v1)/ (t2 t1) = (5.0m/s 15.0m/s)/(5.0s 0)= -2.0m/s2


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Motion in constant acceleration

Many practical situation occur in whichMany practical situation occur in which

acceleration is constant close enough that we canacceleration is constant close enough that we can

assume it is constant.assume it is constant.

This acceleration doesnt change over time and itThis acceleration doesnt change over time and itis calledis called uniformly accelerateduniformly accelerated motion.motion.

in this case,in this case, instantaneousinstantaneous andand averageaverage

accelerationacceleration areare equalequal..


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v related to a and t ( a = constant)v related to a and t ( a = constant)

t1=t0 =0 t2 = t

x1=x0 x2=x

v1=v0 v2= v

Considering all parameters above:

Average velocity :

v = x / t = (x x0) / t -------------------(1)Acceleration:

a = a = ( v v0) /t -------------------------------(2)

then v = v0 + at ---------------(3)


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ExampleExample

The acceleration of a particular motorcycle is 4.0m/s2 and we wish

to determine how fast it will be going after 6.0s.

Solution:

Assuming it starts from rest,.: (v0 = 0), after 6.0s the velocity will be:

From equation (3) :v = v0 + at = (4.0m/s2 )(6.0s) = 24m/s


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x related to a and t ( a = constant)x related to a and t ( a = constant)

From equation (1)

v = x / t = (x x0) / t

Then x = x0 + v t ---------------------------(4)

Because the velocity increase at a uniform rate (linearly), the average velocity

( v ) will be midway between the initial and final velocity.

Then v = (v0 + v ) / 2 --------------------(5)

(5) Into (4)

x = x0 +((v0 + v ) /2) t ----------------------(6)

(3) Into (6)

x = x0 + v0 t + ( at2 )/2 --------------------(7)


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v related to a and x ( a = constant)v related to a and x ( a = constant)From equation (4)

x = x0 + v t ---------------------------(4)

Then (5) into (4)

x = x0 + ((v0 + v ) / 2) t -------------(8)

From equation (3)

t = ( v - v0 ) /a ------------------------(3)

(3) Into (8)

v2 = v02 + 2a (x - x0 )


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Motion in constant accelerationExample:Example:

You are designing an airport for small planes. One kind of airplane that mightYou are designing an airport for small planes. One kind of airplane that might

use this airfield must reach a speed before takeoff of at least 27.8m/suse this airfield must reach a speed before takeoff of at least 27.8m/s

( 100km/h), and can accelerate at 2.00m/s( 100km/h), and can accelerate at 2.00m/s22 . (a) if runaway is 150m long, can. (a) if runaway is 150m long, can

this airplane reach the proper speed to take off? (b) if not, what minimum lengththis airplane reach the proper speed to take off? (b) if not, what minimum length

must the runaway have?must the runaway have?

Solution:Solution:

(a) From equation (9c), v2 = v02 + 2a (x - x0 ) = 0 + 2(2.00m/s2.00m/s

22 )(150m) = 600m)(150m) = 600m22/s/s22

v = 24.5m/s.v = 24.5m/s.

.: this runaway is not sufficient..: this runaway is not sufficient.

(b)(b) (x - x0 ) = (v2 - v0

2 )/2a = ((27.8m/s)2 0) / (2 (2.0m/s2)) = 193m.

knownknown wantedwanted

xx00 = 0= 0 vv

vv00 = 0= 0

x = 150mx = 150m

a = 2.00m/sa = 2.00m/s22


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Question:Question:

How long does it take a car to cross a 30.0m wide intersection after theHow long does it take a car to cross a 30.0m wide intersection after thelight turns green, if it accelerates from rest at a constant 2.00m/slight turns green, if it accelerates from rest at a constant 2.00m/s22 ??


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Graphical analysis of linear motion

Constant velocityConstant velocity

Magnitude of velocity variedMagnitude of velocity varied


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Constant velocity:Constant velocity:

The time t is considered the independent variable and is measuredThe time t is considered the independent variable and is measured

along the vertical axis.along the vertical axis.

The position x, the dependent variable, is measured along vertical axis.The position x, the dependent variable, is measured along vertical axis.

x increases by 10m every second.x increases by 10m every second.

34 t (s)

30m

40m

Position,x

(m)

x = 10m

t = 1s20m

2


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Position vs. time (Constant velocity):Position vs. time (Constant velocity):

The small triangle on the graph indicates the slope of straight line,The small triangle on the graph indicates the slope of straight line,

which is define as the change in the independent variable (which is define as the change in the independent variable (x ).

Slope = (x / t)

Slope = (x / t) =10m / 1s = 10m/s = velocity

+ slope = moving right, - slope = moving left

34 t (s)

30m

40m

Position,x

(m)

x = 10m

t = 1s20m

2


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Graphical analysis of linear motionPosition vs. time (magnitude of velocity varied):Position vs. time (magnitude of velocity varied):

The slope of the curve at any point is defined as the slope of tangent tothe curve at that point.

The tangent is a straight line drawn so it touches the curve only at that

one point but do not pass across or through the curve.

34 t (s)

30m

40m

Position,x

(m)

x = 10m

t = 1s20m

2

tangent


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Since theSince the slope equals to velocityslope equals to velocity, we, we

could reconstruct the v vs. t graph.could reconstruct the v vs. t graph.

We can determine the velocity as a functionWe can determine the velocity as a function

of time using graphical methods, instead ofof time using graphical methods, instead of

using equations.using equations.

This technique is particularly useful whenThis technique is particularly useful when

the acceleration is not constant, for thenthe acceleration is not constant, for then

equations (9) cannot be used.equations (9) cannot be used.


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Velocity vs. time:Velocity vs. time:

t (s)

v(m/s)

05 10 15 20 25 30

5

10

15

If we given v vs. t graph, we can determine the position ,x , as aIf we given v vs. t graph, we can determine the position ,x , as afunction of time.function of time.

Divide the time axis into many subintervals.Divide the time axis into many subintervals. In each interval, a horizontal dashed line is drawn to indicate theIn each interval, a horizontal dashed line is drawn to indicate the

average velocity during that time interval.average velocity during that time interval.

The displacement (change in position) during any subinterval isThe displacement (change in position) during any subinterval isx = vx = vt and total dis lacement after 30s will be sum of 6 rectan les.t and total dis lacement after 30s will be sum of 6 rectan les.


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Velocity vs. time:Velocity vs. time:

t (s)

v(m/s)

05 10 15 20 25 30

5

10

15

If the velocity varies a great deal, it may difficult to estimate v from theIf the velocity varies a great deal, it may difficult to estimate v from thegraph. To reduce this difficulty, narrower subintervals are.graph. To reduce this difficulty, narrower subintervals are.

The result, in any case, is thatThe result, in any case, is that the total displacement between any twothe total displacement between any two

times is equal to the area under the v vs. t graph between these twotimes is equal to the area under the v vs. t graph between these two

times.times.


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Question:Question:

t (s)

v(m/s)

01.0 2.0 3.0 4.0 5.0 6.0

50

100

A space probe accelerate uniformly from 50m/s at t =0s to 150m/s at t =A space probe accelerate uniformly from 50m/s at t =0s to 150m/s at t =

10s. How far did it move between t =2.0s and t = 6.0s?10s. How far did it move between t =2.0s and t = 6.0s?


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Question:Question:

t (s)

v(m/s)

01.0 2.0 3.0 4.0 5.0 6.0

50

100

Solution:Solution:

x = area under v vs. t graph = area of trapezoid = [((70m/s+110m/s))/2]4.0sx = area under v vs. t graph = area of trapezoid = [((70m/s+110m/s))/2]4.0s

= 360m= 360m

Can we use equations (9) to get total displacement?Can we use equations (9) to get total displacement?


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Question:Question:

What does the slope of a velocity/time graphWhat does the slope of a velocity/time graph

represent?represent?

What does the area under a velocity/timeWhat does the area under a velocity/time

graph represent?graph represent?



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S


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KinematicsKinematics deals with description of how object move. Thedeals with description of how object move. The

description of the motion of any object must always be given relativedescription of the motion of any object must always be given relativeto some particularto some particularreference framereference frame..

TheThe displacementdisplacement of an object is the change in position of theof an object is the change in position of theobject.object.

Average speedAverage speed is the distance traveled divided by elapsed time.is the distance traveled divided by elapsed time.

An objectsAn objects average velocityaverage velocity over a particular time intervalover a particular time intervalt is thet is thedisplacementdisplacementx divided byx divided byt:t:

v =v =x /x /tt

Instantaneous velocity,Instantaneous velocity, whose magnitude is the same as thewhose magnitude is the same as theinstantaneous speedinstantaneous speed..

An objects average acceleration over time intervalAn objects average acceleration over time intervalt is:t is:a =a = v/v/ tt

Summary

S


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If an objects moves in a straight line with constantIf an objects moves in a straight line with constant

acceleration, the velocity v and position x are related to theacceleration, the velocity v and position x are related to theacceleration a, the elapsed time t, and initial position xacceleration a, the elapsed time t, and initial position x

00andand

initial velocity vinitial velocity v00, by equations (9):, by equations (9):

Summary

v = vv = v00 + at+ at [ a = constant ]---(9a)[ a = constant ]---(9a)

x = xx = x00 + v+ v00 t + ( att + ( at22 )/2)/2 [ a = constant ]---(9b)[ a = constant ]---(9b)

vv22 = v= v0022 + 2a (x - x+ 2a (x - x00 )) [ a = constant ]---(9c)[ a = constant ]---(9c)

v =v =x /x / t = (x xt = (x x00) / t) / t [ a = constant ]---(9d)[ a = constant ]---(9d)

Engineering Dynamics Linear Motion

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