King Fahd University of Petroleum &

MineralsMechanical Engineering

Dynamics ME 201BY

Dr. Meyassar N. Al-HaddadLecture # 11

Sir Isaac Newton1642-1727

Proposed fundamental laws that are the basis of modern mechanics

3 laws of motionlaw of gravitation

Newton’s lawsLaw of inertia a body in motion will stay in motion and a body at rest will

stay at rest unless acted upon by a net external force.

Law of force-acceleration A particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force

Law of action-reaction for every action, there is an equal and opposite reaction mg = FN

Law of gravitation - all bodies are attracted to one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

221

rmmGF

amF

13.2 The Equation of Motion

• Free-Body diagram (Force Diagram)

• Kinetic diagram (acceleration Diagram)

aF m

13.3 Equation of Motion for a System of Particles

iiii m

m

afF

aF

G

iii

m

m

aF

aF

Internal forces cancel each other

13.4 Equations of Motion: Rectangular Coordinates

• When the net force is projected to separate coordinate axes the Newton’s second law still holds

)kji(kji

aF

zyxzyx aaamFFF

m

zz

yy

xx

maF

maF

maF

Free Body Diagram Method

•Draw each object separately

•Draw all the forces acting on that object

•Get x and y components of all the forces to calculate the net force

•Apply Newton’s second law to get acceleration

•Use the acceleration in any motion analysis and establish a Kinetic Diagram

ma

Normal & Frictional Force

F

Ff

FN

Action-Reaction forces -mg = FN

mg

Static Friction ( s )

• Static friction – parallel force on the surface when there is no relative motion between the 2 objects

• Static friction force can vary from zero to Maximum

The coefficient of static friction is material dependent.

StaticFf = sFN

DynamicFf = kFN

Applied external force

Frict

ion

Kinetic Friction ( k )

• Kinetic friction – parallel force on the surface when there is relative motion between the 2 objects

• Kinetic friction force is always the same

• The coefficient of Kinetic friction

• is material dependent.

NkFF μf

StaticFf = sFN

DynamicFf = kFN

Applied external force

Frict

ion

Spring Force

• Spring force

• k : spring stiffness (N/m)• s : stretched or

compressed length

ksFs

olls

ol l

s

• m = 2 kg• y = 1m• smooth• a = ?

maFmg s sin

2219sin

9053cosx5.0x3coscos

m/s.θmksga

N.θksθFN sc

ma

Example 13-4

0cos sc FN

13537501tan

507507501

1

22

..

θ

m...S

x

ymg

N Nk

Problem

NNN

369020cos81.940

;0 yF

xx maF mamgNk sin

0cos mgN

2/66.5

4020sin81.940)369(25.0

sma

a

x

x

a? =

Example 13-5

m A= 3 kgm B= 5 kgFrom restvB= ? In 2 second

yy maFAaT 32981

Block A

yy maFBaT 52.196

Block B

lss BA 2 02 BA 02 BA aa

taBo

Same

Same ٍshould be

VA? =

2Sc + SA = L 0 = 2ac + aA aA = -2ac

ca

ccx

y

amgTNmaF

NNNF

20030sin25.0:

1699030cos81.9*200:0

Problem

mg Ama

)/(62.42 smsaV AA

)2(12525.1226

12581.9*125

A

AA

aT

aTmaFy

NTsmasma

aaaa

aa

A

A

A

c

cc

c

c

1004)/(777.1)/(888.0

500200250200622

200981)12525.1226(25.849)1()2(

22

Review

• Example 13.1• Example 13.2• Example 13.3• Example 13.4