Kinetics of Particles: Chapter 12 Newton’s 2nd Law

Chapter 12Kinetics of Particles:

Newton’s 2nd Law

2/24/2019 8:13 PM

Mohammad Suliman Abuhaiba, Ph.D., PE

2nd Exam

Saturday

16/3/2019

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2 - 2

Introduction

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• Newton’s 1st and 3rd laws:

Study of bodies at rest (statics)

Study of bodies in motion with no

acceleration

• When a body accelerates

Newton’s 2nd law relates motion of body to

forces acting on it

2

Newton’s 2nd of Motion

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• Newton’s 2nd: If resultant force

acting on a particle is not zero, particle

will have an acceleration proportional

to magnitude of resultant & in

direction of resultant

ma

F

a

F

a

F mass,constant

3

3

2

2

1

1

4

• If force acting on particle is zero, particle will not

accelerate: it will remain stationary or continue

on a straight line at constant velocity.

Newton’s 2nd of Motion

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• When a particle of mass m is acted upon by a

force F, acceleration of particle must satisfy

amF

5

Linear Momentum of a Particle

dt

Ldvm

dt

d

dt

vdmF

Principle of Linear Momentum Conservation:

If resultant force on a particle is zero, linear

momentum of particle remains constant in both

magnitude & direction.

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6

Systems of Units

• International System of Units (SI Units):

length (m), mass (kg), and time (second)

22 s

mkg1

s

m1kg1N1

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7

Systems of Units

ft

slb1

sft1

lb1slug1

sft32.2

lb1lbm1

2

22

• U.S. Customary Units: force (lb), length (ft),

and time (second)

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8

Equations of Motion

Newton’s 2nd law: amF

zmFymFxmF

maFmaFmaF

kajaiamkFjFiF

zyx

zzyyxx

zyxzyx

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9

Equations of Motion

Tangential & normal components

2vmF

dt

dvmF

maFmaF

nt

nntt

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10

Dynamic Equilibrium

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• Alternate expression of

Newton’s 2nd law,

ectorinertial vam

amF

0

• Inertial forces = resistance

of particles to changes in

motion, i.e., changes in

speed or direction.

11

Sample Problem 12.1

A 200-lb block rests on a

horizontal plane. Find the

magnitude of the force P

required to give the block

an acceleration of 10 ft/s2

to the right. The

coefficient of kinetic

friction between the block

and plane is mk 0.25.

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12

Sample Problem 12.3

The two blocks shown start

from rest. The horizontal

plane and the pulley are

frictionless, and the pulley

is assumed to be of

negligible mass.

Determine the acceleration

of each block and the

tension in the cord.

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13

Sample Problem 12.4

The 12-lb block B starts from

rest and slides on the 30-lb

wedge A, which is supported

by a horizontal surface.

Neglecting friction, determine:

a. Acceleration of wedge

b. Acceleration of block

relative to wedge

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14

Sample Problem 12.5

The bob of a 2-m pendulum

describes an arc of a circle

in a vertical plane. If the

tension in the cord is 2.5

times the weight of the bob

for the position shown, find

the velocity and

acceleration of the bob in

that position.

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15

Sample Problem 12.6Determine the rated

speed of a highway curve

of radius = 400 ft banked

through an angle q = 18o.

Rated speed of a banked

highway curve = speed at

which a car should travel if

no lateral friction force is

to be exerted at its wheels

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16

Assignment #12-1

10, 17, 23, 30, 37, 44, 51

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17

Angular Momentum of a Particle

• moment of momentum or angular

momentum of particle about O

VmrHO

• is perpendicular to plane containing r & mvOH

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18


zyx

O

mvmvmv

zyx

kji

H

q q2sin mrvrmrmVHO

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19

• Differentiate angular momentum wrt time:

O

O

MFr

amrVmV

VmrVmrH

• Newton’s 2nd law: sum of moments about O

of forces acting on particle = rate of change of

angular momentum of particle about O

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20

Equations of Motion (Polar Coordinates)

qq

q

qq

rrmmaF

rrmmaF rr

2

2

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21

Conservation of Angular Momentum

When net force acting on particle is directed

toward or away from a fixed point O, particle is

said to be moving under a central force.

Since line of action of central

force passes through O,

and 0 OO HM

constant OHVmr

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22

Conservation of Angular Momentum

Magnitude of angular momentum,

000 sinconstantsin VmrVrmHO

massunit

momentumangular

constant

2

2

hrm

H

mrH

O

O

q

q

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23

Newton’s Law of Gravitation• Gravitational force exerted by sun on a planet or by

earth on a satellite

4

49

2

312

2

slb

ft104.34

skg

m1073.66

ngravitatio ofconstant

G

r

MmGF

• Newton’s law of universal gravitation - two

particles of mass M & m attract each other with

equal & opposite force directed along line

connecting the particles,

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24

Sample Problem 12.7

A block B of mass m can slide freely on

a frictionless arm OA which rotates in a

horizontal plane at a constant rate .

B is released at a distance r0 from O.0q

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25

qqqq errerra r

22

Express as a function of r :

a. Component vr of velocity of B along OA

b. Magnitude of horizontal force exerted on B by the arm OA.

Sample Problem 12.8A satellite is launched in a direction parallel to the

surface of the earth with a velocity of 18820 mph

from an altitude of 240 mi. Determine the velocity

of satellite as it reaches its max altitude of 2340 mi.

Radius of earth is 3960 mi.

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26

Assignment #12-2

66, 74, 81

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27

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Kinetics of Particles: Chapter 12 Newton’s 2nd Law

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