Physics 121Newtonian Mechanics

Instructor: Karine Chesnel

Feb 26th , 2009

Review for Exam 2

Class website:www.physics.byu.edu/faculty/chesnel/physics121.aspx

Mid-term exam 2

• Fri Feb 27 through Tuesday Mar 3

• At the testing center : 8 am – 9 pm

• Closed Book and closed Notes

• Only bring: - Pen / pencil- Calculator- Math reference sheet- dictionary (international)- your CID

• No time limit (typically 3 hours)

Midterm exam 2

Review: ch 5 – ch 8

Ch. 6 Newton’s laws applications• Circular Motion• Drag forces and viscosity• Friction• Fictitious forces

Ch. 8 Conservation of Energy• Mechanical energy• Conservation of energy

Ch. 7 Work and energy• Work• Kinetic energy• Potential energy• Work- kinetic energy theorem

Ch. 5 The Laws of Motion• Newton’s first law• Newton’s second law• Newton’s third law

Summary ofthe Laws of Motions

Third Law: Action and reaction If two objects interact,

the force exerted by object 1 on object 2 is equal in magnitude and opposite in direction

to the force exerted by object 2 on object 1.

First Law: Principle of InertiaIn a inertial frame,

an isolated system remains at constant velocity or at rest

Second Law: Forces and motionIn an inertial frame

the acceleration of a systemis equal to the sum of

all external forcesdivided by the system mass

Fam

m

Fa

Ch.5 Laws of motion 2/26/09

F1 F2

Review of basic forces

• The weight

Object of mass m

gmFg

Fg

• Normal reaction

When two objects are in contact

- N

N

The reaction exerted by the support on the object is NORMAL to the surface

• Force of tension

The spring force tendsto bring the object back to rest

F

• Spring force

F = - k x î

0 x

T

The tension exerted by a rope on the object is ALONG the direction of the rope

Ch.5 Laws of motion 2/26/09

`

Forces of FrictionTwo regimes

mg

RF

f

If one applies a force Fat which point

the system starts to move?

• If F is smaller than a maximum value fmax

then the system does not move

• If F is larger than the maximum value fmax

the system starts to move and the friction is constant

Static regime Kinetic regime

F

f

maxf

kf

Static regime

Dynamic regime

Ch.5 Friction 2/26/09

FrictionSummary of two regimes

Static regime Kinetic regime

F

f

maxf

kf

In the static regime, the magnitude of the friction is equal to the force pushing the object

Ff s

When the system is on the verge to move

Nf ss max,

Static coefficient

of friction

Once the system is moving

Nf kk

Kinetic coefficient of friction

Ch.5 Friction 2/26/09

Resistive forcesLow speed regime

V

F

The equation of the motion is given by

and Vf = mg/b is the terminal speed

Where = m/b the time constant

)1()( /tf eVtV

The motion starts at t = 0 with no initial speedThe speed increases to reach the limit Vf

When t = the speed value isV = (1-1/e) Vf ~ 0.63 Vf

V(t)

Vf

t

(1-1/e)Vf

0

Ch.6 Special applications of Newton’s law 2/26/09

m

bVg

dt

dV

Ch.5&6 Laws of motion 2/26/09

• Define a frame of work that suits with the situation: either Cartesian coordinates (x, y) or polar coordinates (, )

List all the forces applied on the system, for example:- the weight mg - the normal reaction of a support N- a force of tension T- a force of friction f … etc

2. List the forces

3. Apply Newton’s law

Fam

General method

To solve a given problem:

1. Define system

Define the object you will consider and identify its mass m

m

4. Define a frame and project

• Project the Newton’s law along each axis separately.•Be careful with the SIGN!!

• Newton’s second law Fam

Pitfalls to avoid

This is an ABSOLUTE equation (vectors)

. Projection along specific axis

The projection is not an absolute equation: the sign depends on your choice of axis orientation. Be CONSISTENT with your choice of axis!

Tgmam

Vectorialequation

Example

m

T

mg

mgTma Tmgma

Axis choice

m

T

mg

z

Choice 1

m

T

mgz

Choice 2

Ch.5 Laws of motion 2/26/09

Ch.5&6 Laws of motion 2/26/09

• Newton’s second law Fam

Pitfalls to avoid

This is a VECTORIAL equation

• What if forces are in different directions?

Be careful: do not mix forces in different directions!!

Examples

m

T

mg

Take into account the direction, possibly by using inclination angles ().Project Newton’s law along each axis separately

H

mg

R

Tangential and radialacceleration

General case

V1

V2

V3

a

a a

V is tangential to the trajectory

• Tangential accelerationThe sign tells if the particlespeeds up or slows downat= dV/dt

• Centripetal accelerationThe centripetal acceleration

is toward the center of curvatureac= R2 = V2/R

Ch.6 Motion 2/26/09

Problem (Attwood machine)

m1

m2

Two objects of different mass are suspended at each end of a string

with a frictionless pulleyWill the system move?

If so, in which direction and with what acceleration?

Let’s apply Newton’s Lawon each object:

Fam

Object 1: m1a1 = T1 + m1g

Object 2: m2a2 = T2 + m2g

z

Let’s project these equations along z axis

Knowing that T1 = T2 and that a2= - a1 we get

m1a1 = T1 - m1gm2a2 = T2 - m2g

So T1 = m1a1 + m1gT2 = m2a2 + m2g

m1a1 + m1g = -m2a1 + m2g

(m2 - m1) g = (m2 + m1) a1

Ch.5&6 Newton’s law 2/26/09

gmm

mma

12

121

T1

T2

m1g

m2g

m1

m2

Two objects of different mass are suspended at each end of a string

with a frictionless pulleyWill the system move?

If so, in which direction and with what acceleration?

z

We have T1 = T2

and a2 = - a1

gmm

mma

12

121

T1

T2

m1g

m2g

• If m2 > m1 : then a1 > 0the red sphere moves down and green cube moves up

• If m2 < m1 : then a1 < 0the red sphere moves up and green cube moves down

Problem (Attwood machine)

Ch.5 Laws of motion 2/26/09

Work of a force

A particle moves under the action of a force Ffrom initial point A to final point B

dr

F

A

B

The total work done by the force F on the particle from point A to point B is

B

A

B

ABA rdFdWW

.

Ch.7 Work and energy 2/26/09

F

A

Bdr

0.

B

ABA rdFW

If at any time along the path, the force F is perpendicular to the displacement, then:

F

drA

B

cosABBA dFW

If the force is constant and working along a straight line

Conservative force

A force is conservative when: its work does not depend on the path.

The force conserves the energy

• Examples of conservative forces:- Gravity- Elastic force- Gravitational field- Electric force- Magnetic force- any constant force

0. rdFWloop

• Path independenceA

The work done by a conservative force on a closed path is zero

For conservative forces, we can express the work in terms of

potential energy

Ch.7 Work and energy 2/26/09

Gravity potential energy

We can express the work of the weight as a variation of a potential function Ep

mg

B

H

A

)(. BABA zzmgHmgW

BABA mgzmgzW

)( ABBA mgzmgzW

pBA EW

mgzEp

Ch.7 Work and energy 2/26/09

Elastic potential energy

L0

x0

F

)(2

22BABA xx

kW

x

FA

B

)22

( 22ABBA x

kx

kW

pBA EW

2

2

1kxEp

We can express the work of a spring force as a variation of the elastic potential Ep

Ch.7 Work and energy 2/26/09

amF

Work and kinetic energy

B

A

B

ABA rdamrdFW

..

Defining the kinetic energy

2

2

1mVK

Using Newton’s second law

Work- Kinetic energy theorem

ABBA KKKW

Ch.7 Work and energy 2/26/09

Mechanical energy

nccons WWK

ncp WEK

ncp WEK

ncmech WE

ncp WEK )(

pmech EKE

We define the mechanical energy Emech

as the sum of kinetic and potential energies

Ch.7 Work and energy 2/26/09

Closed System with conservative forces only

0 ncmech WE

Fcons

cstEmech

cstEK p

There are no non-conservative forces working

The mechanical energy is constant

ipifpf EKEK ,,

The mechanical energy is conserved between initial and final points

Ch.7 Work and energy 2/26/09

Read Textbook:

Chapter 9

Next Class

Homework assignment:

Today Feb 26th 7pm Problems 8: 5-7

Good luck on You exam!!

Tuesday March 3rd