Tuesday June 14, 2005 1 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt PHYS 1443 – Section 001 Lecture #8 Tuesday June 14, 2005 Dr. Andrew Brandt Accelerated

Tuesday June 14, 2005 1 PHYS 1443-001, Summer I 2005Dr. Andrew Brandt

PHYS 1443 – Section 001Lecture #8

Tuesday June 14, 2005Dr. Andrew Brandt

• Accelerated Frames• Work• Kinetic and Potential Energy


Announcements• Test 2 Thursday 6/16• Homework:

– HW4 on ch5 due Tuesday 6/14 at 8pm– HW5 on ch 6 due Wednesday 6/15 at 6pm


Motion in Accelerated FramesNewton’s laws are valid only when observations are made in an inertial frame of reference. What happens in a non-inertial frame?Fictitious forces are needed to apply Newton’s second law in an accelerated frame.

This force does not exist when the observations are made in an inertial reference frame.

What does this mean and why is this true?

Let’s consider a “free” ball inside a box that is moving under uniform circular motion.

We see that the box has a radial force exerted on it but none on the ball directly

How does this motion look like in an inertial frame (or frame outside a box)?

rF

rHow does this motion look like in the box?

The ball appears to experience a force that moves it in a curved path towards the wall of the box

Why?

According to Newton’s first law, the ball should continue in its original direction but since the box is turning, the ball feels like it is being pushed toward the wall relative to everything else in the box.

v


Non-Inertial Frame

Example of Motion in Accelerated FramesA ball of mass m is hung by a cord to the ceiling of a boxcar that is moving with an acceleration a. What do the inertial observer at rest and the non-inertial observer traveling inside the car conclude? How do their conclusions differ?

m

This is how the ball looks like no matter which frame you are in.

F

Inertial Frame

How do the free-body diagrams look for two frames?

Fg=mgm

T

Fg=mgm

T

Ffic

ac

How are the motions interpreted in these two frames? Any differences?

For an inertial frame observer, the forces being exerted on the ball are only T and Fg. The acceleration of the ball is the same as that of the box car and is provided by the x component of the tension force.

F

In the non-inertial frame observer, the forces being exerted on the ball are T, Fg, and Ffic.

For some reason the ball is under a force, Ffic, that provides acceleration to the ball.

xF

yF

cos

mgT tangac

xF

yF

cos

mgT

sinTmaF ficfic

tanga fic While the mathematical expression of the acceleration of the ball is identical to that of inertial frame observer’s, the cause of the force is dramatically different.

TF g

xma cma sinT

mgT cos 0

ficg FTF

ficFT sin 0

mgT cos 0


Scalar (Dot) Product of Two Vectors • Product of magnitude of the two vectors and the cosine of the

angle between them BA

• Operation is commutative cosBABA

• Operation follows distribution law of multiplication

CABA

• How does the scalar product look in terms of components?

kAjAiAA zyx

kBjBiBB zyx

kBjBiBkAjAiABA zyxzyx termscrosskkBAjjBAiiBA zzyyxx

kkjjii

ikkjji• Scalar products of Unit Vectors

zzyyxx BABABABA

cosBA

cosAB AB

CBA

1 0

Is this a vector or a scalar?

=0


x

y

Work Done by a Constant ForceWork in physics is done only when a sum of forces exerted on an object causes motion of the object.

M

F Free Body

DiagramM

d gMF G

NF

F

Which force did the work?

W

Force F

How much work did it do?

What does this mean?Physical work is done only by the component of the force along the movement of the object.

Unit?Joule)(for J

mN

Work is energy transfer!!

dF cosFd


Example of Work w/ Constant ForceA man cleaning a floor pulls a vacuum cleaner with a force of magnitude F=50.0N at an angle of 30.0o with East. Calculate the work done by the force on the vacuum cleaner as it is displaced by 3.00m to the East.

Does work depend on mass of the object being worked on?

M

F

M

d

W

JW 13030cos00.30.50

Yes

Why don’t I see the mass term in the work at all then?

It is reflected in the force. If the object has smaller mass, its would take less force to move it the same distance as the heavier object. So it would take less work.

dF cosdF


Example of Work by Scalar ProductA particle moving in the xy plane undergoes a displacement d=(2.0i+3.0j)m as a constant force F=(5.0i+2.0j) N acts on the particle.

a) Calculate the magnitude of the displacement and that of the force.

d

F

b) Calculate the work done by the force F.

W

jiji 0.20.50.30.2 )(166100.20.30.50.2 Jjjii

Y

X

d F

Can you do this using the magnitudes and the angle between d and F?

cosdFdFW

22yx dd m6.30.30.2 22

22yx FF N4.50.20.5 22

dF


Work Done by Varying Force• If the force depends on position of the object during the motion

– one must consider work in small segments of the position where the force can be considered constant

– Then add all the work-segments throughout the entire motion (xi xf)

W

f

i

x

xx xFW

0lim

f

i

x

xx

x

F x

( ) f

i

x

ixxW net F dx

– If more than one force is acting, the net work is done by the net force

In the limit where x0

An example of a force that depends on position is the spring force kxFs

The work done by the spring force is

W

f

i

x

xxF dx W

max

0

sxF dx

max

0

xkx dx

21

2kx

xF x


Kinetic Energy and Work-Kinetic Energy Theorem• Some problems are hard to solve using Newton’s second law

– If forces acting on the object during the motion are complicated– Relate the work done on the object by the net force to the change of the

speed of the object

MF

M

d

vi vf

Suppose net force F was exerted on an object over a displacement d to increase its speed from vi to vf.

The work on the object by the net force F is

cos 0W F d ma d ma d ur ur

tvvd if 2

1 a

W

Displacement Acceleration

W

Work 21

2KE mv

Kinetic Energy

Work The work done by the net force causes a change of object’s kinetic energy.

dma

tvv

t

vvm if

if 2

1 22

2

1

2

1if mvmv

t

vv if

22

2

1

2

1if mvmv if KEKE KE

Work-Kinetic Energy Theorem


Example of Work-KE TheoremA 6.0kg block initially at rest is pulled to the East along a horizontal, frictionless surface by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m.

Work done by the force F is

From the work-kinetic energy theorem, we know

Since the initial speed is 0, the above equation becomes:

M F M

d

vi=0 vf

22

2

1

2

1if mvmvW

2

2

1fmvW

fv Solving the equation for vf, we obtain 2W

m

2 363.5 /

6.0m s

W cosF d ur ur

12 3.0cos 0 36 J


Work and Energy Involving Kinetic Friction• How will friction affect the work that can be done by a

force?– Does static friction matter?

M M

d

vf

Frictional force Ffr works on the object to slow it down

The work on the object by friction Ffr is

frW

The final kinetic energy of an object, taking into account its initial kinetic energy, friction force and other source of work, is

Ffr

KE

fKE

t=0, KEi Friction,Engine work

t=T, KEf

No, since there must be motion for work to occur.

cos 180frF d frF d frF d

iKE W frF d

Reduces it, of course!

vi


Example of Work with FrictionA 6.0kg block initially at rest is pulled to the East along a horizontal surface with a coefficient of kinetic friction k=0.15 by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m.

Work done by the force F is

Thus the total work is

M F M

d=3.0m

vf

fv

Work done by friction Fk is

Fk

kkW F d ur ur

g

J26180cos0.38.90.615.0

W

Using work-kinetic energy theorem and the fact that initial speed is 0, we obtain

W Solving the equation

for vf, we obtain

cosdmgk

kF WW )(102636 J

F kW W 21

2 fmv 2W

m

2 101.8 /

6.0m s

coskF d uur ur

FW cosF d ur ur

12 3.0cos 0 36 J vi=0

What’s another way to solvethis problem?


Kinetic Energy at High SpeedThe laws of Newtonian mechanics are no longer valid for objects moving at a speed close to that of light, c. It must be generalized for these special cases. Theory of relativity. The kinetic energy must be

modified to reflect the fact that the object is moving very high speed.

The speed of an object cannot be faster than light in vacuum. Have not seen any particle that goes faster than light, yet.However this equation must satisfy the Newtonian expression for smaller speeds!!

11

12

2

cv

mcK

What does this expression tell you?

1...8

3

2

111

1

142

2

2

2

c

v

c

vmc

cv

mcK

22

22

2

2

1

2

11

2

11 mv

c

vmc

c

vmcK


Potential EnergyEnergy associated with a system of objects Stored energy which has the potential or the possibility to do work or to convert stored E to kinetic energy

What does this mean?

In order to describe potential energy, U, a system must be defined.

What are some other forms of energy?

The concept of potential energy can only be used under the special class of forces called, conservative forces which results in principle of conservation of mechanical energy.

Mechanical Energy

Biological EnergyElectromagnetic

EnergyNuclear Energy

Chemical Energy

ME

These different types of energies are stored in the universe in many different forms!!!

If one takes into account ALL forms of energy, the total energy in the entire universe is conserved. It just transforms from one form to another.

i iKE PE f fKE PE


Gravitational Potential Energy

When an object is falling, a gravitational force, Mg, performs work on the object, increasing its kinetic energy. The potential energy of an object at a height y (its the potential to work ) is expressed as

m

yf

m

mgyigU

What does this mean?

gW The work performed on the object by the gravitational force as the brick goes from yi

to yf is:

mgyU g mgy

i fmgy mgy

Work by the gravitational force as the brick goes from yi to yf is the negative of the change in the system’s potential energy

Potential energy was lost in order for gravitational force to increase the brick’s kinetic energy.

cosgF y ur ur

gF yur ur

gU i fU U

gF y ur ur

Potential energy given to an object by the gravitational field due to its height above the surface of the Earth

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Tuesday June 14, 2005 1 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt PHYS 1443 – Section 001 Lecture #8 Tuesday June 14, 2005 Dr. Andrew Brandt Accelerated