Monday, June 23, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 1 PHYS 1441 – Section 001 Lecture #11 Monday, June 23, 2014 Dr. Jaehoon Yu Newton’s Law.

Monday, June 23, 2014 PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu

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PHYS 1441 – Section 001Lecture #11

Monday, June 23, 2014Dr. Jaehoon Yu

• Newton’s Law of Universal Gravitation• Weightlessness• Work done by a constant force• Multiplication of Vectors• Work-Kinetic Energy Theorem

Announcements• Term exam #2

– In class this Wednesday, June 25– Non-comprehensive exam– Covers CH 4.7 to what we finish tomorrow, Tuesday, June 24– Bring your calculator but DO NOT input formula into it!

• Your phones or portable computers are NOT allowed as a replacement!– You can prepare a one 8.5x11.5 sheet (front and back) of

handwritten formulae and values of constants for the exam no solutions, derivations or definitions!• No additional formulae or values of constants will be provided!


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Newton’s Law of Universal GravitationPeople have been very curious about the stars in the sky, making observations for a long~ time. The data people collected, however, have not been explained until Newton has discovered the law of gravitation.

Every object in the Universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

How would you write this law mathematically?

G is the universal gravitational constant, and its value is

This constant is not given by the theory but must be measured by experiments.

With G gF

Unit? 22 / kgmN

This form of forces is known as the inverse-square law, because the magnitude of the force is inversely proportional to the square of the distances between the objects.

1m 1 22

12

mm

r2m G

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F

Ex. Gravitational AttractionWhat is the magnitude of the gravitational force that acts on each particle in the figure, assuming m1=12kg, m2=25kg, and r=1.2m?

11 2 22

12 kg 25 kg6.67 10 N m kg

1.2 m

81.4 10 N

1 22

mmGr

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Why does the Moon orbit the Earth?

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Gravitational Force and Weight

Since weight depends on the magnitude of gravitational acceleration, g, it varies depending on geographical location.

The attractive force exerted on an object by the Earth

Gravitational Force, Fg

Weight of an object with mass M is W

By measuring the forces one can determine masses. This is why you can measure mass using the spring scale.

Mg

What is the SI unit of weight? N

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W

W

g

Gravitational Acceleration

2EM m

Gr

mg

2EM m

Gr

mg

2EMGr

What is the SI unit of g? m/s2

Gravitational acceleration at distance r from the center of the earth!

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Gravitational force on the surface of the earth:

Magnitude of the gravitational acceleration on the surface of the Earth

g

29.80 m s

GF 2EM m

Gr

2E

E

M mGRmg

2E

E

MGR

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Monday, June 23, 2014

Example for Universal GravitationUsing the fact that g=9.80m/s2 on the Earth’s surface, find the average density of the Earth.

g

Since the gravitational acceleration is

EM

Therefore the density of the Earth is

2E

E

R

MG

2111067.6

E

E

R

M

E

E

V

M

3

2

4E

E

R

GgR

EGR

g

4

3

33611

/1050.51037.61067.64

80.93mkg

Solving for ME

2ER g

G

gF 2E

E

R

mMG mg Solving for g

PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu

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There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.

Satellite in Circular Orbits

cF

EGM

r

What acts as the centripetal force?The gravitational force of the earth pulling the satellite!

2EmM

Gr

2v

mr

v

2 EGMv

r

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Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the earth’s surface.

11 2 2 24

6 3

6.67 10 N m kg 5.98 10 kg

6.38 10 m 598 10 m

Ex. Orbital Speed of the Hubble Space Telescope

v

37.56 10 m s 16900mi h

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2EGM rv

r T

3 22

E

rT

GM

Period of a Satellite in an Orbit

Speed of a satellite

Period of a satellite

22EGM r

r T

2 32 2

E

rT

GM

Square either

side and solve for T2

This is applicable to any satellite or even for planets and moons.

Kepler’s 3rd Law

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Geo-synchronous Satellites

What period should these satellites have?

Satellite TV

Global Positioning System (GPS)

The same as the earth!! 24 hours

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In each case, what is the weight recorded by the scale?

Ex. Apparent Weightlessness and Free Fall

0

14

0


At what speed must the surface of the space station move so that the astronaut experiences a push on his feet equal to his weight on earth? The radius is 1700 m.

Ex. Artificial Gravity

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x

y

Work Done by a Constant ForceA meaningful work in physics is done only when the net forces exerted on an object changes the energy of the object.

M

Fθ Free Body

DiagramM

d

θ

Which force did the work?

W

Force

How much work did it do?

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

Unit? N m

Work is an energy transfer!!

cosFd θ (for Joule)J

Why? What kind? Scalar

Let’s think about the meaning of work!• A person is holding a grocery bag and

walking at a constant velocity.• Are his hands doing any work ON the bag?

– No– Why not?– Because the force hands exert on the bag, Fp,

is perpendicular to the displacement!!– This means that hands are not adding any

energy to the bag.• So what does this mean?

– In order for a force to perform any meaningful work, the energy of the object the force exerts on must change due to that force!!

• What happened to the person?– He spends his energy just to keep the bag up

but did not perform any work on the bag.Monday, June 23, 2014 PHYS 1441-001, Summer 2014

Dr. Jaehoon Yu17


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W cos 0

Work done by a constant force

s cos90

cos180

1

0

1

s


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

angle between them

• Operation is commutative

• Operation follows the distribution law of multiplication

• How does scalar product look in terms of components?

x x y y z zA B i i A B j j A B k k

kkjjii

ikkjji• Scalar products of Unit Vectors 1 0

x xA B y yA B z zA B =0

cross terms


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Example of Work by Scalar ProductA particle moving on 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.

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?

W

22yx dd m6.30.30.2 22

22yx FF N4.50.20.5 22


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Ex. Pulling A Suitcase-on-WheelFind the work done by a 45.0N force in pulling the suitcase in the figure at an angle 50.0o for a distance s=75.0m.

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

W

Yes

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

It is reflected in the force. If an object has smaller mass, it would take less force to move it at the same acceleration than a heavier object. So it would take less work. Which makes perfect sense, doesn’t it?


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Ex. 6.1 Work done on a crateA person pulls a 50kg crate 40m along a horizontal floor by a constant force Fp=100N, which acts at a 37o angle as shown in the figure. The floor is rough and exerts a friction force Ffr=50N. Determine (a) the work done by each force and (b) the net work done on the crate.

What are the forces exerting on the crate?

FG=-mg

So the net work on the crate

Work done on the crate by FG

Fp Ffr

Which force performs the work on the crate?

Fp Ffr

Work done on the crate by Fp:

Work done on the crate by Ffr:

This is the same as

FN=+mg

Work done on the crate by FN


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cosF sθ

cos F sθ

W

Ex. Bench Pressing and The Concept of Negative Work

A weight lifter is bench-pressing a barbell whose weight is 710N a distance of 0.65m above his chest. Then he lowers it the same distance. The weight is raised and lowered at a constant velocity. Determine the work in the two cases.

What is the angle between the force and the displacement?

0 Fs

W 180 Fs

What does the negative work mean? The gravitational force does the work on the weight lifter!


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The truck is accelerating at a rate of +1.50 m/s2. The mass of the crate is 120-kg and it does not slip. The magnitude of the displacement is 65 m. What is the total work done on the crate by all of the forces acting on it?

Ex. Accelerating a Crate

What are the forces acting in this motion?

Gravitational force on the crate, weight, W or Fg

Normal force force on the crate, FN

Static frictional force on the crate, fs


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Ex. Continued…Lets figure out what the work done by each of the forces in this motion is.

Work done by the gravitational force on the crate, W or Fg

Work done by Normal force force on the crate, FN

Work done by the static frictional force on the crate, fs

W sf

W cos 90ogF s 0

W cos 90oNF s 0

ma 2120 kg 1.5m s 180N

sf s

Which force did the work? Static frictional force on the crate, fsHow? By holding on to the crate so that it moves with the truck!


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

– If forces exerting on an 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

MΣF

M

s

vi vf

Suppose net force ΣF was exerted on an object for displacement d to increase its speed from vi to vf.

The work on the object by the net force ΣF isW

2as as

W

Using the kinematic equation of motion

W

Work KE Kinetic Energy

Work Work done by the net force causes change in the object’s kinetic energy.

ma s 2 20

1

2 fm v v 2 20

1 1

2 2fmv mv

2 20

2fv v

2 21 1

2 2f imv mv f iKE KE KE

Work-Kinetic Energy Theorem

cos 0ma s ma s

21

2mv

2 20fv v


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When a net external force by the jet engine does work on an object, the kinetic energy of the object changes according to

W

Work-Kinetic Energy Theorem


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The mass of the space probe is 474-kg and its initial velocity is 275 m/s. If a 56.0-mN force acts on the probe parallel through a displacement of 2.42×109m, what is its final speed?

Ex. Deep Space 1

Solve for vf


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A satellite is moving about the earth in a circular orbit and an elliptical orbit. For these two orbits, determine whether the kinetic energy of the satellite changes during the motion.

Ex. Satellite Motion and Work By the Gravity

For a circular orbit

For an elliptical orbit

No change! Why not?

Gravitational force is the only external force but it is perpendicular to the displacement. So no work.

Changes! Why?Gravitational force is the only external force but its angle with respect to the displacement varies. So it performs work.

Monday, June 23, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 1 PHYS 1441 – Section 001 Lecture #11 Monday, June 23, 2014 Dr. Jaehoon Yu Newton’s Law.

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