Yu 1 PHYS 1441 Section 001 Lecture 10 Thur sday June 19 2014 Dr Jae hoon Yu Uniform Circular Motion Centripetal Acceleration Unbanked and Banked highways Newtons Law of Universal ID: 658924
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Slide1
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
1
PHYS 1441 – Section 001Lecture #10
Thursday, June 19, 2014Dr. Jaehoon Yu
Uniform Circular MotionCentripetal AccelerationUnbanked and Banked highwaysNewton’s Law of Universal GravitationWeightlessnessWork done by a constant force
Today’s homework is homework
#6,
due
11pm
,
Tues
day
, June
24!
!Slide2
Announcements
Quiz 3
Beginning of the class Monday, June 23
Covers CH 4.7 to what we finish todayBring 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!
Mid-term resultClass average: 67/97Equivalent to 69.1/100Previous exam: 61.8/100Top score: 89/97Mid-term grade discussion bottom half of the classThursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu2Slide3
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Reminder: Special
Project
#3
Using the fact that g=9.80m/s
2
on the
Earth’s
surface, find the average density of the Earth.
Use the following information
only but without computing the volume explicitly
The gravitational constant
The radius of the Earth
20 point extra credit
Due: Monday, June 23You must show your OWN, detailed work to obtain any credit!! Much more than in this lecture note!
3Slide4
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Uniform circular motion is the motion of an object traveling at a constant
speed
on a circular path.
Definition of the Uniform Circular Motion
4Slide5
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Let
T
be the period of this motion, the time it takes for the object to travel once around the complete circle whose radius is r.
Speed of a uniform circular motion?
5Slide6
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
The wheel of a car has a radius of 0.29m and is being rotated at 830 revolutions per minute on a tire-balancing machine. Determine the speed at which the outer edge of the wheel is moving.
Ex. : A Tire-Balancing Machine
6Slide7
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Newton’s
Second Law &
Centripetal Force
The
centripetal *
acceleration is always perpendicular to the velocity vector,
v
, and points to the center of the axis (radial direction) in a uniform circular motion.
The force that causes the centripetal acceleration acts toward the center of the circular path and causes the change in the direction of the velocity vector. This force is called the
centripetal force.
Are there forces in this motion? If so, what do they do?
What do you think will happen to the ball if the string that holds the ball breaks?
The external force no longer exist. Therefore, based on
Newton’s
1st law,
the ball will continue its motion without changing its velocity and will fly away following the tangential direction to the circle.
*
Mirriam
Webster: Proceeding or acting in
the
direction toward
the
center or axis
7Slide8
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Ex. Effect of Radius on Centripetal Acceleration
The bobsled track at the 1994 Olympics in Lillehammer, Norway, contain turns with radii of 33m and 23m. Find the centripetal acceleration at each turn for a speed of 34m/s, a speed that was achieved in the
two–man event. Express answers as multiples of g=9.8m/s2.
Centripetal acceleration:
R=33m
R=24m
8Slide9
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Example 5.1: Uniform
Circular Motion
A ball of mass 0.500kg is attached to the end of a 1.50m long cord. The ball is moving in a horizontal circle. If the string can withstand maximum tension of 50.0 N, what is the maximum speed the ball can attain before the cord breaks?
Centripetal acceleration:
When does the string break?
when the required centripetal force is greater than the sustainable tension.
Calculate the tension of the cord when speed of the ball is 5.00m/s.
9Slide10
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
On an unbanked curve, the static frictional force provides the centripetal force.
Unbanked Curve and Centripetal Force
10Slide11
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
On a frictionless banked curve, the centripetal force is the
horizontal component of the normal force. The vertical
component of the normal force balances the car’s weight.
Banked Curves
11Slide12
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Ex.
The Daytona 500
The Daytona 500 is the major event of the NASCAR season. It is held at the Daytona International Speedway in Daytona, Florida. The turns in this oval track have a maximum radius (at the top) of r=316m and are banked steeply, with θ=31o
. Suppose these maximum radius turns were frictionless. At what speed would the cars have to travel around them?
x comp.
x
y
y comp.
12Slide13
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Newton’s
Law of Universal Gravitation
People 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
Unit?
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.
13Slide14
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Ex. Gravitational Attraction
What is the magnitude of the gravitational force that acts on each particle in the figure, assuming m
1
=12kg, m
2
=25kg, and r=1.2m?
14Slide15
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Why does the Moon orbit the Earth?
15Slide16
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
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,
F
g
Weight of an object with mass M is
By measuring the forces one can determine masses. This is why you can measure mass using the spring scale.
What is the SI unit of weight?
N
16Slide17
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Gravitational Acceleration
What is the SI unit of g?
m/s
2
Gravitational acceleration at distance r from the center of the earth!
17Slide18
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Gravitational force on the surface of the earth:
Magnitude of the gravitational acceleration on the surface of the Earth
18Slide19
Thursday, June 19, 2014
Example for Universal Gravitation
Using the fact that g=9.80m/s
2
on the
Earth’s surface, find the average density of the Earth.
Since the gravitational acceleration is
Therefore the density of the Earth is
Solving for M
E
Solving for g
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
19Slide20
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
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
What
acts as
the centripetal force?
The gravitational force of the earth pulling the satellite!
20Slide21
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the
earth’s
surface.
Ex. Orbital Speed of the Hubble Space Telescope
21Slide22
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Period of a Satellite in an Orbit
Speed of a satellite
Period of a satellite
Square either side and solve for T2
This is applicable to any satellite or even for planets and moons.
Kepler’s
3
rd
Law
22Slide23
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
Geo-synchronous Satellites
What period should these satellites have?
Satellite TV
Global Positioning System (GPS)
The same as the earth!!
24 hours
23Slide24
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
In each case, what is the weight recorded by the scale?
Ex. Apparent Weightlessness and Free Fall
0
24
0Slide25
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
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
25Slide26
Thursday, June 19, 2014
Motion in Resistive Forces
Medium can exert resistive forces on an object moving through it due to viscosity or other types frictional properties of the medium.
These forces are exerted on moving objects in opposite direction of the movement.
Some examples?
These forces are proportional to such factors as speed. They almost always increase with increasing speed.
Two different cases of proportionality:
Forces linearly proportional to speed: Slowly moving or very small objects
Forces proportional to square of speed: Large objects w/ reasonable speed
Air resistance, viscous force of liquid, etc
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
26Slide27
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
27
x
y
Work Done by a Constant Force
A meaningful work in physics is done only when the net forces exerted on an object changes the energy of the object.
M
F
θ
Free Body
Diagram
M
d
θ
Which force did the work?
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?
Work is an
energy transfer
!!
Why?
What kind?
ScalarSlide28
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? NoWhy 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.
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
28Slide29
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
29
Work done by a constant force
sSlide30
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
30
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?
Scalar products of Unit Vectors
=0Slide31
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
31
Example of Work by Scalar Product
A particle moving on the xy plane undergoes a displacement
d
=(2.0
i
+3.0
j
)m as a constant force
F
=(5.0
i
+2.0
j
) 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.
Y
X
d
F
Can you do this using the magnitudes and the angle between
d
and
F
?Slide32
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
32
Ex. Pulling A Suitcase-on-Wheel
Find the work done by a 45.0N force in pulling the suitcase in the figure at an angle 50.0
o
for a distance s=75.0m.
Does work depend on mass of the object being worked on?
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?Slide33
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
33
Ex. 6.1 Work done on a crate
A person pulls a 50kg crate 40m along a horizontal floor by a constant force F
p
=100N, which acts at a 37
o
angle as shown in the figure. The floor is rough and exerts a friction force F
fr
=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?
F
G
=-mg
So the net work on the crate
Work done on the crate by F
G
F
p
F
fr
Which force performs the work on the crate?
F
p
F
fr
Work done on the crate by F
p
:
Work done on the crate by F
fr
:
This is the same as
F
N
=+mg
Work done on the crate
by F
NSlide34
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
34
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?
What does the negative work mean?
The gravitational force does the work on the weight lifter!Slide35
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
35
The truck is accelerating at a rate of +1.50 m/s
2
. 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
F
g
Normal force force on the crate,
F
N
Static frictional force on the crate,
f
sSlide36
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
36
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
F
g
Work done by
Normal force force on the crate,
F
N
Work done by the static frictional force on the crate,
f
s
Which force did the work?
Static frictional force on the crate,
f
s
How?
By holding on to the crate so that it moves with the truck!Slide37
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
37
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
M
ΣF
M
s
v
i
v
f
Suppose net force Σ
F
was exerted on an object for displacement d to increase its speed from
v
i
to
v
f
.
The work on the object by the net force Σ
F
is
Using the kinematic equation of motion
Work
Kinetic Energy
Work
Work done by the net force causes change in the
object’s
kinetic energy.
Work-Kinetic Energy TheoremSlide38
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
38
When a net external force by the jet engine does work on
an
object, the kinetic energy of the object changes according to
Work-Kinetic Energy TheoremSlide39
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
39
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
×10
9
m, what is its final speed?
Ex. Deep Space 1
Solve for v
fSlide40
Thursday, June 19, 2014
PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu
40
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.