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PHYS 1443 – Section 003 Lecture #9. Wednesday, Sept. 24, 2003 Dr. Jae hoon Yu. Forces of Friction Uniform and Non-uniform Circular Motions Resistive Forces and Terminal Velocity Newton’s Law of Gravitation. Homework #5 due at midnight next Thursday, Oct. 2!!. - PowerPoint PPT Presentation
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Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
1
PHYS 1443 – Section 003Lecture #9
•Forces of Friction•Uniform and Non-uniform Circular Motions•Resistive Forces and Terminal Velocity•Newton’s Law of Gravitation
Wednesday, Sept. 24, 2003Dr. Jaehoon Yu
Remember the first term exam on next Monday, Sept. 29!!
Homework #5 due at midnight next Thursday, Oct. 2!!
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
2
Forces of Friction
nf ss
Resistive force exerted on a moving object due to viscosity or other types frictional property of the medium in or surface on which the object moves.
Force of static friction, fs:
Force of kinetic friction, fk
The resistive force exerted on the object until just before the beginning of its movement
The resistive force exerted on the object during its movementnf kk
These forces are either proportional to velocity or normal force
Empirical Formula What does
this formula tell you?
Frictional force increases till it reaches to the limit!!
Beyond the limit, there is no more static frictional force but kinetic frictional force takes it over.
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
3
Example w/ FrictionSuppose a block is placed on a rough surface inclined relative to the horizontal. The inclination angle is increased till the block starts to move. Show that by measuring this critical angle, c, one can determine coefficient of static friction, s.
F
Free-bodyDiagram
x
yM a
Fg
nn
F= -Mg
fs=kn
yF
xF
s
Net force
x comp.
y comp.
sf
n
x
y
aM sg fnF
sgx fF sfMg sin 0 ns cMg sin
yMa gyFn cMgn cos 0 gyF cMg cos
n
Mg csin c
c
MgMg
cossin
ctan
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
4
Newton’s Second Law & Uniform Circular Motion
Fr
Fr
r
m
rvar
2
The centripetal acceleration is always perpendicular to velocity vector, v, for uniform circular motion.
The force that causes the centripetal acceleration acts toward the center of the circular path and causes a change in the direction of the velocity vector. This force is called centripetal force.
Are there forces in this motion? If so, what do they do?
rF
What do you think will happen to the ball if the string that holds the ball breaks? Why?
Based on Newton’s 1st law, since the external force no longer exist, the ball will continue its motion without change and will fly away following the tangential direction to the circle.
rmarvm
2
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
5
Example of Uniform Circular MotionA 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?
mFr rvar
2
Centripetal acceleration: rF
When does the string break?
When the centripetal force is greater than the sustainable tension.
Trvm
2
Calculate the tension of the cord when speed of the ball is 5.00m/s.
NrvmT 33.8
5.100.5500.0
22
smmTrv /2.12
500.05.10.50
rmarvm
2
T
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
6
Example of Banked Highway(a) For a car traveling with speed v around a curve of radius r, determine a formula for the angle at which a road should be banked so that no friction is required to keep the car from skidding.
rmvn
2
sin
yF
xF
smhrkmv /14/50
x comp.
y comp.x
y rman sin
sinmgn
0
mgn cos
rmvmgmgn
2
tancos
sinsin
grv2
tan
4.08.950
14tan2
rmvn
2
sin
0 mgn cos
(b) What is this angle for an expressway off-ramp curve of radius 50m at a design speed of 50km/h?
o224.0tan 1
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
7
Forces in Non-uniform Circular MotionThe object has both tangential and radial accelerations.
What does this statement mean?The object is moving
under both tangential and radial forces.
Fr
Ft
F
tr FFF
These forces cause not only the velocity but also the speed of the ball to change. The object undergoes a curved motion under the absence of constraints, such as a string.
22tr aaa How does the
acceleration look?
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
8
Example of Non-Uniform Circular MotionA ball of mass m is attached to the end of a cord of length R. The ball is moving in a vertical circle. Determine the tension of the cord at any instant when the speed of the ball is v and the cord makes an angle with vertical.
Tm
What are the forces involved in this motion?
tF
The gravitational force Fg and the radial force, T, providing tension.
R Fg=mg
At what angles the tension becomes maximum and minimum. What are the tensions?
singat
rF
cos
2
gRvmT
tangential comp.
Radial comp.
sinmg tma
cosmgT rmaRvm
2
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
9
Motion in Resistive ForcesMedium can exert resistive forces on an object moving through it due to viscosity or other types frictional property 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: 1. Forces linearly proportional to speed: Slowly moving or very small objects2. Forces proportional to square of speed: Large objects w/ reasonable speed
Air resistance, viscous force of liquid, etc
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
10
Resistive Force Proportional to Speed Since the resistive force is proportional to speed, we can write R=bv
m
Let’s consider that a ball of mass m is falling through a liquid.R
mgv RFF g This equation also tells you that
0 when , vgvmbg
dtdv
The above equation also tells us that as time goes on the speed increases and the acceleration decreases, eventually reaching 0. An object moving in a viscous medium will obtain speed to a certain speed (terminal speed) and then maintain the same speed without any more acceleration.
What does this mean?
What is the terminal speed in above case?
bmgvv
mbg
dtdv
t ;0
How do the speed and acceleration depend on time?
;0 when 0 ;1
tvebmgv m
bt
The time needed to reach 63.2% of the terminal speed is defined as the time constant, m/b.
0 xF
vmbg
dtdv
dtdvmmabvmgFy
;0 when ; tgagee
mb
bmg
dtdva
tm
bt
vmbge
mb
bmge
mb
bmg
dtdv tt
11
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
11
Newton’s Law of Universal GravitationPeople have been very curious about the stars in the sky, making observations for a long time. But the data people collected have not been explained until Newton has discovered the law of gravitation.
Every particle in the Universe attracts every other particle 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
principle mathematically? 212
21
rmmFg
1110673.6 GG is the universal gravitational constant, and its value is
This constant is not given by the theory but must be measured by experiment.
With G 212
21
rmmGFg
Unit? 22 / kgmN
This form of forces is known as an inverse-square law, because the magnitude of the force is inversely proportional to the square of the distances between the objects.
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
12
It means that the force exerted on the particle 2 by particle 1 is attractive force, pulling #2 toward #1.
More on Law of Universal GravitationConsider two particles exerting gravitational forces to each other.
Gravitational force is a field force: Forces act on object without physical contact between the objects at all times, independent of medium between them.
12221
12 r̂rmmGF
The gravitational force exerted by a finite size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distributions was concentrated at the center.
m1
m2r
F21
F12
12r̂ Two objects exert gravitational force on each other following Newton’s 3rd law.
Taking as the unit vector, we can write the force m2 experiences as
12r̂
What do you think the negative sign mean?
gF
How do you think the gravitational force on the surface of the earth look?
2E
E
RmMG
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
13
Free Fall Acceleration & Gravitational ForceWeight of an object with mass m is mg. Using the force exerting on a particle of mass m on the surface of the Earth, one can get
•The gravitational acceleration is independent of the mass of the object•The gravitational acceleration decreases as the altitude increases•If the distance from the surface of the Earth gets infinitely large, the weight of the object approaches 0.
What would the gravitational acceleration be if the object is at an altitude h above the surface of the Earth?
mg
What do these tell us about the gravitational acceleration?
gF
2E
E
RmMG
g 2E
E
RMG
'mg 2rmMG E 2hR
mMGE
E
'g 2hRMGE
E
Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu
14
Example for Gravitational ForceThe international space station is designed to operate at an altitude of 350km. When completed, it will have a weight (measured on the surface of the Earth) of 4.22x106N. What is its weight when in its orbit?
The total weight of the station on the surface of the Earth is
Therefore the weight in the orbit is
GEF
OF
Since the orbit is at 350km above the surface of the Earth, the gravitational force at that height is
MEE
OF
mg 2E
E
RmMG N61022.4
'mg 2hRmMG
E
E
GEE
E FhR
R2
2
GEE
E FhR
R2
2
N66256
26
1080.31022.41050.31037.6
1037.6