PHY 113 C Fall 2013 -- Lecture 6 19/12/2013
PHY 113 A General Physics I11 AM-12:15 PM MWF Olin 101
Plan for Lecture 6:
Chapters 5 & 6 – More applications of Newton’s laws
1. Friction and other dissipative forces
2. Forces in circular motion
PHY 113 C Fall 2013 -- Lecture 6 29/12/2013
PHY 113 C Fall 2013 -- Lecture 6 39/12/2013
Isaac Newton, English physicist and mathematician (1642—1727)
http://www.newton.ac.uk/newton.html
1. In the absence of a net force, an object remains at constant velocity or at rest.
2. In the presence of a net force F, the motion of an object of mass m is described by the form F=ma.
3. F12 =– F21.
PHY 113 C Fall 2013 -- Lecture 6 49/12/2013
Webassign question from assignment #5
PHY 113 C Fall 2013 -- Lecture 6 59/12/2013
Webassign question from assignment #5 – continued
)m. ˆ4.40+ ̂(1.60
at rest at initially is that kg 1.90 of particle aon act N, ) ˆ6.30 + ˆ(2.70 = and N ) ˆ4.45 + ˆ(2.80 = forces, Two
i
21
jir
jiFjiF
m
m
F1
F2 2
21
21
ttt
ttm
m
ii
i
net
net
avrr
avv
Fa
aFFF
Nˆ75.10ˆ50.5
Nˆ6.304.45ˆ2.702.80
N, ) ˆ6.30 + ̂(2.70 ) ˆ4.45 + ˆ(2.80 = 21
ji
ji
jijiFF
PHY 113 C Fall 2013 -- Lecture 6 69/12/2013
Webassign question from assignment #5 – continued
m
F1
F2
2
21
21
ttt
ttm
m
ii
i
net
net
avrr
avv
Fa
aFFF
sttt
s
/mˆ658.5ˆ895.20
/mˆ658.5ˆ895.2kg90.1
Nˆ75.10ˆ50.5
Nˆ75.10ˆ50.5=
2
21
jiv
jijia
jiFF
PHY 113 C Fall 2013 -- Lecture 6 79/12/2013
Webassign question from Assignment #4
T T
mg mg sin q
aa
amgmTamgmT
22
11
sin q
PHY 113 C Fall 2013 -- Lecture 6 89/12/2013
Newton’s second law aF mnet
applied
g
FnFTF
jF
:surface a tonormal forceSupport :rope masslessin tension todue Force
ˆ :surface sEarth'near Gravity
:known have weForces
n
mg
TaTF
:ssfrictionle is surface Ifmnet
PHY 113 C Fall 2013 -- Lecture 6 99/12/2013
mg
nT
Another example of motion along a frictionless surface
jiF
jnjiT
jnTF
ˆsinˆ cos
ˆ ˆ sinˆ cos
ˆ
mgnTT
nTT
mg
net
net
iF ˆ cos
0sin If
q
q
T
mgnT
net
PHY 113 C Fall 2013 -- Lecture 6 109/12/2013
mg
nT
iF ˆ cos
0sin If
q
q
T
mgnT
net
iclicker exercise:
Assuming T>0, how does the suitcase move along surface? A. It moves at constant
velocity.B. It accelerates.C. Whether it moves or
not depends on magnitude of T.
PHY 113 C Fall 2013 -- Lecture 6 119/12/2013
In presence of friction force:
mg
nT
f
j
iF
ijnTF
ˆsin
ˆ cos
ˆˆ
mgnT
fT
fmg
net
net
q
q
PHY 113 C Fall 2013 -- Lecture 6 129/12/2013
Friction forces
The term “friction” is used to describe the category of forces that oppose motion. One example is surface friction which acts on two touching solid objects. Another example is air friction. There are several reasonable models to quantify these phenomena.
Surface friction:
N
Ff applied
Material-dependent coefficient
Normal force between surfaces
Air friction:
speedhigh at speed lowat
2vKKv
D
K and K’ are materials and shape dependent constants
PHY 113 C Fall 2013 -- Lecture 6 139/12/2013
Models of surface friction forces
(applied force)
surfa
ce fr
ictio
n fo
rce
fs,max=sn
Coefficients s , k depend on the surfaces; usually, s > k
PHY 113 C Fall 2013 -- Lecture 6 149/12/2013
PHY 113 C Fall 2013 -- Lecture 6 159/12/2013
Surface friction models
F-fs = 0 if F < sn=smg
if F>sn=smg , then F-fk=ma (fk= kmg)
mmgFa k
Static friction case:
Kinetic friction case:
PHY 113 C Fall 2013 -- Lecture 6 169/12/2013
mg
f
n
qmg sin q
mg cos q
Consider a stationary block on an incline:
sin 0sin:surface along Forces
cos 0cos:surface tonormal Forces
mgfmgf
mgnmgn
PHY 113 C Fall 2013 -- Lecture 6 179/12/2013
mg
f
n
qmg sin q
mg cos q
Consider a stationary block on an incline:
What happens when f>fs,max?
PHY 113 C Fall 2013 -- Lecture 6 189/12/2013
mg
f
n
qmg sin q
mg cos q
Consider a stationary block on an incline:
What happens when f=fs,max? qq
qqqqq
sincosThen
cos Ifsin 0sincos 0cos
max,
mgmg
mgnffmgfmgfmgnmgn
S
SSS
q tanS
PHY 113 C Fall 2013 -- Lecture 6 199/12/2013
V (constant)
iclicker exercise:Suppose you place a box on an inclined surface as shown in the figure and you notice that the box slides down the incline at constant velocity V. Which of the following best explains the phenomenon:
A. There is no net force acting on the box. B. There is a net force acting on the box.
q
PHY 113 C Fall 2013 -- Lecture 6 209/12/2013
V (constant)
q
mg
n
f=kn
Consider a block sliding down an inclined surface; constant velocity case
qqq
qqqq
sincosThen cos If
sin 0sincos 0cos
mgmgmgnf
mgfmgfmgnmgn
K
KK
q tanK
PHY 113 C Fall 2013 -- Lecture 6 219/12/2013
mg
f
n
qmg sin q
mg cos q
Summary
slip about tojust isblock when tanelocityconstant vat movesblock when tan
S
K
PHY 113 C Fall 2013 -- Lecture 6 229/12/2013
A block of mass 3 kg is pushed up against a wall by a force P that makes an angle of q=50o
with the horizontal. s=0.25. Determine the possible values for the magnitude of P that allow the block to remain stationary.
0cos :forces Horizontal0sin :forces Vertical s
NPmgPN
qqf
mg
N
f
PHY 113 C Fall 2013 -- Lecture 6 239/12/2013
0cos :forces Horizontal0sin :forces Vertical s
NPmgPN
qqf
mg
N
f
N 48.57N 72.31
50cos25.050sinN 8.93
cossin :for Solving
0sincoscos
s
s
ooP
mgPP
mgPPPN
qqq
PHY 113 C Fall 2013 -- Lecture 6 249/12/2013
Models of air friction forces
2
air
:ocitieslarger velFor
:s velocitiesmallFor
DvF
bvF
air
mg
bv
mamgbvv
:0 assuming and asdirection up Denoting
mbteb
mgtv
dtdvmmgbv
S
/1)(
:equation aldifferenti olution to
PHY 113 C Fall 2013 -- Lecture 6 259/12/2013
Recall: Uniform circular motion:
animation from http://mathworld.wolfram.com/UniformCircularMotion.html
PHY 113 C Fall 2013 -- Lecture 6 269/12/2013
Uniform circular motion – continued
ra ˆ
:ison accelerati lcentripeta theanddirection radial in the
onaccelerati then the, If
2
rv
vvv
c
fi
PHY 113 C Fall 2013 -- Lecture 6 279/12/2013
Uniform circular motion – continued
r
ra
ra
ra
ˆ2
ˆ2
ˆ
2
2
2
rf
rT
rv
c
c
c
Trv 2
In terms of time period T for one cycle:
In terms of the frequency f of complete cycles:πfrv
Tf 2 ;1
PHY 113 C Fall 2013 -- Lecture 6 289/12/2013
Uniform circular motion and Newton’s second law
r
ra
aF
ˆ2
rv
m
c
iclicker exercise:For uniform circular motion
A. Newton’s laws are repealedB. There is a force pointing radially outward from
the circleC. There is a force pointing radially inward to the
circle
PHY 113 C Fall 2013 -- Lecture 6 299/12/2013
http://earthobservatory.nasa.gov/Features/OrbitsCatalog/page1.php
Example of uniform circular motion:
PHY 113 C Fall 2013 -- Lecture 6 309/12/2013
Example of uniform circular motion: Consider the moon in orbit about the Earth
NaMF
smRT
a
s. days.T
mR
kgM
cM
Mc
M
M
20
232
6
8
22
100.2 Law Second sNewton'
/1072.22
10362327 :Moon of period Rotational
1084.3 :Earth ofcenter from Distance
1035.7 :Moon of Mass
PHY 113 C Fall 2013 -- Lecture 6 319/12/2013
Example of uniform circular motion:
ra ˆ2
rv
c
r̂
q
sinsin
:law second sNewton' ofcomponent (radial) Horizontal
0cos:law second sNewton' ofcomponent Vertical
22
Lmv
rmvmaT
mgT
c
PHY 113 C Fall 2013 -- Lecture 6 329/12/2013
Example of uniform circular motion:
mgnmgn 0:law second sNewton' ofcomponent Vertical
grμvr
mvmgμnμ
rmvmaf
sss
c
max
2
2
:condition Maximum
:law second sNewton' ofcomponent (radial) Horizontal
PHY 113 C Fall 2013 -- Lecture 6 339/12/2013
Curved road continued:
hrmim/s.smgrμv
mrμ
grμv
s
s
s
/29113 /358.95.0
35 5.0 :Example
max
max
PHY 113 C Fall 2013 -- Lecture 6 349/12/2013
Mass on a swing:
mg
q T
iclicker exercise:Which of these statements about the tension T in the rope is true?
A. T is the same for all q.B. T is smallest for q0.C. T is largest for q0.
LvmmgT
LvmmgT
2
2
:0For
cos
:rope thealongdirection in the laws sNewton'
q
q
L
PHY 113 C Fall 2013 -- Lecture 6 359/12/2013
Newton’s law in accelerating train car
ga
maθTmgT
q
q
tan
sin :direction Horizontal0cos :direction Vertical
PHY 113 C Fall 2013 -- Lecture 6 369/12/2013
Notion of fictitious forces
iclicker questionA. Fiction is for English class and not for
physics class.B. The concept of fictitious forces is a bad
idea.C. The concept of fictitious forces is
necessary for analyzing certain types of problems.
In analyzing motion in an inertial frame of reference, the notion of “fictitious force” does not appear.
PHY 113 C Fall 2013 -- Lecture 6 379/12/2013
Example of analysis of forces Ferris wheel moving at constant speed v
ac
ac
ctop
ctop
agmn
mamgn
cbottom
cbottom
agmnmamgn
PHY 113 C Fall 2013 -- Lecture 6 389/12/2013
Example of analysis of forces Mass moving in vertical circle
Analysis similar to Ferris wheel except that the speed v varies and the magnitude of the centripetal acceleration varies accordingly.