Mechanics 105 Forces of friction (static, kinetic) Uniform circular motion Nonuniform circular motion Velocity dependent forces Numerical methods Fundamental forces Gravitational field More applications of Newton’s laws (chapter five)
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Mechanics 105
Forces of friction (static, kinetic)
Uniform circular motion
Nonuniform circular motion
Velocity dependent forces
Numerical methods
Fundamental forces
Gravitational field
More applications of Newton’s laws (chapter five)
Mechanics 105
FrictionForce acting parallel to an interface that
opposes the relative motion
Static – frictional force opposite to applied force - magnitude fs
where s is the coefficient of static friction and n is the magnitude of the normal forces between the surfaces
the equality holds just as the object starts to slip
nf ss
Mechanics 105
Friction
Kinetic – frictional force opposite to relative motion – magnitude fk
where k is the coefficient of static friction and n is the magnitude of the normal forces between the surfaces
the kinetic frictional force is constant
s and k are constants that depend on the nature of the surfaces
Usually, s > k
nf kk
Mechanics 105
Friction
Note: Static friction is not constant – it is whatever is needed to match the applied force, up to the limit of Sn
As the applied force increases, the static frictional force also increases, until the limit, then the object begins to slide, and the frictional force goes to a constant value
applied force (N)
forc
e o
f fr
icti
on
(N
) static friction: fs=applied force
kinetic friction: fs=constant
Mechanics 105
Friction
ConcepTest Examples Demo
Mechanics 105
Example
m1
m2
m1
T
gm
1
+y
+x
m2
T
gm
2
N
ks ff
or
Mechanics 105
Example
amgmamgmT
gmNf
fgmT
gN-my
amfTx
m
amgmTy
m
k
kkk
s
ks
1122
2
1
2
2,
2
11
1
:0)(a case kinetic For the
:0)(a case static For the
0 )(
)(
:
)(
:
Mechanics 105
Question
What do you call a broken boomerang?
Mechanics 105
Question
What do you call a broken boomerang?
Answer: A stick.
Mechanics 105
Newton’s 2nd law applied to uniform circular motion
A mass in uniform circular motion (speed v) accelerates
according to
This acceleration must be caused by some force
along a direction towards the center of the radius of curvature (r)
r
vac
2
r
vmamF c
2
Mechanics 105
Example: conical pendulum
L
T
gm
rT
gm
cosT
sinT
tantan
:equations two theCombining
sin
:
cos
0cos
:
2
2
rgvrg
v
r
vmTF
x
mgT
mgTF
y
x
y
Mechanics 105
Uniform circular motion
ConcepTest
Mechanics 105
Nonuniform circular motion
If an object changes its speed while in circular motion, there is both a radial and a tangential component to the acceleration, therefore, there will be a radial and tangential force applied.
Example: mass moving in a vertical circle
R
T
gm
T
T
gm
gm
cos
get weequation, second theFrom
cos
sin
2
2
gR
vmT
mgTR
vmmaF
mgmaF
rr
tt
Mechanics 105
Words of wisdom
"If I had only known, I would have been a locksmith."-Albert Einstein
"There is no clearer manifestation of pure evil than teachers giving assignments over holiday breaks."-James Halloran
Mechanics 105
Velocity dependent forces
Two models:
1. Force proprtional to the velocity (viscous, low speed)
b is a constant that depends on the object size and
shape and the medium2. Force proportional to the square of the magnitude of
the velocity (air, high speed)D: drag coefficient: density of airA: cross sectional area of object
vbR
2
2
1AvDR
Mechanics 105
Velocity dependent forces
1. Force proprtional to the velocity
0)()(
or
0)()(
written becan This
motion, theofdirection In the
mgtxbtxm
mgtbvtvm
dt
dvmbvmgmaF
vb
gm
Mechanics 105
Velocity dependent forces
Can solve differential equation
where = m/b is a time constant related to the motion
Or, just find terminal speed (a=0)
)1()( t
eb
mgtv
b
mgv
bvmg
T
0
Mechanics 105
Velocity dependent forces
Force proportional to the square of the magnitude of the velocity
Nonlinear differential equation
Terminal speed:
maAvDmgmaF 2
2
1
motion, theofdirection In the
AD
mgvAvDmg T
20
2
1 2
Mechanics 105
Words of wisdom
"I love deadlines. I like the whooshing sound they make as they fly by."-Douglas Adams
"In a survey taken several years ago, all incoming freshman at MIT were asked if they expected to graduate in the top half of their class. Ninety-seven percent responded that they did."-???
"We made too many wrong mistakes."-Yogi Bera
Mechanics 105
Numerical representations of particle dynamics
Euler method
ttvtxttx
ttatvttv
)()()(
)()()(
Mechanics 105
Fundamental forces of nature
Gravitational: force between any two objects
where G is the universal gravitational constant
Electromagnetic: force between two charged objects (q)
where ke is the Coulomb constant
Nuclear (strong) – short range
Weak – short range
rr
mmGFg ˆ
221
rr
qqkF ee ˆ
221
Mechanics 105
Gravitational field
Field: the effect in a region of space that induces a force on an object
e.g the field (created by a mass) exerts the force on the other masses
rr
GM
m
Fg Eg ˆ
2
Mechanics 105
Last bad joke for this chapter
An atom walking down the street says to its friend “I think I lost an electron. The friend asks “Are you sure?” to which the first atom repiles “Yea, I’m positive.”
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