Download Lecture 19

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lecture 19
Falling with air resistance
dv
a
 g  kv
dt
Falling with air resistance
dv
2
a=
= g - kv
dt
Terminal Velocity with Coffee Filters
mg - Fr = ma
where Fr is the resistance force.
Fr
a=gm
1. A penny and a quarter dropped from a ladder land at the
same time (air resistance is negligible).
2. A coin dropped in a coffee filter from a ladder lands later
than a coin without coffee filter (the terminal velocity is
smaller for larger cross-section area).
3. A quarter dropped in a coffee filter will land faster than a
penny in a coffee filter (the terminal velocity is larger for
larger mass)
4. Two identical coins dropped in coffee filters of different
diameters land at different times (the terminal velocity is
smaller for larger cross-section area).
Resistance force: Fr = gAv
2
A – area of the projectile
For a spherical projectile in air at g = 0.25 N ´ s /m
STP:
2
4
Terminal velocity:
Fr
a=g=0
m
Fr = mg
gAv = mg
mg
vT =
gA
2
A 70-kg man with a parachute: vT ~ 5 m/s
A 70-kg man without a parachute: vT ~ 70
m/s
James Prescott Joule
b. Dec. 24, 1818, Salford, Lancashire, England
d. Oct. 11, 1889, Sale, Cheshire
Discovered some basic laws of electricity and
thermodynamics (Joule ’ s Law and JouleThomson Law); established the basis of the
Law of Conservation of Energy and The
First Law of Thermodynamics
Main occupation and source of funding:
Brewery
Was home-schooled by some of the finest scientists of
his time (including John Dalton)
Since childhood, he was a fearless and meticulous experimenter
As a boy, he hiked in the mountains with a pistol, studying echo and the
speed of sound.
He sounded the depth of Lake Windermere to be 198 ft.
He tortured servants with crazy experiments.
Spent his honeymoon climbing in French Alps and measuring the
temperature at the top and the bottom of waterfalls
1843 (Age 24): Paddle-wheel experiment: mechanical
work can be converted into heat!
Therefore, heat is one of the forms of energy.
It led to the First Law of Thermodynamics
It also showed that energy is conserved.
Energy Conservation Law!
Met with hostility and disbelief.
It took decades before Joule’s
discovery was accepted.
Only one person believed Joule. It was William Thomson (later
Lord Kelvin). They started working together.
As a teenager, Joule was trying to replace steam engines
with electric engines in his brewery.
He failed, but became interested in the connections
between mechanical work, heat, and electricity.
In 1840, at the age of 21, he discovered “Joule’s Law”:
Heat generated in a wire
= Resistance x Current^2
It showed that electricity can be converted into heat!
All forces are
CONSERVATIVE
or
NON-CONSERVATIVE
A force is conservative if:
The work
done
by the force in going


from r1 to r2 is independent of the path
the particle follows
or
The work done by
particle goes from
closed path, back to
the
 force
 when the
r1 to r2 around a
r1 , is zero.
Non-conservative:
doesn ’ t satisfy
above conditions
the
If a force can be written as the derivative of
some function, that force is conservative.
1D case:
dU
Fx  
dx
U(x) is called the  potential energy
function for the force F
If such a function exists, then the force is
conservative
W
conservative
does NOT depend on path!
1D case:
dU
F 
dx
x2
W
conservative
  Fdx 
x1
x2
dU
 
dx   [U ( x2 )  U ( x1 )]
dx
x1
W
con
W
 [U ( x2 )  U ( x1 )]
con
does NOT depend on path!
If Fx(x) is known, you can find the potential energy function as
U ( x)    Fx ( x) dx  C



Force of gravity: F  0i  mg j
Potential energy function: U  mgy  Const
Spring:

 
F  kxi  0 j
2
kx
U


Const
Potential energy function:
2
A particle moves in one dimension under the
influence of a single conservative force given by
F ( x)   x
3
where  is a given constant. Take the potential
energy reference to be at xi=0 such that:
U ( xi  0)  0
and calculate the potential energy function U(x).
or
nc
1 2
If W
 0, KE2  U 2  KE1  U1
Spring problem revisited
A block of mass M is on a horizontal surface and is attached to
a spring, spring constant k. If the spring is compressed an
amount A and the block released from rest, how far from
unstretched position will it go before stopping if there is no
friction between the block and the surface?
How will this answer change is the block is not attached
to the spring??
Have a great day!
Reading: Chapter 9
Hw: Chapter 8 problems and
exercises