Transcript Conservative Force - University College Cork

Quantum Puzzles and their Applications in Future Information
Technologies
A Public Lecture by Prof. Anton Zeilinger
The University of Vienna and The Institute of Quantum Optics and Quantum Information,
Austrian Academy of Sciences
7pm Monday, 10th November 2008
Lecture Theatre Boole 4, University College Cork
Copyright: Jacqueline Godany
Quantum Fair!
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the experts will be there to answer them and guide you through the
amazing quantum world. Refreshments will be provided.
Work and Conservative Forces (recall: Lecture 12)
Definition: Conservative Force
A force is conservative when it does no net work on moving an object around a
closed path (i.e. starting and finishing at same point)
1D:
This is just some function that we
need to find to determine the work.
Conservative Systems
In a conservative system, a function G(r) exists, whereas in a non-conservative system
it does not exist (and the evaluation of the work integral is more complicated).
Definition: Potential Energy
For a uniform gravitational field (y-direction only):
U(r) depends on objects position in the gravitational field.
Gravity exerts a force mg on the basketball. Work is done by
the gravitational force as the ball falls from a height h0 to a
height hf.
W  mgh 0  mgh f  mgs
path travelled doesn’t
matter
Conservative Systems
dv
Since: a 
dt
Since:
ds
v
dt
Since:
Work done in moving a body from A to B
in a conservative force is the change in
kinetic energy of the body.
Conservative Systems
For any conservative force field:
The sum of the two terms remains unchanged throughout the force field
Recall nose basher
pendulum!
where:
: total energy, scalar
: kinetic energy KE, scalar
Principle of
Conservation of
Mechanical Energy
: potential energy PE, scalar
The total mechanical energy (E = KE + PE) of an object remains constant as the object
moves, provided that the net work done by external non-conservative forces is zero.
Conservation of Mechanical Energy
Example
Motorcyclist leaps across cliff. Ignoring air resistance, find speed at which the cycle
strikes the ground on the other side. Use conservation of KE + PE expression.
Einitial  E final 
1 2
1
mv0  mgh0  mv 2f  mgh f
2
2
 v f  v02  2 g h0  h f 
vf 
38ms 
1 2


 2 9.81ms 2 70m  35m   46ms 1
Conservation Laws
Two universal conservation laws:
1. Conservation of angular momentum
L  r  mv   mrv sin   constant
(assuming that there are no external torques on the system)
2. Conservation of mechanical energy
1 2
E  mv  U ( x), U(x) is the potential energy
2
(assuming no friction or other non-conservative forces present)
Differentiate the energy (in 1D) with respect to time:
since v = dx/dt and a = dv/dt
in a conservative systems
Conservation of Energy
The conservation of energy can be used to solve problems in mechanics where Newton's
Laws cannot. The system must be conservative, i.e. no non-conservative forces present.
Example of conservation of energy: free fall
no initial velocity
finite initial velocity
v
Friction – presence of non-conservative force
vo
N
N
v
FR
FR

x
W
y component:
: kinetic friction coefficient
mgcos

W = mg
y
FR contains no y component:
Note: acceleration of skateboarder purely in x direction
x component:


since FR = mRN
since
v 2  v02  2as
Friction – Work and Power
Loss of energy (energy is not conserved, since friction is present):
DE = DKE + DPE
using
0
since y = s sin
In the absence of friction:
and energy conserved.
Define the work done per time interval as
This is the power generated:
The average power, P, is the average rate at which work, W, is done, and it is obtained by dividing
W by the time, t, required to perform the work. SI Unit: J/s = Watt (W)
Momentum Conservation (p123 M&O’S, p201 C&J)
A
B
acceleration
Consider two objects, A and B moving
in opposite directions. Mass of A is
mA, mass of B is mB Velocity of A is vA,
velocity of B is vB
From Newton’s 3rd law
Rate of change of
linear momentum, pA and pB.
Conservation of Momentum

The total linear momentum of an isolated system remains constant (i.e. is conserved).
An isolated system is one for which the vector sum of the average external forces
acting on the system is zero.
momentum before interaction = momentum after interaction
Example: Consider an explosion
p  mv  0 since object stationary
p2
p3
p1
p4
before
after
p5
If there is no external force, than the momenta before and after have to be the same
Conservation of Momentum
Example: Consider a two body explosion, e.g. a gun being fired
v
M+
m
Before (b)
There are no external forces:
The magnitude of V depends on
the energy put into the system:
m
After (a)
M
V
Conservation of Momentum*
Example continued
Kinetic energy of m:
Kinetic energy of M:
if
Summary
Definition: Conservative Force
A force is conservative when it does no net work on moving an object around a
closed path (i.e. starting and finishing at same point)
Principle of
Conservation of
Mechanical Energy
In words: the total mechanical energy (E = KE + PE) of an object remains constant as
the object moves, provided that the net work done by external non-conservative
forces is zero.
Power generated:
The average power, P, is the average rate at which work, W, is done, and it is obtained by dividing
W by the time, t, required to perform the work. SI Unit: J/s = Watt (W)
Conservation of momentum: momentum before interaction = momentum after interaction