Transcript Potential Energy - McMaster University

Potential Energy
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Work and potential energy
Conservative and non-conservative forces
Gravitational and elastic potential energy
Physics 1D03 - Lecture 22
mg
Gravitational Work
To lift the block to a height y
requires work (by FP :)
FP = mg
WP = FPy
= mgy
When the block is lowered,
gravity does work:
y
mg
Wg1 = mg.s1 = mgy
or, taking a different route:
y
Wg2 = mg.s2 = mgy
s1
s2
Physics 1D03 - Lecture 22
Work done (against gravity) to lift the box is “stored” as
gravitational potential energy Ug:
Ug =(weight) x (height) = mgy
(uniform g)
When the block moves,
(work by gravity) = P.E. lost
Wg = -DUg
• The position where Ug = 0 is arbitrary.
• Ug is a function of position only. (It depends only on the
relative positions of the earth and the block.)
• The work Wg depends only on the initial and final heights,
NOT on the path.
Physics 1D03 - Lecture 22
Example
• A rock of mass 1kg is released from rest from a 10m
tall building. What is its speed as it hits the ground ?
• The same rock is thrown with a velocity of 10m/s at
an angle of 45o above the horizontal. What is its
speed as it hits the ground.
Physics 1D03 - Lecture 22
Conservative Forces
path 1
A force is called “conservative”
if the work done (in going from A
to B) is the same for all paths
from A to B.
B
A
path 2
W1 = W2
An equivalent definition:
For a conservative force, the
work done on any closed path
is zero.
Total work is zero.
Physics 1D03 - Lecture 22
Concept Quiz
The diagram at right shows a
force which varies with position.
Is this a conservative force?
a)
b)
c)
d)
Yes.
No.
We can’t really tell.
Maybe, maybe not.
Physics 1D03 - Lecture 22
For every conservative force, we can define a potential energy
function U so that
WAB = -DU = UA -UB
Note the negative
Examples:
Gravity (uniform g) : Ug = mgy, where y is height
Gravity (exact, for two particles, a distance r apart):
Ug = - GMm/r, where M and m are the masses
Ideal spring: Us = ½ kx2, where x is the stretch
Electrostatic forces (we’ll do this in January)
Physics 1D03 - Lecture 22
Non-conservative forces:
• friction
• drag forces in fluids (e.g., air resistance)
Friction forces are always opposite to v (the direction
of f changes as v changes). Work done to overcome friction is
not stored as potential energy, but converted to thermal energy.
Physics 1D03 - Lecture 22
Conservation of mechanical energy
If only conservative forces do work,
potential energy is converted into kinetic
energy or vice versa, leaving the total
constant. Define the mechanical energy E
as the sum of kinetic and potential energy:
E  K + U = K + Ug + Us + ...
Conservative forces only: W = -DU
Work-energy theorem:
W = DK
So, DK+DU = 0; which means that E
does not change with time:
dE/dt = 0
Physics 1D03 - Lecture 22
Example: Pendulum
L
The pendulum is released from
rest with the string horizontal.
a) Find the speed at the lowest
point (in terms of the length L
of the string).
vf
Physics 1D03 - Lecture 22
Example: Pendulum
The pendulum is released from
rest at an angle θ to the
vertical.
a) Find the speed at the lowest
point (in terms of the length L
of the string).
θ
vf
Physics 1D03 - Lecture 22
Example: Block and spring
v0
A block of mass m = 2.0 kg
slides at speed v0 = 3.0 m/s
along a frictionless table
towards a spring of stiffness k
= 450 N/m. How far will the
spring compress before the
block stops?
Physics 1D03 - Lecture 22