Sect. 7.7: Conservative & NonConservative Forces
Conservative Forces
• Conservative Force  The work done by that force
depends only on initial & final conditions & not on path
taken between the initial & final positions of the mass.

A PE CAN be defined for conservative forces
• Non-Conservative Force  The work done by that
force depends on the path taken between the initial &
final positions of the mass.
 A PE CANNOT be defined for non-conservative
forces
• The most common example of a non-conservative
force is FRICTION
• Conservative
forces:
A PE CAN
be defined.
• Nonconservative
forces: A PE
CANNOT be
defined.
Friction is nonconservative.
Work depends on the path!
• If several forces act (conservative & nonconservative):
The total work done is: Wnet = WC + WNC
WC = work done by conservative forces
WNC = work done by non-conservative forces
• The work-kinetic energy theorem still holds:
Wnet = K
• For conservative forces (by definition of PE U):

OR:
WC = -U
KE = -U + WNC
WNC = K + U
 In general,
WNC = K + U
Work done by non-conservative
forces = total change in KE
+ total change in PE
Mechanical Energy & its Conservation
• GENERAL: In any process, total energy
is neither created nor destroyed.
• Energy can be transformed from one
form to another & from one body to
another, but the total amount remains
constant.
 Law of Conservation of Energy
• In general, we found:
WNC = K + U
• For the Special case of conservative
forces only  WNC = 0
 K + U = 0
 Principle of Conservation of
Mechanical Energy
• Note: This is NOT (quite) the same as the Law of
Conservation of Energy! It is a special case of this
law (where the forces are conservative)
Conservation of Mechanical Energy
• For conservative forces ONLY! In any process
K + U = 0
Conservation of Mechanical Energy
• Define mechanical energy:
EK+U
Conservation of mechanical energy
 In any process, E = 0 = K + U
OR:
E = K + P = Constant
In any process, the sum of the K and the U is
unchanged (energy changes form from U to K or K to
U, but the sum remains constant).
• Conservation of Mechanical Energy

K + U = 0
E = K + U = Constant
OR
For conservative forces ONLY (gravity, spring, etc.)
• Suppose, initially: E = K1 + U1
& finally:
E = K2+ U2
E = Constant

K1 + U1 = K2+ U2
A powerful method of calculation!!
• Conservation of Mechanical Energy

OR
K + U = 0
E = K + U = Constant
• For gravitational PE:
Ug = mgy
E = K1 + U1 = K2+ U2
 (½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2
y1 = Initial height, v1 = Initial velocity
y2 = Final height, v2 = Final velocity
 U1 = mgh
K1 = 0
The sum remains constant
 K + U = same
as at points 1 & 2
K1 + U 1 = K 2 + U 2
0 + mgh = (½)mv2 + 0
v2 = 2gh
 U2 = 0
K2 = (½)mv2
Example
• Energy “buckets” are for
visualization only! Not real!!
• Speed at y = 1.0 m?
• Conservation of
mechanical energy!
 (½)m(v1)2 + mgy1
= (½)m(v2)2 + mgy2
(Mass cancels in equation!)
y1 = 3.0 m, v1 = 0
y2 = 1.0 m, v2 = ?
Find: v2 = 6.3 m/s
 PE only
 Part PE, part KE
 KE only
Conceptual Example
• Who is
Both
traveling
start
faster at the here! 
bottom?
• Who gets to
the bottom first?
• Demonstration!
Frictionless
water
slides!
Example: Roller Coaster
• Mechanical energy conservation! (Frictionless!)
 (½)m(v1)2 + mgy1
= (½)m(v2)2 + mgy2
Only height differences matter!
Horizontal distance doesn’t matter!
(Mass cancels!)
• Speed at bottom?
y1 = 40 m, v1 = 0
y2 = 0 m, v2 = ?
Find: v2 = 28 m/s
• What y has
v3 = 14 m/s? Use:
(½)m(v2)2 + 0 = (½)m(v3)2 + mgy3 Find: y3 = 30 m