1. Sect. 7.7: Conservative & Non-Conservative Forces

2. Conservative Forces  A PE CAN be defined for 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

3. Conservative
forces:
A PE CAN
be defined.
Nonconservative
forces: A PE
CANNOT be
defined.

4. Friction is nonconservative.
Work depends on the path!

5. Wnet = K WC = -U  KE = -U + WNC OR: WNC = K + U
If several forces act (conservative & non-conservative):
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):
WC = -U
 KE = -U + WNC
OR: WNC = K + U

6.  In general,
WNC = K + U
Work done by non-conservative forces = total change in KE
+ total change in PE

7. 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

8.  Principle of Conservation of Mechanical 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)

9. 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).

10.  K1 + U1 = K2+ U2  K + U = 0 A powerful method of calculation!!
Conservation of Mechanical Energy
 K + U = 0
OR E = K + U = Constant
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!!

11. OR E = K + U = Constant  K + U = 0
Conservation of Mechanical Energy
 K + U = 0
OR 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

12. K1 + U1 = K2 + U2 0 + mgh = (½)mv2 + 0 v2 = 2gh  U1 = mgh K1 = 0
The sum remains constant
 K + U = same
as at points 1 & 2
K1 + U1 = K2 + U2
0 + mgh = (½)mv2 + 0
v2 = 2gh
 U2 = 0
K2 = (½)mv2

13. Example Conservation of mechanical energy!  (½)m(v1)2 + mgy1
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

14. Conceptual Example Who is traveling faster at the bottom? Who gets to
the bottom first?
Demonstration!
Both
start
here! 
Frictionless
water
slides!

15. Example: Roller Coaster
Mechanical energy conservation! (Frictionless!)
 (½)m(v1)2 + mgy Only height differences matter!
= (½)m(v2)2 + mgy 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