Example: Toy Car Going through Loop
Conservation of Energy
Given:
mcar = 0.3 kg
k = 1560 N/m
rloop = 0.96 m
!mv02 + mgh0 + ! k(x0)2 + Win
= !mvf2 + mghf + ! k(xf)2 + Eloss
Where is the datum for
gravitational potential energy?
A. 
B. 
C. 
D. 
Lowest point
Highest point
Jupiter
Wherever you want
Where is the datum for
elastic potential energy?
A. 
B. 
C. 
D. 
Required:
Amount spring has to
be pulled back so car
makes it through loop
without falling off.
The car starts 0.62 m from the bottom of the loop.
There is a constant friction force of 0.84 N as the car rolls.
Beginning position
Undeformed position
Final position
Wherever you want
Could the problems of this module have been
solved using F=ma and kinematics?
A. Yes
B. No
C. Maybe
EF 151 Spring, 2011 Lecture 3-4
1
Kinetic Energy at Very High Speeds
EF 151 Spring, 2011 Lecture 3-4
Example: Spring, Friction, Gravity, …
Given: System starts from at rest
As we approach speed of light, c
= 3x108 m/s
1.50 m
Proton of mass 1.7x10-27 kg in accelerator
Kinetic Energy
Newtonian
Quantum
0.1c
1.23E-12 J
1.24E-12 J
0.5c
3.07E-11 J
3.80E-11 J
0.9c
9.94E-11 J
3.18E-10 J
0.99c
1.20E-10 J
1.49E-09 J
EF 151 Spring, 2011 Lecture 3-4
k = 1200 N/m
Required: Maximum distance mass
slides down incline
&
#
1
K = mc 2 $
' 1!
$
!
2
% 1 ' (v / c )
"
Speed
2
60°
μk = 0.40
Unstretched spring
length = 1.40 m
3
EF 151 Spring, 2011 Lecture 3-4
4