7. One-dimensional systems
Work and energy for one-dimensional systems
x2
For one-dimensional motion
W  x1  x2    Fx  x dx
x1
work is path independent.
This follows from the fact that
 f x dx    f x dx
x2
x1
x1
x2
U ( x)    Fx  x dx
It is possible to introduce
potential energy:
x2
x1
Equilibrium: dU dx  0
2
Stable equilibrium: d 2U dx  0
2
Unstable equilibrium: d 2U dx  0
1
Example (Stability of a cube balanced on a cylinder):
U ( )  mgh  mg r  b  cos   r sin  
2b
dU d  mg r cos   b sin  

dU d  0    0
h
r
d 2U d 2  mg r  b  cos   r sin  
d 2U   0  d 2  mg r  b 
If r>b then equilibrium is stable
If r<b then equilibrium is unstable
2
Integration of equation of motion in one-dimension
1
2
mx 2  U  x   E dx dt 
2 m E  U x  
m x
t
2 x0
dx
 t0
E  U x 
Two arbitrary constants in
this solutions are energy E
and initial time t0
Limits of motion (turning points): U  x   E
A finite motion in 1-D is oscillatory –
motion between two turning points.
Period:
T E   2m 
x2  E 
x1  E 
dx
E  U x 
3
Example (Period of oscillation of a simple pendulum):
U    mgl cos 
dx  ld
l
 0  max   E  mgl cos  0
T  2m 
x2  E 
x1  E 


x
l
0
dx
ld
2l 0
d
 2m 
2
 0
g 0 cos   cos  0
E  U x 
mgl cos   mgl cos  0
l 0
d
l 1 du
  1  cos   1    T  4 
4
2
2
0
g
g 0 1  u 2
0  
1
2
2
T  2
l
g
l 0
d
l
T 2

T

4
K sin  0 2 

2
2
0
g
g
sin  0 2   sin  2 
sin  2 
sin  
sin  0 2 
K k   
 2
0
d
1  k 2 sin 2 
Complete elliptic
integral of the first
kind
4
Spherical coordinates
 x  r sin  cos 

 y  r sin  sin 
 z  r cos 


dr  drrˆ  rdˆ  r sin ˆ
f ˆ 1 f ˆ 1 f
f  rˆ  

r
r 
r sin  
5
Central forces
 

Definition: F r   f r rˆ
  
Example: F r  
rˆ
2
r
• A central force that is conservative is spherically symmetric
• A central force that is spherically symmetric is conservative
 
U ˆ 1 U ˆ 1 U


F r   f r rˆ  U  rˆ


 U r   U r  
r
r

r sin  

0
0
 
U r 
F r   rˆ
 f r rˆ
r

 
r
F r   f r rˆ  f r  
r
Fx   f r x 
  f r   r
 
 
x 
  xy 
y y  r 
r  r  y
r 



Fx Fy
f r   Fy
  F z 

0

r  x
y
x


6