Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 19, 20, 21
Quiz
A block of mass m is attached to a vertical spring,
spring constant k. The spring is unstretched at y=0.
A
If the spring is compressed an amount A and the block
released from rest, how high will it go?
All forces are
CONSERVATIVE
or
NON-CONSERVATIVE
A force is conservative if:
The work
done
by the force in going


from r1 to r2 is independent of the path
the particle follows
or
The work done by
particle goes from
closed path, back to
the
 force
 when the
r1 to r2 around a
r1 , is zero.
Non-conservative:
doesn’t
satisfy
above conditions
the
If a force can be written as the derivative of
some function, that force is conservative.
1D case:
Fx  
dU
dx
U(x) is called the  potential energy
function for the force F
If such a function exists, then the force is
conservative
W
conservati ve
does NOT depend on path!
1D case:
F 
x2
W
conservati ve


dU
dx
Fdx 
x1
x2



x1
dU
dx
dx   [U ( x 2 )  U ( x1 )]
W
con
W
 [U ( x 2 )  U ( x1 )]
con
does NOT depend on path!
If Fx(x) is known, you can find the potential energy function as
U ( x )    F x ( x ) dx  C



Force of gravity: F  0 i  mg j
Potential energy function: U  mgy  Const
Spring:



F   kx i  0 j
Potential energy function:
U 
kx
2
2
 Const
A particle moves in one dimension under the
influence of a single conservative force given by
F ( x)   x
3
where  is a given constant. Take the potential
energy reference to be at xi=0 such that:
U ( xi  0 )  0
and calculate the potential energy function U(x).
W
nc
1 2
 KE 2  U 2   KE 1  U 1 
or
If W
nc
1 2
 0 , KE 2  U 2  KE 1  U 1
Spring problem revisited
A block of mass M is on a horizontal surface and is attached to
a spring, spring constant k. If the spring is compressed an
amount A and the block released from rest, how far from
unstretched position will it go before stopping if there is no
friction between the block and the surface?
How will this answer change is the block is not attached
to the spring??
Block of mass m has a spring connected to
the bottom. You release it from a given
height H and want to know how close the
block will get to the floor. The spring has
spring constant k and natural length L.
H
y=0
ENERGY DIAGRAMS
Potential Energy Diagrams
• For Conservative
forces can draw
energy diagrams
• Equilibrium points
– Motion will move
“around” the
equilibrium
– If placed there with
no energy, will just
stay (no force)
Fx  
dU
dx
0
Stable vs. Unstable Equilibrium
Points
The force is zero at both maxima and minima
but…
– If I put a ball with no velocity there would it stay?
– What if it had a little bit of velocity?
A particle moves along the x-axis while
acted on by a single conservative force
parallel to the x-axis. The force
corresponds to the potential-energy
function graphed in the Figure. The
particle is released from rest at point A.
a)What is the direction of the force on the
particle when it is at point A?
b) At point B?
c) At what value of x is the kinetic energy
of the particle a maximum?
d) What is the force on the particle when
it is at point C?
e) What is the largest value of x reached by
the particle during its motion?
f) What value or values of x correspond to
points of stable equilibrium?
2 or 3D cases:
If
or
Fx  

dU
F  
dr
U ( x, y , z )
x
then
; Fy  
U ( x, y , z )
y
; Fz  

U ( r2 )
U ( x, y , z )
z
 



dU
W   F  d r      d r    dU   U ( r2 )  U ( r1 ) 
L

dr
U (r )
1
W
con


 [U ( r2 )  U ( r1 )]
Several dimensions: U(x,y,z)
Fx  
U ( x, y , z )
x
; Fy  
U ( x, y , z )
y
; Fz  
U ( x, y , z )
z
Partial derivative is taken assuming all other arguments fixed
Compact notation using vector del, or nabla:

 
   
F   U ,  
i 
j
k
x
y
z

dU
Another notation: F  

dr
Geometric meaning of the gradient  U
Direction of the steepest ascent;
Magnitude  U : the slope in that direction

F   U : Direction of the steepest descent

Magnitude F : the slope in that direction
http://reynolds.asu.edu/topo_gallery/topo_gallery.htm
:
Given the potential energy function
U ( x, y )  
Gm 1 m 2
1
(x  y )2
2
2
find the x and y components of the corresponding
force.
Have a great day!
Reading: Chapter 9
Hw: Chapter 8 problems and
exercises