Spring Work
F varies with x: F = - k x
Spring Force
Hooke’s Law gives the force
W=F*x only good if F constant.
F=-kx
Approximate with series of steps
Work is sum of areas of rectangles
= area under curve.
k is spring constant
• F is the restoring force
• F is in the opposite direction of x
• k depends on how the spring was
formed, material from which it
was made, thickness of the wire,
etc.
Linear spring is simple
A=½Bh
W = ½ xmax Fmax
= ½ k x2
= work done on spring
Potential Energy in a Spring
PE s =
• Wnc = (KEf – KEi) + (PEgf – PEgi) + (PEsf – PEsi)
– PEg is the gravitational potential energy
– PEs is the elastic potential energy associated with a
spring
– PE will now be used to denote the total potential energy
of the system
1 2
kx
2
Elastic Potential Energy
– related to the work required to
compress spring from its
equilibrium position to some
final, arbitrary, position x
Area unde
r cu
Work = Potential Energy
Initial and Final Kinetic Energies=0
Work with Spring + Gravity
Conservation of Energy: Spring + Gravity
rve
• The PE of the spring is added to both sides of the
conservation of energy equation
• The same problem-solving strategies apply
(KE + PE g + PE s )i = (KE + PE g + PE s )f
Spring + Gravity-Conservative System
(KE + PE g + PE s )i = (KE + PE g + PE s )f
– PEg is the gravitational potential energy
PE g = mgh
– PEs is the spring potential energy
1
PEs = kx 2
2
Nonconservative Forces and Energy
Wnc = ( KEf − KEi ) + (PEi − PEf )
Wnc
or
= (KEf + PEf ) − (KEi + PEi )
The energy can either cross a boundary or the
energy is transformed into a form of nonmechanical energy such as thermal energy
(friction or direct collisions)
1