Elastic Energy
Compression and Extension
It takes force to press a
spring together.
It takes force to extend a
spring.
More compression requires
stronger force.
More extension requires
stronger force.
Spring Constant
The distance a spring moves
is proportional to the force
applied.
Fx
The ratio of the force to the
distance is the spring
constant (k).
k F/x
F
x
Hooke’s Law
The force from the spring
attempts to restore the
original length.
This is sometimes called
Hooke’s law.
The distance x is the
displacement from the
natural length, L.
L
L+ x
L-x
F kx
Scales
One common use for a
spring is to measure weight.
The displacement of the
spring measures the mass.
Fs = -k(-y)
Fg Fs
-y
mg k ( y )
m (k / g ) y
Fg = -mg
Stiff Springs
Two spring scales measure
the same mass, 200 g. One
stretches 8.0 cm and the
other stretches 1.0 cm.
What are the spring
constants for the two
springs?
The spring force balances
the force from gravity: F = 0
= (-mg) + (-kx).
Solve for k = mg/ (–x).
x is negative.
Substitute values:
(0.20 kg)(9.8 m/s2)/(0.080 m)
= 25 N/m.
(0.20 kg)(9.8 m/s2)/(0.010 m)
= 2.0 x 102 N/m.
Force and Distance
The force applied to a spring
increases as the distance
increases.
The product within a small
step is the area of a
rectangle (kx) Dx.
The total equals the area
between the curve and the x
axis.
F
F = kx
Dx
x
Work on a Spring
For the spring force the force
makes a straight line.
The area under the line is
the area of a triangle.
F
Fs=kx
1
Fs x
2
1
Ws (kx) x
2
1
Ws kx2
2
Ws
x
x
Elastic Work
• W = (1/2) kx2
• The work done by the spring
as it compresses is negative.
Ws = (1/2)ky2
Fs = -k(-y)
-y
The elastic force exerted
by a spring becomes work.
Like gravity the path taken
to the end doesn’t matter.
• Spring force is conservative
• Potential energy, U = (1/2)kx2
Springs and Conservation
The spring force is
conservative.
• U = ½ kx2
The total energy is
A 35 metric ton box car
moving at 7.5 m/s is brought
to a stop by a bumper.
The bumper has a spring
constant of 2.8 MN/m.
• E = ½ mv2 + ½ kx2
• Initially, there is no bumper
• E = ½ mv2 = 980 kJ
v
x
• Afterward, there is no speed
• E = ½ kx2 = 980 kJ
• x = 0.84 m
Energy Conversion
A 30 kg child pushes down
15 cm on a trampoline and is
launched 1.2 m in the air.
What is the spring constant?
Initially the energy is in the
trampoline.
• U = ½ ky2
Then the child has all kinetic
energy, which becomes
gravitational energy.
• U = mgh
The energy is conserved.
• ½ ky2 = mgh
• k = 3.1 x 104 N/m
0
