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.
Fx

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