Lecture 3-2 Springs and SHM
Springs and Simple Harmonic Motion
Spring force depends on displacement:
Review - characteristics of SHM
(13-13)
Here, k is spring constant, different for every spring.
Combining with Newton’s second law gives:
(13-14)
Restoring force - cause of motion
For SHM, force is proportional to displacement
A mass is attached to a vertical spring. When it is pulled down 6.0
cm and released, it starts upward with an acceleration of 40 cm/s2.
What is the period of the motion? With what speed does it pass the
equilibrium point?
Add a Constant Force
Energy and Simple Harmonic Motion
Energy and Simple Harmonic Motion
Potential energy of mass on a spring:
As sin2θ + cos2θ = 1, the sum of the kinetic and potential
energies is constant:
(13-18)
(13-23)
And Kinetic energy:
The total energy varies from being all potential (at extremes of
motion) to all kinetic (when spring is neither stretched nor
compressed):
(13-19)
Substituting for x and v:
(13-20)
(13-21)
Kinetic,
Potential, and
Total Energy,
Figure 13-8
A hockey puck of mass 350 g moves horizontally with speed v =0.88
m/s on a frictionless surface toward the end of a relaxed spring for
which k=140N/m, and compresses the spring. By how much is the
spring compressed? How would your answer change if the mass of the
puck were doubled?
Other Applications of SHM
Besides springs, there are many other systems
that exhibit simple harmonic motion. Here are
some examples:
Questions
If the amplitude of a simple harmonic motion
is doubled, what happens to the maximum
kinetic energy?
If adding a constant force to a spring force
means the motion remains periodic with the
same frequency, is the same true when the
force is aligned in a different direction than
the spring force?
Here’s why SHM is so general:
Potential well - Figure 13-11