PHYSICS 133 EXPERIMENT 7
SIMPLE HARMONIC MOTION
Introduction
In this lab, we study the phenomenon of simple harmonic motion for a mass-and-spring
system and for a variety of pendulums. In a linear mass-spring system, the physical basis
for this kind of motion is that the restoring force
F
exerted on a mass
m
that has been
displaced a distance
x
from equilibrium must be proportional to –
x
. This relationship may
be written
F
= -
kx
(1)
where
k
is a constant that characterizes the stiffness of the spring. A large value of
k
would indicate that the spring is difficult to stretch or compress. In the case of a simple
pendulum, there is no spring, and
k
is replaced by the quantity (
mg/L
)
,
where
m
is the
mass of the pendulum bob,
g
is the acceleration due to gravity,
L
is the length of the
pendulum, and
x
represents the (small) lateral displacement of the bob.
Eq. (1) can be
generalized to represent other physical situations. For example, the displacement might
be given in terms of an angle, in which case the restoring variable would be a torque. See
your textbook for examples.
Using Newton’s second law,
F
=
md
2
x
/
dt
2
, we can write Eq. (1) as a differential equation,
d
2
x
/
dt
2
= - (
k
/
m
)
x
(2)
This equation has as a possible solution the sinusoidal oscillation
x = A
cos
ωt,
which you
can verify by direct substitution in Eq. (2).
Here
A
is the amplitude and
ω
= √ (
k/m
) is the
circular frequency. The frequency depends on physical characteristics of the system. For
example,
ω
= √ (
k/m
) for a linear mass-spring system and
ω
= √ (
g/L
) for small
oscillations of a simple pendulum. Since the period
T
= 2
π
/
ω
, we have
T
= 2
π
√ (
m/k
) for
the mass-spring system and
T
= 2
π
√ (
L/g
) for the simple pendulum, respectively.
Equipment
•
1 air track with glider/spring oscillator,
•
Low-friction pulley and thread,
•
small masses,
•
photogate,
•
magnets,
•
1 simple pendulum,
•
1 meter stick pendulum, ring and disk pendulums
•
1 computer and interface box.
Method