PASCO ME-9502 Statics System - Manual (Page 71)

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M o d e l N o . M E - 9 5 0 2 E x p . 1 1 C : S i m p l e H a r m o n i c M o t i o n – B e a m o n a S p r i n g
0 1 2 - 1 2 8 7 6 B 67
Exp. 11C: Simple Harmonic Motion–Beam on a Spring
Equipment Needed
Theory
Imagine a horizontal beam that is supported by a hinge at one
end and a vertical spring at the other end. If the beam is dis-
place, the spring exerts a restoring force, F = -kx, to return the
beam to its equilibrium position. The beam will oscillate up
and down with a period, T
beam
.
For a mass on a spring, the period, T, is as follows:
where M is the oscillating mass and k is the spring constant. What is the period for a beam on a spring?
The beam rotates at the hinge as the spring oscillates up and down. The force of the spring on the beam, F = -kx,
produces a torque on the beam. Let L
lever
be the length of the lever arm of the beam. The torque due to the spring
is
= FL
lever
. A net torque causes angular acceleration,
, that is directly proportional to the torque, , and
inversely proportional to the moment of inertia, I. That is,
or
 = I. Setting the two expressions for torque equal to each other gives FL
lever
=
I or -kxL
lever
=
I where x is
the displacement of the spring up and down as it oscillates.
The angular acceleration,
, and the tangential (linear) acceleration, a
T
, of the beam are related. The tangential
acceleration, a
T
=
r where r is the radius of rotation. In this case, the radius of rotation is the lever arm, L
lever
, so
a
T
=
L
lever
, or
 = a
T
/L
lever
.The expression becomes:
Solving for a
T
gives:
This expression has the form of a
T
=
 2
x, where
 is the angular frequency, so  is:
Since the angular frequency,
 = 2/T, the period, T = 2/or
Item Item
Statics Board Mounted Spring Scale
Mass and Hanger Set Balance Arm and Protractor
Stopwatch (ME-1234) Thread
Hinge
Beam
Spring
L
lever
Figure 11.3: Beam on a Spring
T
M
k
----- =


I
-- =
kxL
lever
–
a
T
L
lever
--------------I =
a
T
kL
lever
2
I
--------------------x =

kL
lever
2
I
-------------------- L
lever
k
I
-- = =
T
2

L
lever
-------------- I
k
-- =