®
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
--
=
Summary of Contents for ME-9502
Page 1: ... PASCO Mechanics Statics System ME 9502 Instruction Manual 012 12876B 012 12876 ...
Page 4: ... Statics System iv 012 12876B ...
Page 20: ...Statics System Exp 2 Adding Forces Resultants and Equilibriants 16 012 12876B ...
Page 24: ...Statics System Exp 3 Resolving Forces Components 20 012 12876B ...
Page 28: ...Statics System Exp 4 Torque Parallel Forces 24 012 12876B ...
Page 32: ...Statics System Exp 5A Center of Mass 28 012 12876B ...
Page 36: ...Statics System Exp 5B Equilibrium of Physical Bodies 32 012 12876B ...
Page 44: ...Statics System Exp 7 The Inclined Plane 40 012 12876B ...
Page 50: ...Statics System Static Friction on an Inclined Plane 46 012 12876B ...
Page 60: ...Statics System Exp 10 Simple Harmonic Motion The Simple Pendulum 56 012 12876B ...
Page 66: ...Statics System Exp 11A Simple Harmonic Motion Physical Pendulum 62 012 12876B ...
Page 70: ...Statics System Exp 11B Minimum Period of a Physical Pendulum 66 012 12876B ...
Page 76: ...Statics System Exp 11C Simple Harmonic Motion Beam on a Spring 72 012 12876B ...
Page 84: ...Statics System Exp 13 Simple Machines The Inclined Plane 80 012 12876B ...
Page 94: ...Statics System Technical Support 90 012 12876B ...