3B Scientific Pohl's Torsion Pendulum User Manual
3B SCIENTIFIC3B SCIENTIFIC
3B SCIENTIFIC®
3B SCIENTIFIC3B SCIENTIFIC
PHYSICSPHYSICS
PHYSICS
PHYSICSPHYSICS
U15040 Torsion pendulum according to Professor Pohl
Operating instructions
12/03 ALF
9
8
7
6
5
4
bl bm bn bo bp
bq br bs
1 Exciter motor
2 Control knob for fine adjustment of the exciter voltage
3 Control knob for coarse adjustment of the exciter voltage
4 Scale ring
5 Pendulum body
6 Coil spring
7 Pointer for the exciter phase angle
8 Pointer for the pendulum’s phase angle
9 Pointer for the pendulum’s deflection
3
bl Exciter
bm Eddy current brake
bn Guide slot and screw to set the exciter amplitude
2
bo Connecting rod
bp Eccentric drive wheel
bq 4-mm safety socket for exciter voltage measurement
1
br 4-mm safety sockets for the exciter motor power supply
bs 4-mm safety sockets for the eddy current brake power
supply
The torsion pendulum may be used to investigate free,
forced and chaotic oscillations with various degrees of
damping.
Experiment topics:
• Free rotary oscillations at various degrees of damp-
ing (oscillations with light damping, aperiodic oscillation and aperiodic limit)
• Forced rotary oscillations and their resonance
curves at various degrees of damping
• Phase displacement between the exciter and reso-
nator during resonance
• Chaotic rotary oscillations
• Static determination of the direction variable D
• Dynamic determination of the moment of inertia J
1. Safety instructions
• When removing the torsional pendulum from the
packaging do not touch the scale ring. This could
lead to damage. Always remove using the handles
provided in the internal packaging.
• When carrying the torsional pendulum always hold
it by the base plate.
• Never exceed the maximum permissible supply
voltage for the exciter motor (24 V DC).
• Do not subject the torsional pendulum to any un-
necessary mechanical stress.
2. Description, technical data
The Professor Pohl torsional pendulum consists of a
wooden base plate with an oscillating system and an
electric motor mounted on top. The oscillating system
is a ball-bearing mounted copper wheel (5), which is
connected to the exciter rod via a coil spring (6) that
provides the restoring torque. A DC motor with coarse
and fine speed adjustment is used to excite the torsional pendulum. Excitement is brought about via an
eccentric wheel (14) with connecting rod (13) which
6
unwinds the coil spring then compresses it again in a
periodic sequence and thereby initiates the oscillation
of the copper wheel. The electromagnetic eddy current brake (11) is used for damping. A scale ring (4)
with slots and a scale in 2-mm divisions extends over
the outside of the oscillating system; indicators are
located on the exciter and resonator.
The device can also be used in shadow projection demonstrations.
A DC power supply unit for the torsional pendulum
U11755 is required to power the equipment.
Natural frequency: 0.5 Hz approx.
Exciter frequency: 0 to 1.3 Hz (continuously adjustable)
Terminals:
Motor: max. 24 V DC, 0.7 A,
via 4-mm safety sockets
Eddy current brake: 0 to 24 V DC, max. 2 A,
via 4-mm safety sockets
Scale ring: 300 mm Ø
Dimensions: 400 mm x 140 mm x 270 mm
Ground: 4 kg
2.1 Scope of supply
1 Torsional pendulum
2 Additional 10 g weights
2 Additional 20 g weights
3. Theoretical Fundamentals
3.1 Symbols used in the equations
D = Angular directional variable
J = Mass moment of inertia
M = Restoring torque
T = Period
T0= Period of an undamped system
Td= Period of the damped system
= Amplitude of the exciter moment
M
E
b = Damping torque
n = Frequency
t = Time
Λ = Logarithmic decrement
δ = Damping constant
ϕ
= Angle of deflection
ϕ
= Amplitude at time t = 0 s
0
ϕ
= Amplitude after n periods
n
ϕ
= Exciter amplitude
E
ϕ
= System amplitude
S
ω0= Natural frequency of the oscillating system
ωd= Natural frequency of the damped system
ωE= Exciter angular frequency
ωE
= Exciter angular frequency for max. amplitude
res
Ψ0S= System zero phase angle
3.2 Harmonic rotary oscillation
A harmonic oscillation is produced when the restoring
torque is proportional to the deflection. In the case of
harmonic rotary oscillations the restoring torque is
proportional to the deflection angle ϕ:
M = D ·
ϕ
The coefficient of proportionality D (angular direction
variable) can be computed by measuring the deflection angle and the deflection moment.
If the period duration T is measured, the natural resonant frequency of the system ω0 is given by
ω
= 2 π/T
0
and the mass moment of inertia J is given by
D
2
ω
=
0
J
3.3 Free damped rotary oscillations
An oscillating system that suffers energy loss due to
friction, without the loss of energy being compensated
for by any additional external source, experiences a
constant drop in amplitude, i.e. the oscillation is
damped.
At the same time the damping torque b is proportional
to the deflectional angle
.
ϕ
.
The following motion equation is obtained for the
torque at equilibrium
.
..
JbD⋅+⋅+⋅=
ϕϕϕ
0
b = 0 for undamped oscillation.
If the oscillation begins with maximum amplitude
ϕ
at t = 0 s the resulting solution to the differential equation for light damping (δ² < ω0²) (oscillation) is as follows
–δ ·t
ϕ
ϕ
=
· e
· cos (
ω
0
d
· t)
δ = b/2 J is the damping constant and
2
ωωδ
=−
d
2
0
the natural frequency of the damped system.
Under heavy damping (δ² > ω0²) the system does not
oscillate but moves directly into a state of rest or equilibrium (non-oscillating case).
The period duration Td of the lightly damped oscillating system varies only slightly from T0 of the undamped
oscillating system if the damping is not excessive.
By inserting t = n · Td into the equation
–δ ·t
ϕ
ϕ =
and ϕ =
ϕ
tain the following with the relationship
ϕ
ϕ
· e
· cos (
ω
0
for the amplitude after n periods we ob-
n
n
0
δ
−⋅
n
=⋅
eT
d
d
· t)
ω
= 2 π/T
d
d
and thus from this the logarithmic decrement Λ:
Λ
=⋅ =⋅
δ
T
d
ϕ
1
In In
n
ϕ
n
0
ϕ
n
=
ϕ
n+1
0
7
Loading...