Experiment Instructions

TM 600 Centrifugal Force

 TM 600 CENTRIFUGAL FORCE
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Experiment Instructions

Last modification by: Dipl.-Ing. (FH) Peter Mittasch

This manual must be kept by the unit.

Before operating the unit:

- Read this manual.
- All participants must be instructed on
handling of the unit and, where appropriate,
on the necessary safety precautions.

Version 0.3 Subject to technical alterations

i
 TM 600 CENTRIFUGAL FORCE

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Intended Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Structure of the Safety Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Safety Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Description of the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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3.1 Design of the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Operation of the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 First Start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1 Movement of a Body on a Circular Path . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Dynamic of a Body on a Circular Path. . . . . . . . . . . . . . . . . . . . . . . . 11

5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.1 Aim of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Perform Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.3 Centrifugal force as a function of speed . . . . . . . . . . . . . . . . . . . . . . 14
5.3.1 Evaluation of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4 Centrifugal Force as a Function of Mass. . . . . . . . . . . . . . . . . . . . . . 16
5.4.1 Evaluation of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . 17
5.5 Centrifugal Force as a Function of Radius . . . . . . . . . . . . . . . . . . . . 18
5.5.1 Evaluation of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.1 Technical Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2 List of Symbols of Formulae and Units . . . . . . . . . . . . . . . . . . . . . . . 21

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 TM 600 CENTRIFUGAL FORCE

1 Introduction

The movement of a body on a circular path is

encountered in all fields of technology.
Circular motion and its effect in the form of the
centrifugal force produced is observed on all
rotating machines.
v • On centrifuges centrifugal force is used to
m
separate materials of differing densities.
F • On turbines the blades are subjected to cen-
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r trifugal force.
• In car tyres the centrifugal unbalance forces
produce vibration.
Fig. 1.1 Depending on application, centrifugal forces are
either desirable, e.g. on a centrifuge/centrifugal
clutch, or they have an undesirable effect as is the
case with the unbalance forces of a turbine rotor.
The Centrifugal Force Apparatus TM 600 per-
mits experimental investigation of the physical
laws of such circular motion, for example the rela-
tionship between the radius, mass and speed of
the body.
Thanks to its simple, compact, clear-cut design,
the unit is suitable not only for demonstrating the
effects involved, but also for use by trainees.
The digital speed and force displays make for
simple yet precise evaluation of the experiments
performed.

1 Introduction 1
 TM 600 CENTRIFUGAL FORCE

2 Safety

2.1 Intended use

The unit is to be used only for teaching purposes.

2.2 Structure of safety instructions

The signal words DANGER, WARNING or

CAUTION indicate the probability and potential
severity of injury.
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An additional symbol indicates the nature of the

hazard or a required action.

Signal word Explanation

Indicates a situation which, if not avoided, will result in

DANGER death or serious injury.

Indicates a situation which, if not avoided, may result in

WARNING death or serious injury.

Indicates a situation which, if not avoided, may result in

CAUTION minor or moderately serious injury.

Indicates a situation which may result in damage to

NOTICE equipment, or provides instructions on operation of
the equipment.

2 Safety 2
 TM 600 CENTRIFUGAL FORCE

Symbol Explanation

Rotating parts

Notice

2.3 Safety Instructions

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WARNING
Rotating parts.
Risk of hand injuries.
• Do not use the unit without protective lid.

NOTICE
Make sure that locking pin (1) of body is prop-
erly engaged in hole in arm and that safety
1 catch (2) is correctly positioned. Body could
otherwise come loose and fly off.

NOTICE
Do not overload the unit.
• Maximum force: 25N.
• Never attach more than one body.

2 Safety 3
 TM 600 CENTRIFUGAL FORCE

3 Description of the Unit

3.1 Design of the Unit

r 3 2

m 5
4
6 4 3 2 1
w
12
13 7
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5 14 2 3

8 9 10 11

1 Protective lid 8 Force display

2 Crank lever 9 Speed display
3 Arm, rotating 10 Motor switch
4 Thrust rod 11 10-turn-potentiometer
5 Mass bodies 12 Bending bar
6 Safety catch 13 Displacement measuring system
7 Retaining ring 14 Engaging holes

Fig. 3.1 Design of the unit

The main part of the unit is an arm (3) which

rotates about a vertical axis. Bodies (5) of varying
mass m with radius r are attached to the arm.
For measuring the centrifugal force F exerted by
the body of mass m (5), the arm is supported such
that it can move in radial direction by two guides.

3 Description of the Unit 4

 TM 600 CENTRIFUGAL FORCE

The centrifugal force is transmitted by way of a

crank lever (2) and a thrust rod (4) located in the
axis of rotation to a clamped bending bar (12).
The deformation w of the bar, which is propor-
tional to the centrifugal force, is measured using a
displacement measuring system (13) and the
centrifugal force is displayed directly in N via a
digital instrument (8). Before starting a measuring
series, reset the display using the RST button.
Various engaging holes (14) enable the bodies (5)
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to be attached with the desired radius.

The speed of the arm (3) can be infinitely adjusted
with the speed-controlled drive motor.
The speed is set using a 10-turn
potentiometer (11) and indicated by way of a
digital rev counter (9) in min-1.
A transparent protective lid (1) covers the rotating
parts whilst measurement is in progress. The
apparatus can only be started up when the protec-
tive lid (1) is located in its retaining ring (7).

3 Description of the Unit 5

 TM 600 CENTRIFUGAL FORCE

3.2 Operation of the Unit

• Pull locking pin when removing mass body.

Locking pin

• Fold up locking catch at end of arm and slip off

body.
Locking catch A safety catch (9) on the arm ensures that a
body cannot fly off if not properly engaged.
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Fig. 3.2 Remove mass body

• Fit mass body.

– Select body (54g, 79g, 105g).

105g 79g 54g

Fig. 3.3 Mass bodies

– Attach body to arm and engage at desired

125mm
100mm
75mm
50mm
25mm

radius (25mm, 50mm, 75mm, 100mm,

125mm).

– Make sure locking pin engages properly.

Fig. 3.4 Fit mass body

3 Description of the Unit 6

 TM 600 CENTRIFUGAL FORCE

• Fit protective lid.

NOTICE
When the forced cut-off has been triggered by
raising the protective lid, the motor must be
restarted.
Fig. 3.5 Fit protective lid

NOTICE
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The following mass/radius combinations are

recommended for smooth running at high speeds.
• m = 100g, r = 75 mm
• m = 75g, r = 100 mm
• m = 50g, r = 125 mm
The system is then more or less completely
balanced.

3.3 First Start-up

• Remove transportation safeguard.
1

2 – Loosen the two M6 hexagon socket-head

bolts (1) and remove the red securing
block (3). In doing so, fold up the safety
3
catch (2).

Fig. 3.6 Remove transportation

safeguard

3 Description of the Unit 7

 TM 600 CENTRIFUGAL FORCE

4 Basic Principles

The basic principles set out in the following make

no claim to completeness. For further theoretical
explanations, refer to the specialist literature.

4.1 Movement of a Body on a Circular Path

A technically important special case relating to the

movement of bodies is that of movement on a cir-
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cular path which involves motion on a curved path

with constant radius of curvature or path
radius r.
In order to be able to derive the acceleration for a
point on a circular path, the position of the point P
y
is defined by its localized vector r . In terms of
cartesian path coordinates, the localized vector
can be expressed as follows
v

r  cos 
P x r  cos 
r= = (4.1)
r  sin 
y r  sin 
r


x
where r is the radius of the circular path and  the
angle of rotation at time t.
Fig. 4.1 Path coordinates
The velocity vector v is obtained by differenti-
aion in terms of time t

·
x· – r    sin  · – sin 
v = r· = = = r   (4.2)
y· · cos 
r    cos 

4 Basic Principles 8
 TM 600 CENTRIFUGAL FORCE

The vector

– sin 
et et = (4.3)
y cos 
cos 

– sin  P
has the direction of a tangent to the circular path
 and is known as tangential unit vector with a
value of 1.
x
Substituting the angular velocity  for  gives the
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path velocity or circumferential velocity

Fig. 4.2 Tangential unit vector

vt = r   (4.4)

The acceleration vector is obtained by further

differentiation

·· ·2
x·· – r    sin  – r    cos 
a = r·· = = (4.5)
y·· ·· ·2
r    cos  – r    sin 

and can be expressed as the sum of two vectors

·· – sin  · 2 – cos 
a = r  + r   (4.6)
cos  sin 

The vector in the first summand is again the tan-

gential unit vector in tangential direction.

4 Basic Principles 9
 TM 600 CENTRIFUGAL FORCE

The vector

– cos 
en = (4.7)
y – sin 

– cos 
P
en points towards the centre of the circular path and
– sin 

is referred to as the perpendicular unit vector.

x Thus there are two acceleration components act-

ing on a given point in a circular path:
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Fig. 4.3 Perpendicular unit vector • Path acceleration in tangential direction

·· ·
at = r   = r   (4.8)
y

at • Normal or radial acceleration towards the

centre of the circle

P
·2 2
an an = r   = r   (4.9)

The angular velocity  in rad/s is calculated

Fig. 4.4 Acceleration from the speed n in min-1

n
 = 2
------------------ = 0,1047  n (4.10)
60

Given movement with constant angular velocity

 = const (constant speed) the angular accelera-
tion and also the tangential acceleration is zero.
All that remains is the normal acceleration an.

4 Basic Principles 10
 TM 600 CENTRIFUGAL FORCE

4.2 Dynamic of a Body on a Circular Path

A body on a circular path is thus subject to accel-

eration even at constant velocity (speed) with the
acceleration being directed towards the centre of
the circle.
The production of such acceleration requires
external force acting in the direction of accelera-
tion, i.e. towards the centre of the circle.
Applying the fundamental law of dynamics
(Newton’s 2nd law) to the centre of gravity gives
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m  an =  Fi = F (4.11)

The only external force acting here is the so-

called centrifugal force F.
With angular velocity  the centrifugal force is
Mass m

2
an
F F = mr (4.12)

r
It can be seen that the centrifugal force increases
linearly with mass m and path radius r.

Fig. 4.5 Forces at body

The angular velocity  is such that four times the
force is required to keep the body on its circular
path if the speed is doubled

4 Basic Principles 11
 TM 600 CENTRIFUGAL FORCE

The following example should serve to clarify this

relationship:
The performance of a turbine is to be enhanced
by increasing the speed.
A new, tougher material provides a 20 % increase
in strength, thus permitting a 20% increase in cen-
trifugal force.

2
F new  new
------------ = 1,2 = ------------
- (4.13)
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F old 2
 old

F new
------------ = 1,2 = 1,095 (4.14)
F old

Despite 20% more strength, speed and perfor-

mance can only be increased by 9,5%.

4 Basic Principles 12
 TM 600 CENTRIFUGAL FORCE

5 Experiments

The selection of experiments makes no claims of

completeness but is intended to be used as a
stimulus for your own experiments.
The results shown are intended as a guide only.
Depending on the construction of the individual
components, experimental skills and environmen-
tal conditions, deviations may occur in the experi-
ments. Nevertheless, the laws can be clearly
demonstrated.
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5.1 Aim of the Experiment

The following experiments are designed to sub-

2
stantiate the relationship F = m  r   estab-
lished in Chapter 4.2, Page 11, between centrifu-
gal force, mass, radius and speed.

5.2 Perform Measurement

• Before starting a measuring series, reset the

force display using the RST button.
• Set speed potentiometer (2) to zero first and
then switch on motor (1).
• Approach desired speed with speed
potentiometer (2).
• Note down force (4) and speed (3) indicated.
4 3 2 1

Fig. 5.1

5 Experiments 13
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5.3 Centrifugal force as a function of speed

To investigate the dependence on speed, select

body m = 105g and radius r = 100mm.
Increase speed from 100...450min-1 in incre-
ments of 50min-1.

The expected centrifugal force can be calculated

from the speed in min-1 as follows:
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2
2 2
0,105  0,1   - 2
 r    n - = -------------------------------------
F = mr = m
2
------------------------------- 2
n (5.1)
2
30 30

2
F = 0,000115  n

Measured and calculated results are compared in

the following table.

Deviation
Speed n in Measured Calculated referenced to
min-1 force F in N Force F in N end value 25N
in %
100 0,9 1,15 -1,0
150 2,3 2,59 -1,2
200 4,5 4,60 -0,4
250 7,2 7,19 0,04
300 10,4 10,35 0,2
350 14,3 14,09 0,8
400 18,5 18,40 0,4
450 23,6 23,29 1,2
Tab. 5.1 Speed-dependence of centrifugal force for
m = 105g
r = 100mm

5 Experiments 14
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5.3.1 Evaluation of the Experiment

The graph on the next page clearly shows the

square relationship between speed and centrifu-
gal force. The differences between measurement
and calculation can be viewed as minimal.

24
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20

16
Centrifugal force F in N

12

Calculated
4
Measured

0
0 100 200 300 400 500

Speed n in min-1

Fig. 5.2 Centrifugal force as a function of speed

5 Experiments 15
 TM 600 CENTRIFUGAL FORCE

5.4 Centrifugal Force as a Function of Mass

To investigate the dependence on mass, prese-

lect speed n = 300min-1 and radius r = 100mm.
The three bodies m = 54g, 79g and 105g are
utilised one after the other.

The expected centrifugal force is calculated from

the mass m in g as follows:
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2 2
2 2
   n  m- = 0,1    300 -
F = r    m = r-------------------------------
2
-----------------------------------
2
m (5.2)
2
30 30

F = 0,0987  m

Measured and calculated results are compared in

the following table.

Deviation
Mass m Measured Calculated referenced to
in g force F in N Force F in N end value 25N
in %
54 5,2 5,33 -0,52
79 7,7 7,79 -0,36
105 10,4 10,36 0,16
Tab. 5.2 Mass- dependence of centrifugal force for
r = 100mm
n = 300min-1

5 Experiments 16
 TM 600 CENTRIFUGAL FORCE

5.4.1 Evaluation of the Experiment

The graph clearly shows the linear relationship

between mass and centrifugal force.

12

10
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8
Centrifugal force F in N

0
0 20 40 60 80 100

Mass m in g

Fig. 5.3 Centrifugal force as a function of mass

5 Experiments 17
 TM 600 CENTRIFUGAL FORCE

5.5 Centrifugal Force as a Function of Radius

To investigate the dependence on radius, prese-

lect speed n = 300min-1 and mass m = 105g.
The body is consecutively inserted into the
engaging holes at 25mm, 50mm, 75mm, 100mm
and 125mm.

The expected centrifugal force is calculated from

the radius r in mm as follows:
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2 2
2 2
0,105    300
   n  -r = ------------------------------------------
F = m r = m
2
------------------------------- 2
r (5.3)
2
30 30

F = 0,1036  r (5.4)

Measured and calculated results are compared in

the following table.

Deviation refer-
Radius r Measured Calculated
enced to end
in mm force F in N force F in N
value 25N in %
25 2,4 2,59 -0,76
50 5,1 5,18 -0,32
75 7,8 7,77 0,12
100 10,4 10,36 0,16
125 13,2 12,95 1,0
Fig. 5.4 Radius-dependence of centrifugal force
m = 105g
n = 300min-1

5 Experiments 18
 TM 600 CENTRIFUGAL FORCE

5.5.1 Evaluation of the Experiment

The graph on the next page clearly shows the

linear relationship between radius and centrifugal
force.

14

12
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10
Centrifugal force F in N

0
0 50 100 125

Radius r in mm

Fig. 5.5 Centrifugal force as a function of radius

5 Experiments 19
 TM 600 CENTRIFUGAL FORCE

6 Appendix

6.1 Technical Data

Dimensions
Length x width x height 420mm x 400mm x 270 mm
Weight approx. 23 kg

Connections
Power supply 230V, 50 Hz
Nominal consumption (power) 0,3 W
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Alternatives optional, see type plate

Physical parameters
Path radii 25mm, 50mm, 75mm, 100mm, 125 mm
Rotation masses 54g, 79g, 105 g
Speed range 0...500 min-1
(0...52 rad/s)
max. path velocity 6,5 m/s

Force measurement
3 1/2 digit LCD
Measuring range 0...25 N
Resolution 0,1 N

Speed mesurement
8 digit LCD
Measuring range 0...500 min-1
Resolution 0,1 min-1

6 Appendix 20
 TM 600 CENTRIFUGAL FORCE

6.2 List of Symbols of Formulae and Units

Symbols of Mathematical / physical quantity Unit

formulae

an Normal acceleration m/s2

at Tangential acceleration m/s2

F Force, centrifugal force N

m Mass kg

n Speed rpm

r Radius mm
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t Time s

vt Path, circumferential velocity m/s

x Coordinate m

y Coordinate m

 Mathematical constant pi = 3,14 -

 Angle rad

 Angular velocity rad/s

Vektors:

a Acceleration m/s2

en Perpendicular unit vektor -

ef Tangential unit vektor -

r Localized vektor m

v Velocity m/s

6 Appendix 21