Download 3.1 TQ Centrifugal Force Apparatus

Contents
1.
Introduction ...................................................................................................................................... 2
2.
Theory ............................................................................................................................................... 3
3.
Equipment/Apparatus....................................................................................................................... 6
3.1
TQ Centrifugal Force Apparatus.................................................................................................... 6
3.2
The Electronic tachometer............................................................................................................ 7
3.3
The Speed Control Unit ................................................................................................................. 7
4.
Procedure .......................................................................................................................................... 8
5.
Precautions taken ............................................................................................................................. 9
6.
Results ............................................................................................................................................. 10
7.
Conclusion ....................................................................................................................................... 15
8.
References ...................................................................................................................................... 16
1. Introduction
A centripetal force acts towards the centre of a circular path and keeps the body moving in this
path; otherwise the body would observe a straight line motion. In line with Newton's second law
of motion, the centripetal force also produces an acceleration having the same direction it has.
By definition, there exists a direct relationship between the centripetal force of a body and the
square of its speed and an inverse relationship between the force and radial distance from the
center of motion.
Centripetal forces are responsible for the orbiting of the sun by the earth and other satellites.
Other examples include motion of a car round a curved track, the frictional force between the tire
and the ground serves the purpose.
The study of centripetal force is of vital importance due to its impact in many engineering
designs and applications. For modern machinery, rotational speed is of vital importance and also,
centrifugal force resulting from slightly displaced parts can result in failure of vital components.
The study of centripetal forces also helps in the design of curved roads, which are banked so that
for a particular range of velocity deemed safe, cars can drive through the tracks without skidding
off.
2. Theory
A body moving along a curved path experiences changes to its acceleration. This means at each
instantaneous point along this path, the particles has a component of acceleration perpendicular
to the path, even if its speed is constant.
Consider a body moving in circle with uniform speed about a fixed point O,
Fig 1.0 Circular motion
If the body moves from A to B through an angle,
(measured in radians), its angular speed, ω,
about O is defined as the change in angle per second. If t is time taken by the object to move
from A to B,
Also, it should be noted that the time taken for a complete revolution in a circular path, T
Assuming, s is the length of circular distance between A and B,
Dividing both sides of the equation by the time taken to move from A to B, t
This implies that the tangential velocity (v), is
Fig 2 Acceleration in a circle
Considering the scenario above ,assume the body has moved a small radial distance
,the
velocities at X and Y are different even though their magnitude are the same and equal to
tangential velocity, v due to the fact that they have different directions . Therefore, acceleration
occurs as there is a change in velocity.
When t is small,
small and the velocity change acts towards the centre.
Therefore,
Taking limits as
A Body moving in a circular path of radius r with constant speed v has an acceleration directed
towards the centre and by extension experiences a force which is also directed towards the
centre. This force is called the centripetal force.
From Newton's 2nd law of motion,
Since for circular motion,
Centripetal Force,
3. Equipment/Apparatus
3.1
TQ Centrifugal Force Apparatus
The TQ Centrifugal Force Apparatus is designed to demonstrate the relationship between
centrifugal force, mass of a body, its distance from the axis and its angular velocity. It is made up
of two pivoted counterbalanced bell-cranks housed in slideable blocks as shown above. Different
combinations of masses which can be increased by 25g are pivoted while the slideable blocks are
held in place by locating pins. Four equally spaced holes allow each block fit in five different
radial positions in each end of the horizontal rotating member. A transparent safety dome covers
the rotating assembly. Removal of the dome disconnects the motor from power supply.
3.2
The Electronic tachometer
The tachometer measures rotational speed using optical sensor mounted on the experimental
apparatus.
3.3
The Speed Control Unit
The E67 Speed Control Unit produces a variable DC output voltage which can be used to drive
the horizontal rotating member of the centrifugal force apparatus motors.
4. Procedure
1. The locating pins on the sliding blocks were raised and the blocks positioned so that they
were both on the same distance from the centre. The pins were then pushed down to
locate the blocks firmly on the horizontal member. The distance from the axis to the
pivots of the bell-cranks was measured.
2. A mass of 25g was screwed unto each vertical arm of the two bell cranks then, a mass of
175g was then screwed on the horizontal arm of the two bell-cranks. It was made sure the
magnitude of the masses on the respective arms of the bell cranks was equal.
3. The safety dome was replaced and the motor started and observed using the speed control
unit. The speed was slowly increased until the bell-cranks were flung outwards with an
audible click. The approximate speed at which it occurred was noted. Also; the
movement of the bell-cranks was observed from a position level with the plane of
rotation.
4. The speed the bell-cranks were decreased until they returned to their original positions,
then the speed was increased very slowly and the reading repeated. The speed indicated
on the Tachometer at the instant when the upper arm of the bell-cranks moved outwards
was recorded. If the bell-cranks did not move simultaneously, the speed when the first
one moved was recorded.
5. The magnitude on each horizontal arm of the bell cranks was reduced by 25g while the
body of mass on the vertical arm constant was kept constant .Steps three and four was
repeated and more results down to the 50g mass were obtained.
6. The value of the horizontal masses on the bell cranks was then kept at 175g while the
value of the vertical mass was increased by 25g to 50g .Steps three and four was
repeated.
5. Precautions taken
1. Placing the dome on the centrifugal force apparatus before putting it on.
2. Fixing the masses securely on the axis.
3. Accurate taking of readings on the tachometer.
6. Results
Table 1: Results for varying speed and mass for a constant radius of 8cm
r = 8cm
ma = 25
ma = 50
ma = 75
Centrifugal
mb (g) Force mbg (N) Rev/min ω2 (rad/s)2
Rev/min ω2 (rad/s)2 Rev/min ω2 (rad/s)2
175
1.715
230
581
185
376
162
288
150
1.47
215
507
170
317
147
237
125
1.225
195
417
158
274
137
206
100
0.98
175
336
140
215
122
163
75
0.735
152
254
120
158
107
126
Table 2: Results for varying speed and radius for a constant mass of 50 g
ma = 50
r = 8 cm
r = 9.5 cm
r = 11cm
Centrifugal
mb (g) Force mbg (N) Rev/min ω2 (rad/s)2
Rev/min ω2 (rad/s)2 Rev/min ω2 (rad/s)2
175
1.715
179
352
165
299
150
247
150
125
100
75
1.47
1.225
0.98
0.735
162
150
135
124
288
247
200
169
150
140
130
110
247
215
185
133
140
130
120
100
215
185
158
110
Graphs of radial force (F = Mbg) against ω2 as shown in figure 3. At constant upper mass ma, a
straight line profile passing through origin was obtained demonstrating the linear relationship
between radial force and angular velocity. The deviation of the linear theoretical profile is
attributable to experimental errors, measurement errors and wears on the apparatus due to
friction.
Fig. 3: Variation F with speed and mass at constant radius (r= 8cm)
The radial force also varies linearly with the upper mass ma as shown in figure 4. This was
determined by reading values of F from each line on figure 1 for a particular value of angular
speed ω2. The values are shown in table 3.
Table 3: Values of F at constant speed
ω2 = (300rad/s)
F
ma
0.84
25
1.28
50
1.8
75
Fig 4: Variation of mass at constant speed (r = 8cm, ω2 =300 (rad/s)2)
Varying the radius and maintaining the upper mass ma, constant at 50g, the force was plotted
against angular speed as shown in figure 5. The result shows a linear profile demonstrating that
radial force is directly proportional to angular speed.
Fig. 5: Variation of F with speed and radius at constant mass (ma = 50g)
At a constant speed (ω2 = 200 (rad/s)2), the corresponding values of F on each of the lines in
figure 3 was determined as shown in table 4.
Table 4: Values of F at constant speed
ω2 = (200rad/s)
F
r (cm)
1
8
1.16
9.5
1.32
11
A graph of F against radius r was plotted. The obtained profile, figure 6 shows that radial force is
directly proportional to radius.
Fig. 6: Variation of radius at constant speed (Ma = 50g, ω2 = 200 (rad/s2)
7. Conclusion
The complete series of results affirm the theoretical relationship between radial force and each
of the independent variables therefore validating the equation of radial force, F = mω2r.
8. References
Nelkon, Parker. (1958). Advanced Level Physics. Oxford: Heinemann Educational Publishers.
School of Mechanical and Manufacturing Engineering:centrifugal force apparatus. (n.d.). Retrieved
November 2, 2012, from School of Mechanical Engineering DCU Ireland Website:
http://www.dcu.ie/mechanical_engineering/technical/undergrad/mechanics/centrifugal_apparatus/cen
trifugal_apparatus.shtml
School of Mechanical and Manufacturing Engineering:speed control unit. (n.d.). Retrieved November 2,
2012,
from
School
of
Mechanical
Engineering
DCU
Ireland
Website:
http://www.dcu.ie/mechanical_engineering/technical/undergrad/mechanics/speed_control_unit/speed
_control_unit.shtml
School of Mechanical and Manufacturing Engineering:tachometer unit. (n.d.). Retrieved November 2,
2012,
from
School
of
Mechanical
Engineering
DCU
Ireland
Website:
http://www.dcu.ie/mechanical_engineering/technical/undergrad/mechanics/tachometer_unit/tachome
ter_unit.shtml
Young, F. (1996). University Physics. Massachusetts: Addison-Wesley Publishing Company Inc.