Centrifugal force with Cobra4 TEP
Related topics
Centrifugal force, centripetal force, rotary motion, angular velocity, apparent force.
Principle
A body with variable mass moves on a circular path with adjustable radius and variable angular velocity.
The centrifugal force of the body will be measured as a function of these parameters.
Equipment
1 Cobra4 Wireless-Link
1 Cobra4 Wireless manager
1 Cobra4 Sensor-Unit Force ± 40 N
1 Software Cobra4 – multi-user licence
1 Centrifugal force apparatus
1 Cart for measurements and experiments
1 Holding pin
1 Laboratory motor, 220  V AC
1 Gearing 30/1, for 11030-93
1 Bearing unit
1 Driving belt
1 Support stainless steel, l = 250  mm, d = 10  mm
1 Right angle clamp PHYWE
1 Tripod base PHYWE
1 Support base DEMO
1 Bench clamp, PHYWE
1 Fish line, l = 100  m
2 Weight, 150  g, for 11060-00
2 Slotted weight, 50  g, black
Additional materials:
PC, Windows XP or higher
Alternatively:
1 Laboratory motor 110  V AC
12601-00
12600-00
12643-00
14550-61
11008-00
11060-00
03949-00
11030-93
11029-00
02845-00
03981-00
02031-00
02040-55
02002-55
02007-55
02010-00
02090-00
11060-01
02206-01
11030-90
Fig. 1: Setup of the experiment
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TEP Centrifugal force with Cobra4
Tasks
Determination of the centrifugal force as a function
1. of the mass,
2. of the angular velocity,
3. of the distance from the axis of rotation to the centre of gravity of the car.
Set-up and procedure
Perform the experimental set-up as shown in Fig.1. Tie the fishing line to the end of the measuring car facing the rotational axis, threading it beneath the deflection pulley and through the eye. Attach the line to the Sensor-Unit. Make sure that the movable axis of the force meter, the fishing line and the eye are lined up horizontally.
Turn the motor on to begin rotation of the centrifugal force apparatus. Load the “Centrifugal force” experiment. (Experiment > Open experiment). All pre-settings that are necessary for measured value recording are now carried out. To measure the force with the Newton sensor start rotation of centrifugal force apparatus. Start the measurement ( ).
After some time (approx. 10 periods) you can terminate the measurement ( ) and send the results to measure (Fig. 2)
Start a measurement: The periodic fluctuation of force as a function of time is displayed (see Fig. 3).
Fig. 2: Saving measurements.
There is a pumping motion at the force probe, and the eye on the centrifugal force apparatus also moves back and forth slightly when the apparatus is viewed from the side. This eccentricity is intentional; it allows the rotational velocity to be determined. Now, you can count the oscillations lying between two vertical time markings on the graph and calculate the period from this.
Determination of the centrifugal force as a function of the mass
Fig. 3: Typical measurement curve of the central force as a function of time.
The experimental car is gradually loaded with the additional weights. The centrifugal force apparatus with a constant angular velocity is rotated with each given mass m . The force occurring Fz is determined with the help of the interface for the different masses (see Fig. 4).
The car is connected below the pulley to the Sensor-Unit
Force by a thread.
Now a constant angular velocity is set to the motor, which remains constant during the entire course of the experiment. The force Fz experienced by the measuring car without any additional weight is also determined.
Determination of the centrifugal force as a function of the angular velocity
The mass of the experimental car stays constant in this part of the experiment. A predetermined radius (e.
g. r =
20 cm) is marked with a piece of adhesive tape. At different angular velocities, the respective forces Fz are measured.
The angular velocity is calculated from the time of rotation
T .
Fig. 4: Rotating reference system.
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Centrifugal force with Cobra4 TEP
Determination of the centrifugal force as a function of the distance of the axis of the rotational centre and the mass of the measuring car
The mass of experimental car remains constant. A fixed angular velocity is set to the motor. The radius of the orbit r of the measuring car is increased and the respective centrifugal force Fz and the radius r are measured.
After this, you have to determine the average value of the force and the corresponding values of mass, angular velocity and radius for every single measurement. These values are used to create manually data to receive the relationships between the central force and the corresponding parameters.
Therefore, chose ”measurement” and ”enter data manually” to create different graphs.
The results are shown in Figs. 5 –8.
Fig. 5: Typical evaluation of central force as a function of Fig. 6: Typical evaluation of central force as a function of
mass. angular velocity.
Fig. 7: Typical evaluation of central force as a function of
the square of angular velocity.
Fig. 8: Typical evaluation of central force as a function of
radius.
Theory and evaluation
In the reference system which rotates with the angular velocity 𝜔 , the equation of motion of a mass point
(mass m , position vector 𝑟 ) reads: 𝑚 = −∇𝑈 + 𝑚𝑟 × ⃗⃗ × (𝑟 × 𝜔 (1) 𝑑𝑡 𝑑𝑡
The external force field U (gravitational field) is compensated by the track, the angular velocity 𝜔 is constant, and the car is at rest in the rotating reference system ( U = 0; 𝑣 = const. = 0; 𝜔 = const.).
From (1), there only remain the centrifugal force and the compensating force 𝐹 :
⃗⃗ × (𝑟 × 𝜔 .
Since 𝑟 ┴ 𝜔 it follows that
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TEP Centrifugal force with Cobra4
|𝐹 | = 𝑚𝜔
2
∙ 𝑟 . (2)
Figure 3 shows an example of the central force Fz measured for a constant mass and constant rotational velocity.
The angular velocity 𝜔 is calculated from the period T as follows: 𝜔 = 2𝜋/𝑇
A (punctiform) object moving uniformly on a circular path constantly changes the direction of its speed, since the velocity vector is always directed tangentially to the circular path. An object with a mass m moving on a circular path with radius r and with an angular velocity 𝜔 is accelerated at each moment towards the centre of the circle by 𝑎 𝑧
= −𝑎 𝑟
= 𝑟 ∙ 𝜔
2
(4) (centripetal acceleration).
According to Newton’s Basic Law of Dynamics, a force directed towards the centre of a circle must act upon the body, the so called centripetal force:
𝐹 𝑧
= 𝑚 ∙ 𝑎 𝑧
= 𝑚 ∙ 𝑟 ∙ 𝜔
2
(5)
The slope m of the regression line displayed in Fig. 7 is used to derive the proportionality factor 𝑚 ∙ 𝑟 =
0.0253
kg ⋅ m.
Figure 4 is an example of a measurement for the relation between central force 𝐹𝑧 and mass 𝑚 with a constant track radius and constant angular velocity 𝜔 .
The resulting line has a slope corresponding to the product 𝑟 ∙ 𝜔
2
.
Figure 8 shows central force 𝐹 𝑧
as a factor of the track radius 𝑟 , with constant mass 𝑚 and constant angular velocity. The resulting line has a slope corresponding to the product 𝑚 ∙ 𝜔
2
.
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