Introduction In the everyday language use, the centrifugal force is often referred to as the cause of the occurring force during a uniform non-linear motion. Situated in a moving object that changes its direction, you can feel the centrifugal force pushing you to the outside. Actually, the centrifugal force is a pseudo force that occurs in a moving system, only. An observer outside of this system is not able to experience that force. A mass can move on a circular path because of the centripetal force, which drives the moving object continuously to the centre of the orbit. Without the centripetal force, the object would move on a linear pathway. A body inside of the moving object would continue its motion on a linear path because of its inertia, but experiences the centrifugal force because of the acceleration of the surrounding object. The centrifugal force has the same absolute value as the centripetal force but shows in the opposite direction. The mass in motion as well as its radius and its angular velocity influence the centripetal force. In the experiment, the centripetal force is investigated by use of a force sensor that is situated close to the centre of rotation and connected to the mass by a fishing line that keeps it on its orbit. Keywords Centripetal force, centrifugal force, pseudo force, orbit, rotation, angular velocity. Educational objective In this experiment, the influence of the quantities angular velocity ω, mass m and radius r on the centripetal force F=m ω 2 r will be understood. Fig. 1: Experiment set-up P6000660 PHYWE Systeme GmbH & Co. KG All rights reserved 1
Centripetal and centrifugal force Equipment 1 Cobra4 Wireless Manager 12600-00 1 Cobra4 Wireless-Link 12601-00 1 Cobra4 Sensor-Unit Force ±40 N 12643-00 1 Support rod, stainless steel, l = 250 mm, d = 10 mm 02031-00 2 Right angle clamp PHYWE 02040-55 1 Fishing line, d = 0,5 mm, l = 100 m 02090-00 1 Cart for measurements and experiments 11060-00 1 Holding pin 03949-00 2 Weight, 150 g, for cart 11060-01 2 Slotted weight, 50 g, black 02206-01 1 Centrifugal force apparatus 11008-00 1 Driving belt 03981-00 1 Bearing unit 02845-00 1 Laboratory motor, 220 V AC 11030-93 1 Gearing 30/1, for laboratory motor 11029-00 1 Tripod base PHYWE 02002-55 1 Support base DEMO 02007-55 1 Software measure Cobra4 14550-61 Additionally required 1 PC with USB interface, Windows XP or higher Tasks 1. Determination of the relation between centripetal force F and angular velocity ω. 2. Determination of the relation between centripetal force F and the rotating mass m. 3. Determination of the relation between centripetal force F and the radius r of the circular movement. 1) Different angular velocities: set-up and procedure Set-up the experiment as shown in Fig. 1, but without the experimental motor. The Sensor-Unit Force 40 N has to be attached to the support rods in such a way that the hook with the attached fishing line is at the same height as the cart. The line should be parallel to the plane and run below the pulley without touching it. In the first experimental part, the additionally applied masses and the length of the fishing line can be chosen freely. To obtain the angular velocity from the force measurement, the entire set-up should possibly be tilted horizontally, slightly (compare with observations in Fig. 2). Start the PC and Windows. Plug the Cobra4 Wireless Manager into the USB port of the PC. Turn on the Cobra4 Wireless-Link with the connected Cobra4 Sensor-Unit Force 40 N. Start the software package measure on the PC. Load the experiment Centripetal force (Experiment > open experiment > ). All pre-settings required for a direct recording will be loaded. 2 PHYWE Systeme GmbH & Co. KG All rights reserved P6000660
If necessary, tare the sensor before starting. Therefore, double click on the corresponding measurement channel and click on the button tare. Manually rotate the centrifugal force apparatus in such a way that the measuring range of the sensor is not exceeded and start the recording of the measurement in measure l. After the centrifugal force apparatus has come to a stop, end the measurement n and transfer the data to the main programme for further evaluation. 1) Observation and results Fig. 2 shows the course of the centripetal force after having pushed the centrifugal force apparatus until just before it comes to a stop. The blue curve corresponds to a horizontally aligned set-up whereas the red curve was determined with a slightly tilted set-up. The latter shows the force alternating around a decreasing mean value. This oscillation can be used to determine the current angular velocity for each period. If the oscillations are hard to recognise, the measurement should be repeated with a slightly bigger tilt. Fig. 2: Progress of the force (for cart mass m=453 g, radius r=30 cm) from the jolt of the centrifugal force apparatus until short before its stop. Measured with a horizontally aligned set-up and for a tilted set-up. 1) Evaluation The periodic time of the oscillation can easily be determined with the measuring tool and thereof, the angular velocity can be calculated. By determination of the mean value over one period, the corresponding force can be determined. Plotting of these values against the squared angular velocity ω 2 will show a curve as shown in Fig. 3 (acquire the measured values manually). The marked linear regression shows that P6000660 PHYWE Systeme GmbH & Co. KG All rights reserved 3
Centripetal and centrifugal force the values are approximately on a straight line. The centripetal force F therefore is proportional to the squared angular velocity ω 2, and for a constant radius r also proportional to the square of the orbital speed v. Fig. 3: Centripetal force depending on the square of the angular velocity ω 2. 2) Mass dependency: set-up and procedure To examine the dependency of the centripetal force on the mass of the moving object, the laboratory motor has to be coupled to the set-up as shown in Fig. 1. This way, the several measurements can be performed with the same angular velocity. After setting a proper angular velocity, it must not be changed during the consecutive measurements of this experimental part. Switch off the motor directly between two measurements without adjusting its rotational speed. Set the centrifugal force apparatus to rotate by the motor and start the recording in measure l, as soon as the displayed force remains roughly constant. Stop the measurement n after about 20 s and add a new measurement (repeated measurement). Repeat the measurement with different cart weights and note the respective masses. Also regard the mass of the empty cart, not only the additional weights. Finally, transfer all measurements to the main programme. 4 PHYWE Systeme GmbH & Co. KG All rights reserved P6000660
2) Observation and results Fig. 4 shows the centripetal force F for the empty cart (m=53 g) and with up to four additional weights in steps of 100 g, each. Except for slight periodic oscillations because of a not thorough horizontal alignment, the force remains constant during a measurement and increases for larger masses. Fig. 4: Centripetal force for five different masses m and a radius r=30 cm. P6000660 PHYWE Systeme GmbH & Co. KG All rights reserved 5
Centripetal and centrifugal force 2) Evaluation In the range showing the constant force except for the periodic oscillations a mean value can be calculated. If you plot the mean values of the centripetal force F from Fig. 4 manually against the mass m, you should obtain a result as shown in Fig. 5. These data points also are on an approximate straight line, indicating that the centripetal force is proportional to the mass m for a constant radius r and a non-varying angular velocity ω. Fig. 5: Centripetal force depending on the mass m. 3) Radius dependency: set-up and procedure Analogous to the investigation of the mass dependency, the dependency of the centripetal force on the radius of the circular motion can be examined as well. For this purpose, the angular velocity is kept constant again during the entire series of measurements by the use of the laboratory motor. The distance r between the centre of the cart and the rotational axis is modified by the use of different lengths of fishing line. 3) Observation and results Fig. 6 shows the results of six measurements with different radii r. The centripetal force F oscillates again because of a not thorough horizontal set-up periodically around a mean value. For bigger radii, the force increases. 6 PHYWE Systeme GmbH & Co. KG All rights reserved P6000660
Fig. 6: Centripetal force for six different radii r and mass m=453 g. 3) Evaluation In the range showing the constant force except for the periodic oscillations a mean value can be calculated. If you plot the mean values of the centripetal force F from Fig. 6 manually against the radius r, you should obtain a result as shown in Fig. 7. These data points also tend to be on a straight line, indicating that the centripetal force is proportional to the radius r for a constant mass m and a non-varying angular velocity ω. Fig. 7: Centripetal force depending on the radius r. P6000660 PHYWE Systeme GmbH & Co. KG All rights reserved 7