Teacher s notes 19b An investigation into the energy changes occurring in a pendulum swing

Transcription

1 Sensors: Loggers: Rotary Motion Any EASYSENSE Physics Logging time: 5 seconds Teacher s notes 19b An investigation into the energy changes occurring in a pendulum swing Read The relationship between the length of a pendulum and its period will be known. Students may well know that a simple pendulum is an example of Simple Harmonic Motion. A Rotary Motion sensor enables us to log the angular position of the pendulum throughout its swing. The graph formed shows a sine wave, the creation of a smooth wave shape from a back and forward motion can be a valuable teaching step. The visual conversion of a change of distance into an up and down sine graph makes the use of the Rotary Motion sensor especially worthwhile. Knowing the length of the pendulum, and the mass of the bob, the data logging software enables us to calculate the following quantities from the angular data:- Angular displacement Linear displacement Velocity Acceleration Kinetic Energy Potential Energy Total Energy in the system All these quantities can be plotted against time, so the phase relationships are clearly demonstrated. The energy curves show clearly the transfer from PE to KE and back again. The Total energy is shown to be constant and the conservation of energy is visually demonstrated. If it can be arranged, a demonstration of a really big pendulum is useful in setting the scene, a large ball suspended from string (rope) connected to the ceiling or in a stairwell makes a dramatic starting point. Apparatus 1. An EASYSENSE logger. 2. A Smart Q Rotary Motion sensor set to the pendulum range. 3. Pendulum mass. 4. Thin string for the pendulum. 5. Mass for pendulum bob (at least 100 g). 6. Dynamics System pillar and a large bracket OR retort stand and clamps. Set up of the software Use the setup file 19b Pendulum amplitude RMS. If you wish to set the software and logger up manually use the following details. Recording method Recording time Intersample time Trigger when Graph 5 s 20 ms Pendulum rises above 0 degrees T19b- 1 (V2)

2 A pendulum of about 65 cm will give about three periods in 5 seconds. That is sufficient for analysis. If a longer pendulum is used change the recording time to 10 seconds using the New logging wizard. Pr oduct code 3280 Rotary Motion Sensor Alternatives for creating the pendulum. Attaching the pendulum to the Rotary Motion sensor spindle. Notes Make sure that the pendulum can swing freely; a long pendulum will give better results. There are some holes on the pulley wheel attached to the RMS. A string can be passed through them and attached to the central screw. See the diagrams above for details. We suggest the use of thin string for the pendulum. We found it gave the cleanest swing. The pendulum bob should be a regular shape e.g. a sphere or cylinder. Preferably a dense metal e.g. steel. There needs to be some method of attaching the string to the bob so that it does not start its own oscillation during the pendulum swing. A screw hook screwed in along the axis of the bob, or in the case of a cylinder a thin hole drilled along the axis which has the string threaded down it. Students should measure length of pendulum from centre of fixing screw on the RMS to the centre of the pendulum mass (bob). T19b- 2 (V2)

3 It is best to create a marker to indicate the start point. The pendulum range reads ±20 degrees, pendulum theories work best for angles of less than 15 degrees. If you move the pendulum more than 20 degrees the sensor will record 20 until the angle falls below 20. This will present itself on the curve as a flat top to the sine wave drawn / plotted. The values of the energies (PE, KE and Total Energy) will be small. Typically up to 0.03 J. It depends upon the length of the pendulum, the angle of the swing, and the mass of the bob. Within the confines of a school/college lab it is sensible to have a length of less than a metre m length and a 100 g bob gives energies of about J with a 15 swing (the pendulum bob supplied with the Dynamics system is approximately 100 g). The Rotary Motion sensor coupled with the EasySense software is more than capable of accurately calculating these energies. However when using metres and kg as units it is necessary to use 5 decimal places in the calculations, so that the energies are quoted to at least three significant figures. Results and analysis The data will be logged as the angular displacement of the pendulum, from the vertical. Despite zeroing the pendulum when stationary there is normally still a significant zero error. This would have a significant effect when calculating the potential and kinetic energies. The calculations will have to proceed as follows: 1. Produce the angular data to 5 decimal places 2. Correct for any zero error 3. Convert the angular data in degrees to Radians 4. Calculate the Linear displacement 5. Calculate the velocity 6. Calculate PE, and KE 7. Calculate the Total energy in the system by adding together the PE and KE. The sample data was logged using a pendulum of length = m, and a bob of mass kg. The above screen shows the data as collected Using Post-log Functions the following calculations have been carried out to obtain the next screen:- 1. Pendulum data into 5 decimal places. a) Select Formula. b) Select ax + by. c) Select channel x Pendulum. d) Rename the data channel e.g. Pendulum 5 dec. e) Select Dec places = 5; a = 1; b = 0; Channel limits: Manual, Min = -0.4, Max = 0.4. T19b- 3 (V2)

4 2. Pendulum tared to eliminate the zero error. a) Select Preset Function. b) Select General then Tare. c) Select the data channel to be tared e.g. Pendulum 5 dec. d) Name the new data channel e.g. Pendulum tared; e) Dec places = 5; Tare value = your value. (The tare value is half the difference between the successive max and min peaks. Use Show Statistics (Analysis menu) for a quick way to do this. 3. Convert Degrees to Radians. a) Select Preset Function b) Select Conversions then Convert degrees (angles) to Radians. c) Select the source channel e.g Pendulum tared. d) Name the new data channel e.g. Radians e) Dec Places = 5; select Channel limits: Manual, Min = -0.4; Max = 0.4. The screen shows Pendulum degrees, Pendulum 5 dec, Pendulum tared and Radians. There are now four data sets. The only one needed from now on is Radians so use Delete Data Sets (Edit menu) to delete the data sets in degrees. Use Save As to save the now altered data. Do make sure you use a new file name if you wish to keep the original data. Using Post-log Functions calculate the following quantities: 1. Pendulum - Linear Displacement: use Preset; Motion. 2. Velocity: use Preset; Motion. 3. Kinetic Energy: use Preset; Energy. 4. Pendulum Potential Energy: use Preset; Energy. 5. Total Energy: use Formula: ax + by. Notes: Units - all data uses metres, kilograms, seconds, and joules as appropriate. In all cases select Decimal Places = 5 Set Channel limits to Manual and insert the appropriate limits (suggested limits in the table below) Minimum Maximum Pendulum - Linear Displacement Velocity Kinetic Energy Pendulum Potential Energy Total Energy T19b- 4 (V2)

5 The screen above shows the Radians, Linear displacement and Velocity curves. Note that the velocity curve is 90 (π/2) out of phase with the Radians and Linear displacement curves. There are irregularities visible in the velocity curve. This is due to small variations in the swing of the pendulum, particularly near the middle of each swing, where the pendulum is moving near its maximum velocity. It would be possible to remove these irregularities with smoothing, but that would cause a reduction in the peak value of the velocity, and a greater reduction in the peak values of the KE curve. This screen shows the end result of the calculations, with all the curves visible. The screen is rather cluttered. It is possible to show / hide curves by right clicking on the graph area, select Show or Hide Channels, and select the channels you wish to show. T19b- 5 (V2)

6 The graphs show quite clearly that: The Velocity, PE, KE and Total energy channels are displayed. 1. The Total energy is constant. 2. The Peak value of the KE and PE curves are the same. This leads to a clear demonstration of the conservation of energy actually happening and being logged. The exchange between KE and PE is obvious in the visual display. The noise in the Total energy relates directly to the irregularities in the swing of the pendulum, which is magnified in the differentiation process to produce the velocity data. The biggest dip in the Total energy is highlighted on the screen and is obviously linked directly to the largest irregularity in the velocity curve. T19b- 6 (V2)