Physics Laboratory 4

The radial and tangential acceleration of a pendulum

Note: In this lab students will work in pairs to take data.  The two students may analyze their data individually or together.  However each student must hand in a lab report describing the results and answering questions in his or her own words.

Objectives:

In this lab students will use two PASCO acceleration sensors to measure the radial and tangential acceleration of a chosen point on a compound pendulum.  The compound pendulum consists of a rod and a holder for the acceleration sensors.  The acceleration sensors are mounted as shown in the figure above.  The pendulum can rotate about an axis passing through one end of the rod.  A marker on each acceleration sensor indicates the position of the single chip accelerometer, which is the heart of the acceleration sensor.  The markers on the two sensors line up.  They are the same distance r from the pivot point.  Students will measure the radial and tangential acceleration of the marked points. 

Front View Back View

asensor.gif (35334 bytes)

The PASCO acceleration sensor is designed for use with any PASCO computer interface and the Data Studio software.  It can measure accelerations ranging up to 5g with an accuracy of 0.01g (g = acceleration of gravity, 9.8 m/s2).  The sensor produces a bipolar output that may vary from +5g to -5g, depending on the direction of acceleration.  Each acceleration sensor has two built-in features for configuring it to a particular application:

  1. a tare button, used to set the output of the sensor to 0 regardless of the acceleration being applied, which allows the effect of the earth’s gravitational field to be nulled;
  2. a filter with two settings, "slow" or "fast", setting the frequency response of the acceleration sensor to a range suitable for the application.

For this experiment students will select the "slow" setting. .

The acceleration sensor is an electronic device, but a simple mechanical model illustrates its operation.  Assume a ball is suspended in a box between two identical springs as shown in the figure below.  One end of the box is marked with a + sign and the other with a - sign.

When the box is horizontal and at rest, both springs have the same length.  When the box is vertical and at rest, with the + end up, the spring connected to the + end is stretched and the spring connected to the - end is compressed.  A sensor outputs a positive voltage proportional to the amount of stretch of the spring connected to the + end.  If the box is vertical, but the - end is up, the sensor outputs a negative voltage proportional to the amount of stretch of the spring connected to the negative end.

When the box is accelerating in a direction indicated by an arrow pointing from - to +, the spring connected to the positive end will stretch and the spring connected to the negative end will be compressed.  The sensor outputs a positive voltage, indicating positive acceleration.  When the box accelerates in the opposite direction, the sensor outputs a negative voltage, indicating negative acceleration.

In this experiment the marked points on the acceleration sensors move along a circular arc centered on the pivot point.  The speed and direction of motion of each point are changing, the sensors are accelerating.  If the acceleration sensors are mounted as shown in the figures above and zeroed when the pendulum is hanging motionless, sensor 1 measures a quantity a1 = -(v2/r) + g(1 - cosq) and sensor 2 measures a quantity a2 = -(dv/dt) - gsinq as a function of time, as the pendulum swings back and forth.  From these readings students will deduce the acceleration of the chosen point on the pendulum.  The radial component is given by ar = v2/r and the tangential component is given by at = dv/dt, where v is the component of the velocity of the chosen point along the tangential direction and r is the distance of the chosen point from the pivot point.  To deduce ar and at from their data, students must know the angle q.  They will use a rotary motion sensor to measure q as the pendulum swings back and forth.

We consider q to be positive, if the pendulum is displaced towards the right, and we consider the tangential components of the velocity and acceleration to be positive, if they point in the counterclockwise direction.

Students will use the Data Studio program to take data.  They will graph ar and at versus q and study the behavior of the radial and tangential acceleration as a function of deflection angle.

Equipment needed

pendulum with two acceleration sensors
rotary motion sensor
photogate and photogate stand
clamp & rod

Procedure

Make sure the Science Workshop Interface is on.  If not, turn it on and restart the computer.
Use the clamp to mount the rotary motion sensor and the pendulum to the table as shown.

Plug the acceleration sensor 1 (front) into analog channel A and the acceleration sensor 2 (back) into analog channel B of the Science Workshop interface.
Plug the rotary motion sensor into digital channels 1 (yellow) and 2 (black).  
Make sure the pendulum can swing freely.  Wrap the wires to the acceleration sensors so they swing freely with the pendulum and do not hit anything.  Then let the pendulum hang motionless, and position the photogate so that the tip of the plug to sensor 2 just blocks the photogate. 

Make sure the pendulum can swing freely through the photogate.  Do not be sloppy with your setup, a sloppy setup will produce bad data.
Plug the photogate into digital channel 3.

Make sure the pendulum is hanging perfectly still.  Open the Data Studio program.  
In the "Sensors" window find the icon for the acceleration sensor and drag it onto analog channel A of the picture of the SW interface. 
Double click on the acceleration sensor icon.  The "Sensor Properties" window will open.
Under the "General" tab, choose "Sample Rate" 50Hz, fast.
Under the "Measurement" tab, make sure that only acceleration (m/s/s) is checked.
Drag another icon for the acceleration sensor onto analog channel B, double click the icon and choose the same sensor properties as above.
Drag the icon for the rotary motion sensor onto digital channel 1, double click the icon, choose sample rate, 50Hz and measurement, angular position, Ch1&2 (deg).
Drag the icon for the photogate onto digital channel 3, double click the icon and choose measurement, Time In Gate, Ch3 (s).
Click "Options" in "Experiment Setup".  Choose Delay Start, Data Measurement, Time In Gate, Ch3 (s) is above 0s.  Choose Automatic Stop, none.  Click "OK". 
Make sure the pendulum is hanging perfectly still.  Once more double click on the acceleration sensor 1 icon.  Click the calibration tab and then press the tare button on the side of the acceleration sensor (front).  The "Value" that is displayed in the gray area should now be very close to zero.
Make sure the pendulum is hanging perfectly still.  Once more double click on the acceleration sensor 2 icon.  Click the calibration tab and then press the tare button on the side of the acceleration sensor (back).  The "Value" that is displayed in the gray area should now be very close to zero.
Click the calculate button on the menu bar.  
Into the definition textbox type at=-a2-9.8*sin(theta).  Click "Accept".
Click "Please define variable a2" and choose "Data Measurement, Acceleration ChB".
Click "Please define variable theta" and choose "Data Measurement, Angular Position Ch1&2".
Click the "Deg" button because the angles are measured in degrees.  Click "Accept".
Click "New" to define another calculation.
Into the definition textbox type ar=-a1+9.8*(1-cos(theta)).  Click "Accept".
Click "Please define variable a1" and choose "Data Measurement, Acceleration ChA".
Click "Please define variable theta" and choose "Data Measurement, Angular Position Ch1&2".  Click "Accept".
Close the calculator.
From the "Displays" window drag the "Graph" icon onto the calculation "at=-a2-9.8*sin(theta)".  Graph 1 will open. 
This graph will display the tangential acceleration at as a function of time.  If at is positive, the direction is counterclockwise (towards the right), if at is negative, the direction is clockwise (towards the left).
Now drag the "Graph" icon onto the calculation "ar=-a1+9.8*(1-cos(theta))".  Graph 2 will open.
This graph will display the radial acceleration ar as a function of time.  If ar is positive, the direction towards the pivot point (the center of the circle),
Move the pendulum to the right and hold it somewhere around 30o.
Click "Start" and then release the pendulum.  Let it swing for approximately 3 seconds and then click "Stop".
Repeat a few times to produce a plots which are as smooth as possible.  Spikes in that plot are produced if the pendulum is vibrating, wobbling, or when the cables interfere with the motion.
Keep only the plots of your best run.

Open Microsoft Word and prepare a report using the template shown below.

Name:
E-mail address:

Laboratory 4 Report

In a few sentences summarize the experiment.

Show your plots of ar and at versus time.
What do these plots tell you?  Discuss the behavior of ar and at  as a function of time.
Go back to Data Studio, and drag "Angular Position, Ch1&2 (deg)" onto the x-axes of graphs 1 and 2.  This will produce plots of ar and at versus q.  Show the graph of ar and at versus q in your Word document .
What do these graph tell you?  Discuss the behavior of ar and at  as a function of q.

Print out your Word document, and hand it to your lab instructor, or save your Word document (your name_lab4.doc) and attach it to an e-mail message to your lab instructor.