Class Notes 070918

Calibrating the Rubber Bands

Mark your rubber bands so you can identify which is which.

 
Make a chain of rubber bands and paper clips. 
 
Hang the bag containing two dominoes from the chain and measure the lengths of the four rubber bands.  It's best if you actually mark a piece of paper at the ends of each rubber band so you have a record of their actual lengths and not a potentially incorrect measurement.
 
Repeat using six dominoes.
 
Repeat using ten dominoes.
 
Sketch a graph indicating number of dominoes vs. rubber band length.
 
You don't need to follow all the instructions in the posted version of the experiment, but you should take a look at the instructions for setting up the system and making the graph at
 
http://www.vhcc.edu/dsmith/forms/ph1_rubber_band_calibration.htm 
 
(path is
http://vhmthphy.vhcc.edu/ > Physics I > Physics I Homepage > Assts
 
then either search for 'Rubber Band Calibration' or scroll to Assignment 6, where you will find the link to the experiment).
 
Equilibrium of Three Rubber Bands (first trial)

Going back to the experiments we did with three rubber bands:

 
  • Measure the x and y coordinates of the endpoints of each rubber band.
  • For each rubber band find the change `dx in the x coordinate, and the change `dy in the y coordinate, as you move from the point nearer the paper clip to the point further from the paper clip.  Also sketch an arrow from the first point to the second; this arrow represents the length vector of the rubber band.
  • Use the Pythagorean Theorem to find the length of each rubber band (i.e., the length is sqrt( (`dx)^2 + (`dy)^2 ). ).  This length is the magnitude of the length vector.
  • Find the angle made by the vector with the positive x direction, using the rule theta = arcTan(`dy / `dx), plus 180 deg if `dx happens to be negative.  This angle specifies the direction of the length vector.

You will get three lengths and three angles, specifying the magnitude and direction of each of the three length vectors.

Each length will represent the length of one of the rubber bands.  Using your calibration graphs, determine for each rubber band the number of dominoes (to the nearest .1 domino) it would take to stretch the rubber band to this length. 

The force you obtained for each rubber band is the magnitude of the force vector for that rubber band, representing the force it exerts on the paper clip.  The force vector acts in the same direction as the length vector you found previously; that is, the angles for the force vectors are the same as they are for the length vectors.

If a vector has magnitude F and angle theta, as measured from the positive x axis, then that vector has x and y components F_x and F_y as given below:

  • F_x = F cos(theta)
  • F_y =F sin(theta)

You have sin and cos buttons on your calculator.  If theta is in degrees, then you need to be sure your calculator is in degree mode (on most calculators press the Mode button and make sure the line reading Degrees Radians (Grads) is set to Degrees.  That line might be in a different order and Grads might or might not be present, depending on your calculator; in any case it should be easy to select Degrees.

If your angle is correct, including the correct quadrant (e.g., a third-quadrant angle will be between 180 deg and 270 deg), then your F_x and F_y results will be correct, complete with the appropriate + and - sign.

Equilibrium of Three Rubber Bands (second trial)

The setup is the same as for the first trial, but the following restrictions apply:
  • One rubber band will be in the negative y direction.
  • The rubber band in the first quadrant must be at least 10 degrees from the x axis and at least 10 degrees from the y axis, but cannot be within 10 degrees of the 45 degree position.  That is, the angle must be between 10 deg and 35 deg, or between 55 deg and 80 deg.

The steps of the analysis are identical to those of the first trial.  Find the magnitude and angle of each length vector, the magnitude and angle of each force vector, and the x and y components of the force vector.

Slingshot behavior of rubberBand-and-domino system. 

Making a 'slingshot' out of two of your rubber bands, you can 'sling' a domino from the system.  The energy in this domino can be calculated from is mass, using one of two measurements:

  • You can measure the horizontal range of the domino as a projectile if you 'sling' it off the edge of a table.
  • You can measure how far the domino slides along a tabletop or along the floor.

In both cases you don't want to waste energy by allowing the domino to spin too quickly.  Practice your release a bit to minimize rotation of the domino.

The best release for a given rubber band length is the one that results in the greatest sliding distance or the greatest horizontal range.  You should record the 'best' result you obtain for each given length.

Proceed as follows:

  • Using your system, stretch the rubber bands to their two-domino length and measure the horizontal range when projected from tabletop, and the sliding distance along the tabletop.
  • Repeat using the 6-domino length.
  • Repeat using the 10-domino length (sliding distance for this trial might exceed the length of the table).

Graph sliding distance vs. projectile range.  You may assume that (0, 0) is a point on the graph.

Areas of Force vs. Length Trapezoids

Your graph of rubber band force (in dominoes) vs. length (in cm) can be roughly represented by three trapezoids, each with a slope and an area.

  • For example if you observed lengths of 8.0, 9.1 and 10.4 cm when suspending 2, 6 and 10 dominoes, respectively, you will obtain one trapezoid with 'altitudes' 2 dominoes and 6 dominoes and width 1.1 cm (from 8.0 cm to 9.1 cm is a width of 1.1 cm), and another with 'altitudes' 6 dominoes and 10 dominoes and width 1.3 cm.
  • The rise of the first trapezoid is 6 dominoes - 2 dominoes = 4 dominoes and its run is 9.1 cm - 8.0 cm = 1.1 cm, so its slope is 4 dominoes / 1.1 cm = 3.6 dominoes / cm, approximately.
  • The second trapezoid has a rise of 10 dominoes - 6 dominoes = 4 dominoes and its run is 10.3 cm - 9.1 cm = 1.3 cm, so its slope is 4 dominoes / 1.3 cm = 3.1 dominoes / cm, approximately.
  • The first trapezoid is easily formed into an equal-area rectangle with altitude (6 dominoes + 2 dominoes) / 2 = 4 dominoes and width 9.1 cm - 8.0 cm = 1.1 cm, giving it an area of 4 dominoes * 1.1 cm = 4.4 domino * cm.
  • The equal-area rectangle for the second trapezoid has altitude (10 dominoes + 6 dominoes) / 2 = 8 dominoes and width 10.3 cm - 9.1 cm = 1.3 cm, giving it an area of 8 dominoes * 1.3 cm = 10.4 domino * cm.

The slope of each trapezoid is (change in force) / (change in length), which is by definition the average rate of change of force with respect to length, in units of dominoes / cm. 

  • On the first trapezoid, for example, we would say that force changes, on the avearge, by 3.6 dominoes per centimeter of length.
  • We would expect that adding 3.6 dominoes would therefore increase the length of a rubber band in this range by 3.6 cm, and that to increase the length by 1 cm we would have to increase the length by 1 / 3.6 cm = .29 cm.

The area of each trapezoid will turn out to be a measure of the energy required to stretch the rubber band between the two given lengths.

  • For example the energy required to stretch the rubber band in our example from 2 cm to 6 cm is about 4.4 domino * cm; to stretch from 6 cm to 10 cm the energy is about 10.4 domino * cm.

When springing back from one length to the other an ideal rubber band will release an amount of energy equal to the energy required to stretch it; however actual rubber bands deviate a bit from ideal behavior.

Your observations of the 'slingshot' behavior will be used to determine how much energy each rubber band system releases.