Objectives

• Determine the relation between force and extension of a spring and a rubber band

Introduction

A number of physical systems exhibit a linear restoring force when displaced from their equilibrium position. This is described by the relation;

F = -ks    Hooke's Law

            F is the restoring force of the system
            k is a proportionality constant often referred to as the spring constant
            s is the displacement from equilibrium position. 
               Displacement s=P-P₀  where P₀ is equilibrium position (no force acting at P₀)
               and P is displaced position.

Notice this relation is linear (having the appearance of y=mx+b where m=slope and b=0). Applying twice the force will double the displacement. Larger values of k, the spring constant, represent stiffer systems that are harder to stretch.

Hooke's law describes a spring over a limited range of displacements (the relation will not hold if a spring is stretched too far). In this lab you will also consider how well it holds for rubber bands.

References

• Hooke's law and molecular forces
• Rubber bands

• Richard Feynman on Rubber Bands
  Nobel Physicist 3 min talk on rubber bands

Prelab

(Pre-Labs must be submitted to the Physics lab on separate paper (not in your notebook) before or at the beginning your lab period. See Lab Points)

1. Sketch how a graph will appear for applied force and length of a spring with force plotted along the x-axis.
        - On the graph, indicate zero force and zero spring length. (Note, a spring will have some length with zero force)

2. For a graph of applied force along x-axis and length along y-axis,
slope =  length / force.  Write down an equation of how this graph slope relates to spring constant k. Do this by writing an equation (k =) indicating the relationship with the slope (m)

3. Draw a sketch of the experiment "Determining ceiling height with a laser" (from your first lab period). On the sketch clearly show what you measured (showing data, not results which are calculated). Then show your calculation for determining the ceiling height from these measurements. (removing and submitting a notebook page is not acceptable)

Procedure Intro

In this experiment, you will be:

• working separate from your partner
• analyzing a spring and elastic band that is different from your partner's
• collecting data with both a mechanical spring scale and a digital force sensor
• recording data into a table as well as onto a graph at the same time

In working with the digital force sensor

• The software is run from a link under the Physics lab web browser favorites, ForceIntro
• The software is designed to give correct readings when the range switch on the force sensor is set to 10 N
• The sensor needs to be set to read zero for zero force. Do this by pressing the "zero" button in the software with no force applied. 
• You are encouraged to become accustomed with the software and try different things with it. This software will be used in a variety of experiments

In working with the mechanical spring scale

• The scale has an unusual increment that is difficult to read. Take your time with making measurements even to the extent of recording in your notes what you saw.
• Use the Newton measurements of the scale

Of note

• Although your experiment is expected to follow a linear relation (Hooke's law), this may in fact only hold in a limited region.
• Always separate analysis from data. Record data as honestly and completely as possible. Then, as a separate step, consider the data in context of your objective (the analysis or results step).
• Position is data as it is directly measured. Displacement is a result, calculated from two position measurements. 

Procedure

1. Select a spring and a rubber band
These will be analyzed in this experiment. Take note of some of their physical characteristics (width, length, color). Notice that other student's equipment is different. 

2. Experiment setup
    Setup the experiment as in the picture with either the mechanical force scale and the spring or the digital force sensor and the elastic band. You will be measuring position and applied force. The ruler is taped down (use masking tape) in an arbitrary location.
    The setup is arranged such that when no force is applied, your measured position will be something other than zero (zero force must not yield a zero position reading). Position can be measured from any reference point (zero position can be anywhere).
    Record, in your notes, the setup with enough information such that someone could replicate your work (including what spring was used, how the equipment was setup, if digital - how it was connected to the computer and the software used). Test your setup taking note of the maximum and minimum data that is possible for both position and force.

3. Prepare to take data
Prepare a table in your notes to record position and force. Prepare a graph for plotting force and position with force along the x-axis. Scale your graph from your test maximum/minimum data in simple increments (such as multiples of 1, 2, 4 or 5).

4. Take data
For each trial (force applied), record the measured values into your data table and also onto your graph. Data does not need to be taken in equal increments or in sequence. Use the graph to help you in determining where additional data would be useful. Repeating measurements can simply be added to the data table and added to the graph (do not erase any data).

5. Collect a minimum of 5 data points.  

6. Repeat with the other force sensor and object
Use separate data tables and graphs in your notes. On concluding this procedure, you will have two separate graphs with the actual data values in your notes.  One data set and graph for the mechanical force sensor and the spring and another data set and graph for the digital force sensor and the elastic band.

Analysis

1. Draw a best linear fit line on each graph
A best fit is a line that best represents the data points. The best fit does not have to touch every data point and does not have to go through the first and last points. Often the beginning and end parts of the data/graph may exhibit some nonlinearity would be excluded from a linear fit.

2. Determine the slope of each graph
A slope calculation can be done right on the graph. Slope is calculated using widely spaced locations on the drawn line (not from the data points plotted). Use the point format of slope
                       y₂-y₁
        slope =   ------
                       x₂-x₁
with two points from your line (avoiding your data points). Units form part of the slope value. Always calculate slope to a decimal value (do not present slope as a fractional value).

3. Determine the spring constant k for each graph
State the value of k on the graph. Spring constant has units.

4. Summarize in your notebook
Clearly note the slopes and spring constants determined for the spring and elastic used.  Write a concluding statement.

Analysis Theory

• In plotting position with force along the x-axis, the theory of Hooke's law in relating to the graph becomes;