Basic Physics II:  Lab 2 − Springs and Simple Harmonic Motion

Background

The most common way of measuring a force is to use a spring.  When we apply a force to solid objects, they deform, or in the case of a spring their length changes.  For many materials the deformation is proportional to the applied force, as long as the deformation is not too large, and they will return to their original shape when the force is removed.  If this is the case, we say the deformation is an elastic deformation.  (If the deformation is too large they deform permanently, or even break.)  If the deformation is not too large, we can write

Force = (constant) ×(deformation).

For a spring, the deformation we measure is a change in length, or how much it stretches.  If we denote the stretch as x, then

F = kx                                                                                                                            1

where k is called the force constant, or spring constant and its units are N/m.  Here F is the force applied to the spring by an external agent.  (Most of the time books talk about the force the spring applies to the external agent.  That equation is Fby spring = ­­­− kx.)  In this experiment you will measure the force constant for a spring.  If we know k, we can measure an unknown force by measuring x and using eqn. 1 above.  Most devices that measure weight or force are based on this principle.  Can you think of a device that measures weight that does not do it this way?  Note that measuring the mass of an object is not quite the same as measuring its weight.

        You could measure k by applying a known force, F, measuring the deformation and saying using k = F/x.  However, we want to know that eqn. 1 works for a wide range of forces, so we measure the stretch or x for several different known forces.  Then we will plot them on a graph of force vs. x.  (Force is on the vertical axis and x is on the horizontal axis.)  This should look like a straight line and the slope of the line will be k. 

Part I:  Measuring the Force Constant

        The experimental set up is shown at the right.  Attach the spring to the clamp on the ring stand and position it so the spring hangs over the edge of the table.  The “narrow” portion of the spring should be at the top.  Put the weight hanger on the bottom of the spring.  Now lower the clamp so that the bottom of the weight hanger is at the top of the meter stick, i.e. at 0m.  This is your first x.  Add a 40g to the hanger and measure the position of the bottom of the hanger; this is your second x.  Again increase the mass by 40g and repeat the measurement to get the third x.  Repeat until you have ADDED 200g on the hanger.  (Don’t add more than 200g to the hanger!  You will need to put some mass on the base of the ring stand to keep it from tipping as you add weight to the spring.)  Each time you make a measurement, record it.  Remember to convert the mass to the corresponding weight in N, i.e. W = Mg, and put the distance x in meters.  Use g=9.80m/s2.


Plot the force or weight vs. the stretch or x.  Draw a trendline through the points, and calculate the slope.  (Use the Linera Regression function in the spreadsheet to calculate the slope and get the uncertainty in the slope.)  Note that the point x=0 and F=0 is a measured point.

        The springs you will use are over wound, so you will have to apply some force to them so the individual coils of the spring do not touch each other.  Usually the weight hanger (m = 50g) is sufficient to do this.  Make sure you can see between the coils of the spring before you start.  In measuring the force constant you do not have to start x = 0 where there is no force on the spring.  This is because eqn. 1 is linear.  Letting x = 0 when the force is 0.49N will not affect the slope of the F vs. x line, which is DF/Dx.  (However, when considering the energy stored in the spring, you need to choose x = 0 as the unstretched length if you want to use PE = ˝(kx2).  The potential energy is a nonlinear function of x.)  

Questions

  1. Use your “measured” k to calculate the force that would stretch the spring 0.62m.
  2. If the spring were stretched .09m by a force, how much added force would it take to stretch the spring from 0.09m to 0.47m?
  3. How accurate do you think your measured k is?  (Why?)
  4. How much variation is there in the distance the spring stretches each time you add 40g?  (This is the variation in the change of x each time you add 40g.)

 

Sample Data Table for the Spring  (Put this in your spreadsheet.)

X (cm)

X (m)

Total Added Mass (grams)

Total Added Mass (kg)

Force (Weight) in N

0cm

0m

0g

0kg

0N

 

 

 

 

 

Part II:  Cyclical Motion

        In this part you will look at the cyclical motion that results when you attach a mass to a spring, displace the system from equilibrium and let it go.  Use the same spring and ring stand that you used in part I.  Now attach the weight hanger of 50g (don’t use the hangers here, but use the “special” masses given to you) to the spring and displace it upward from the equilibrium position about and release it.  It should oscillate up and down.  This is cyclical motion, similar to the earth revolving about the sun.  The time for one complete cycle, i.e. going from the top to the bottom and back to the top, is called the period of the motion, T.  The number of cycles in one second is the frequency, f.  Just as for circular motion, f = 1/T.  You are to measure the period of the motion and calculate the frequency. 

1.     First measure and record the mass of the spring, Ms, and the mass of the weight hanger, Mw, you are going to attach to the spring.

2.     Attach the mass hanger to the spring and lift the mass slightly above the equilibrium position and let it go.

3.     Measure and record the time it takes for the spring and mass to oscillate through 20, 30 or 40 cycles.  (The total time measured should be greater than 25s.  For the smallest added mass you may need to count for 40 cycles to get this, but 20 cycles should be enough for the largest added mass.)  Use this to calculate the period T and the frequency f.  Do this 2 times and average the periods.

4.     Add 40g to the hanger and repeat the measurement. 

5.     Repeat # 4 until you have added 200g to the hanger.

        The theoretical prediction is that the Period, T, is related to the mass and the spring constant by

                                                                                                                  2.

k is the spring constant you measured in part I above.  M = Mw if Mw >> Ms.  Otherwise you have to take Ms into account.  One would expect that M = Mw + Mo where Mw is the mass of the weight you hang on the spring and Mo is the effective mass of the spring.  If the spring’s deformation is uniform under its own weight, a “simple” calculation predicts that M = Mw + Ms/3, or Mo = Ms/3.  One can test this prediction by squaring equation 2 so that one has

                                                                                                                    3.

When you plot T2 vs Mw you should find that the slope is 4p2/k.  The intercept should be 4Mop2/k.  Do this plot and find the k you would get from this, i.e. k=4p2/slope.  Then use the k from part I and the intercept from part II to find Mo.  Compare this k to the one you found in part I and compare Mo to its predicted value of Ms/3.  Are they within one standard deviation of each other?

        You might want to set up tables like the one below for your initial measurements. 

 

Number of cycles

Total time (s)

Period (s)

Frequency (Hz)

T2 (s2)

Mass (Mw in kg)