In this experiment, you measure  the parameters --mass m  and  spring constant k --- of an oscillating system and check whether the  period  T is as theory predicts. Consider  the mechanical oscillator illustrated in figures  11-1 and 11-2 of the textbook (Giancoli) .  At equilibrium, the mass  has zero force on it; there   the spring is neither stretched or compressed. When x is negative, the spring is compressed. Otherwise it is stretched.  F_x = -kx is the x-component force causing the mass to return to the origin (x = 0) at all times;  when the mass moves away from the origin, it slows down and when it moves toward the origin it speeds up.  

This explains all kinds of oscillators, whether they  be molecules vibrating back and force in the warm pen you are using to  write your lecture notes,   air molecules oscillating back and force as my voice travels across the lecture room,   or  an  “up and down” moving  piece of  string    supporting  a transverse standing wave such as on a guitar string. Such oscillations are everywhere, including   your quartz watch.  

In  Chapter 11,   Newton’s second Law is applied to a mass m attached to a spring. According to  Newton, the resultant force vector on an object of mass m at a given moment  is m times the acceleration.  In Chapter 11, we explored this law  in  the many examples in which the force  points towards x = 0. In this experiment you will test the theory for the special case of a vertically oscillating mass-spring system. The x component of force F_x = -kx leads to the result that

x= Acos(2πt/T) ,

where A is the amplitude and T is the period.  See section 11-3, especially figure 11-6 and equation 11-8c.  On page 292, we discovered   :

T _TH= 2π· (1)
 


Note that you may choose the positive x direction to be downward which means x is positive when the vertical spring is stretched.  To test the theory, you will measure the parameters (Mass m,  Spring Constant k )  that appear  in the theoretical expression for  T to obtain a theoretical value  T_Th. ---see equation 1.  By a timing procedure, you will directly measure an experimental value T_EX, and then compare  T_Th  with T_EX . (See attached data sheet. )  In particular,  you do the following for the mass-spring system:

(a) Measure the parameters m and k that appear in your expression for the period in  equation (1) and  use them to compute T_TH.

(b) Measure the time t for a known number of cycles n (n = 10) and compute the experimental value of the period T_EX = t/n.  You will measure Tex  5 times and compute the average T_{EX_BEST}. From  this, you will also compute R/N (N= 5) like you did in the  lab on centripetal acceleration; this gives the statistical uncertainty in multiple measurements  when the random errors are larger than the instrument error in a single, “simple”  measurement.   The instrument   error is ΔT _inst =  0.001 s /n  or  0.0001 s/n  ( where n = 10),  depending on the timer resolution.  The uncertainty  ΔT_EX   you choose will either be   ΔT _inst or R/N , which ever is larger.  See attached data sheet. In either case, the uncertainty is rounded to just one digit and we round the average (“best”) value to the same decimal place.

(c)  Compute the uncertainties in the values T_EX and T_TH, then test the proposition that T_TH = T_EX using the criteria we used in centripetal acceleration lab:

|T_EX – T_TH| < 2(ΔT_EX + ΔT_TH),

PROCEDURE

As in figure 11-3, you will hang a tapered spring vertically from a support with the small end of the spring at the top. This spring taper direction is important  for an accurate reading. The narrowing of the spring’s diameter toward the top allows the spring to stretch uniformly along its entire   length. The narrowing toward the top means the upper half of the spring is stiffer than the lower half. This stiffness “gradient”  is necessary because  the  upper part  supports the weight of the lower part and so must be stiffer if  it is to have the same  stretch per unit length as the lower.

(Q-1) You will add a mass of value m = 0.30 kg as suggested in figure 11-3 and allow the system to oscillate  up and down.  Remember, the hanger already has mass 50 g, so you will add 0.250 kg. The formula for the period is given by equation (1) but there is one complication that has to do with the correct value of the mass. While the system vibrates up and down, different parts of the spring move at different speeds. The effective mass m_s,Eff  of the spring adds to the system’s inertia and  kinetic energy storage capacity. It  turns out that the mass m includes  both 0.30 kg and one- third of the mass of the spring; a calculus derivation shows that m_s,Eff =  m_s/3, where m_s is the spring mass.  You may enter this expression under (Q-1) of the attached  data sheet.

(Q-2)    Beginning  with  equation (1), it is easy to derive  (using   Math 1 and 3 methods)
the expression for the uncertainty in the theoretical  value T_TH of the period :





The expression contains the “best “ values from discussions below.     Δm and Δk  are the uncertainties in the oscillating mass m and spring constant k discussed below.

(Q-3)  To find the value of the spring constant k, use a sliding caliper jaw to record the position x’ of the bottom of the weight hanger with weights of mass M = 100, 200, 300 , 400 and 500 grams (g) suspended from the spring.  Plot values of force in newtons  vs  x’ in meters and find the spring constant k in N/m with uncertainty Δk.  
Note: The uncertainties in x’   and the hanging masses M should be very small . Thus, the  precision of your value of k will be limited by the graphing process. Thus, you can proceed without using error bars which will be discussed in a future lab. The least squares fit computational method for determining the slope will also be addressed later.  The following suggestions are for graphically measuring, or estimating, the slope k:

(1)  Do not try to make the line pass through any particular data point. Sometimes a line passes through none of these points.  On the other hand, when the  points have very little “scatter” ,  the line may seem to go through all the points.    Since you  give  greater weight to points with short or no error bars, you will make an extra effort to pass the  line close to these points.
(2) Extend the line beyond the data points to the right and /or left.
(3) Determine a right triangle, called the slope triangle. Do this by choosing two points on the graph, indicating them with arrows, that approximately span the domain of the x’ data.  Compute ΔF_s  and Δx’.
Compute the slope k_best from their ratio—k = ΔF_s  / Δx’ . Follow these guidelines

·         (a) Begin and end the Δx’ interval on major grid vertical lines; major grid lines are  the darker lines that divide the smallest grid divisions into groups of 5 or 10.  Do not choose a vertical line that passes through or very close to a data point. The Δx’ interval should be rather wide, about as wide as the domain of your  plotted x’ values. Draw arrows at the two intersection points—to be used in computing the slope---where the graph passes through the two vertical lines.

·         (b) Do not use values from the data table to compute the slope even if they are from data points that appear to be on or near the line. Use the two values located at the arrows you drew in (a). They are determined by the line representing the best information from ALL your data.

·          (c )In your slope(k)  computation,  show the beginning and ending values of x’ and F_s that you read from the graph at the two arrows, by substituting them into your expression used to find the slope. Note: slope  = k_best  = rise/run= ΔF_s  / Δx’.


Since the uncertainties in ΔF_s  and Δx’ are so small, the precision is limited by the drawing process. In this case, you can compute the uncertainty in the slope k like this:
Δk = lc_y/ Δx’, where lc_y is the least count of the grid for the y (F_s) axis and Δx’ is the base of the slope of the triangle used to find   k_best  .

(Q-4) Weigh the spring to find m_s  and compute the effective mass m_s,Eff.  Enter the value m_o = 0.30 kg.
Compute the total mass m = m_s,Eff  + m_o. Use this value along with k_best   to find T _TH = 2π·



Compute the uncertainty ΔT_TH assuming an uncertainty of  1 g for your mass measurements.  See (Q-2) and (Q-3). 

(Q-5) Suspend  0.30 kg from the spring, displace it slightly and determine the experimental period T_EX. 

Make at least 5 trials and count at least 10 cycles for each trial:

You will measure T_EX  5 times and compute the average T_EX-BEST. From  this, you will also compute R/N, where N = 5;  this gives the statistical uncertainty of multiple measurements  when the random errors are larger than the instrument error in a single, “simple”  measurement.   The instrument   error is ΔT _inst =  0.001 s /n or 0.0001 s/n  ( where n = 10),  depending on the timer resolution.  The uncertainty  ΔT_EX   you choose will either be   ΔT _inst or R/N , which ever is larger.  In either case, the uncertainty is rounded to just one digit and we round the average (“best”) value to the same decimal place.

(Q-6)  Compare the discrepancy between the theoretical and experimental values to the sum of their uncertainties.