The experiment involves the determination, of the effective mass of a spring (ms) and the spring constant (k).

                                                     

From the equation it can be shown that:

 

Gradient of line = 4π2 

                               k

∴k = 4π2     

        grad’

To find the value of ms the intercept’s on the y-axis and x-axis can be used can be shown that, on the y-axis where the x value of the graph is zero:

C=4π2ms

k

(where C is the value of y on the y-intercept)

∴ms = C×k

             4π2

 

Alternatively you can use the x-value on the x-axis to find the value of ms:

On x-axis, T2=0

∴0=4π2m+4π2ms

   k         k

∴0=m+ms

m = -ms

This formula can also be used to check the value of ms, with that of the other equation.  They should equal each other, to a certain degree of accuracy.

Apparatus

Stopclock, slotted masses, clamp stand, spring, counterbalance weight.

Apparatus Diagram

Results Section

Characteristics of Instruments

Stopclock

Range 00.00 - ∞ seconds,

Resolution 00.01 seconds,

Sensitivity 00.01 seconds,

Mean zero error 00.00 seconds,

Uncertainty 00.10 seconds.

(Masses have a percentage uncertainty of 5%)

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Trial Readings

These were the trial readings taken before the experiment, they were taken to help in the decision of the size of the limits.  The readings are shown in the order that they were taken.  They were taken in this order, to check that the masses, which were to be used were usable.   i.e. the smaller masses did not oscillate to quickly to be measured and that the larger masses did not damage the spring.  Hence the order of the readings.

Main Readings

Procedure/Method

When planning the experiment, there were two methods of obtaining information about the spring, which were discussed one being the one decided upon.  The other being a way of finding out the spring constant (k), by just measuring the extension (x) of the spring, with different sized weights attached to the spring. Which would have given a straight-line graph going through the origin, where the gradient of the line would give the value k.

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As shown in the graph below:

Graph of Force/N against  Extension/m for a spring

This experiment was chosen over the other one, as not only can k be found but ms can also be found.  

The procedure in the experiment was as follows:

1. The apparatus was set up as shown in the diagram, with the counterbalance weight in position as shown, to ensure safety by stopping the clamp and stand from falling over.  
2. Amass was positioned on the spring as shown.  The mass was then started oscillating and the time for a given number of oscillations was measured.  The measurements were taken from a fixed fiducial point (about 4cm from the masses), with the observers eyes level with the point, to ensure there was no parallax error. The readings were taken on the way up for one set of results and on the way down for the next set of results (as shown below).  Observers took it in turn to measure the time of the oscillations to ensure any errors due to reaction times were averaged out.  
3. Once one set of oscillations was recorded then another mass was positioned on the spring (after checking no damage had been done to the spring, to make sure no miss-readings were taken).  Then another set of readings was taken until all the readings in the table had been filled.  
4. Then the apparatus was disassembled and put safely away.  

Calculations/Derived Quantities

 k can be found using the formula,                         k = 4π2     

                                                                                     grad’

k =    4π2

          1.8564

k = 21.3

 

The value of ms can be found using the formula,     ms = −m

           ms = 0.025 kg

and a rough check can be made ms  using the formula,

       

ms = C×k

         4π2

ms = 0.0456×21.27

          4π2

ms = 0.025 kg

(N.B. it is a rough check as it does not take into account any uncertainties in k.)

To find the uncertainty in k and in ms error bars are draw on the graph, the y error-bars were found by

adding, or taking away the value of ΔT2/S2 to/from the value of T2/S2, these error-bars were then plotted on the graph.  The value of the x error-bars were found, by finding the percentage error in the value of the mass on the spring, at that particular point on the graph.

The furthest error-bars were then joined, to find the gradient of the lines between them.  These gradients were then used to calculate the uncertainties of k and ms in the following way:

 

The value of k is worked out using the higher gradient,

k = 4π2

                                                                                     grad’

 

k =    4π2

          2.3581

k = 16.7 N/m

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Then value of k is worked out using the lower gradient,

 

k = 4π2

                                                                                     grad’

 

k =    4π2

          1.3389

k = 29.5 N/m

Then the difference between these figures and that of the average value of k are found,

diff = 29.5 – 21.3

 

diff = 8.2

diff2 = 21.3 − 16.7

diff2 = 4.6

Then the average value of these results must be found,

av = 4.6 + 8.2

        2

av = 6.4

This is the uncertainty in k so it can be said that,

k = 20 +/− 6 N/m

Similarly the uncertainty of ms can be found by using the value of ms which applies to the lines joining up the error bars,

First the lower value of and m are used,

 ms = −m

ms = 0.14

Then the higher value of m are used,

ms = −m

ms = −0.0507

The differences between these values and the average value for ms are found,

diff = −0.05  − 0.025

diff = 0.075

diff2 = 0.14 – 0.025

diff2 = 0.115

The average of these two values is then found,

av = 0.115 + 0.075

         2

av = 0.095 kg

This is the uncurtainty in ms so it can now be said that,

ms = 0.025 +/− 0.1 kg

To find the value of xT av/S the following formula was used, xTav /S = ∑xT

                                                                   3    

To find the value of T/S the following formula was used, T/S = xTav/S

                                           x

To find the value of T2/S2 the following formula was used, T2/S2 = (T/S)2

To find the value of % unc in xT the following formula was used, % unc in xT = ΔxT/S × 100

                                                                                                   T/S

To find the value of % unc in T2 the following formula was used, % unc in T2 = 2 × % unc in xT2

To find the value of ΔT2/S2 the following formula was used, ΔT2/S2 = T2/S2 × % unc in xT2 

                                                            100

Conclusion

The graphs show that the formula, T2= 4π2m+4π2ms   does indeed give a straight-line graph of form Y = mx + c.

                                                                                                    k         k

The graphs show that the value of ms = 0.025 +/− 0.1 kg  and the value of k = 20 +/− 6 N/m      

The objective of this experiment was to determine ms and k of a spring which has been done within the limits, 0.125 and − 0.075 kg for ms and  26 and 14 N/m for k.

Critical Analysis of Results

The experiment could have been improved in the following ways:

1. More readings for each mass could have been taken to get a more accurate mean value.  
2. A greater range of masses could have been used to get a more accurate trend line for the graph.
3. There could have been more time for the experiment, so that the readings etc would not be so rushed
4. The bounces of the spring could have somehow been restricted to the vertical.
5. The measurement of time could be improved, by using a light beam connected to a timer to measure the time.  The apparatus could be set up so that at the bottom of one oscillation the weight would break the beam and start the timer then at the start of the next the weight would break the beam and stop the clock.  This would give a more accurate measurement of time.