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Investigating the vibration of a mass-spring system
Section A: Determining the spring constant
Section B: Qualitative analysis
Section C: Quantitative analysis
Section D: Quantitative analysis – using logs
Section E: Evaluation
Introduction
In this experiment, you will investigate how the time period T of oscillating mass-spring system varies with
the mass applied m. You will then determine the spring constant k of the spring.
Equipment provided
· Clamp and stand
· Meter rule
· Wire and snips to make a fiducial mark
· Stopwatch
· Spring
· Slotted hanger with masses (to suit stiffness of springs: 10g, 50g, or 100g discs)
Diagram
Procedure
Safety
1. Take care that the clamp stand is secure on the bench and will not topple as masses are added.
2. Take care to avoid a mass falling off the spring and either causing damage/harm by the falling mass
or by the flying spring.
Section A: Determining the spring constant
Method
Set up a metre rule vertically alongside the spring. Attach a short horizontal wire to the lower end
of the spring so that you can measure to the same position on the spring each time.
Measure and record the original length and loaded length L of the spring.
Add a mass to the spring, carefully so that the mass-spring system does not oscillate, and record the
new length.
Repeat the procedure until the spring is maximally stretched without being overstretched.
(Overstretched means that the spring will be permanently distorted in length).
1. Identify the techniques that could be used to perform the investigation as accurately as possible.
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Table of results
2. Repeat the above steps until you have 6 different values of m. Record all your measurements in a table.
Also include in your table values of force applied, F, and the extension of the spring, x
Table 1
m (kg) L (m) x (m) F (N)
0.000 0.021
0.100 0.125 0.104 0.981
0.200 0.168 0.147 1.962
0.300 0.208 0.187 2.943
0.400 0.249 0.228 3.924
0.500 0.289 0.268 4.905
0.550 0.312 0.291 5.396
Graph 1
3. Plot a graph of force on the y axis and extension on the x axis. Draw a line of best fit.
4. Determine the gradient of the line.
Gradient = 23.827
0.05 0.1 0.15 0.2 0.25 0.3 0.350
1
2
3
4
5
6
f(x) = 23.8269575168258 x − 1.51283715968527
Force vs Extension
For
ce (
N)
x (m)
The force applied and the extension are related by the equation;
F=kx
where k is the spring constant.
5. Determine the value of k.
k= F/x
k=m and as m=d(F)/d(x)
k=23.827
k = 23.827 Unit Nm -1
6. Justify the number of significant figures for k.
As force is recorded to 3 & 4 s.f. and extension is recorded to 3 s.f. so k can only be
justified to 3s.f. as this is the minimum amount of s.f. used.
Section B: Qualitative analysis
Method
Repeat the loading of the spring, but this time, for each mass m, gently displace the mass-spring
system by a few cm and release so that it is oscillating. Set up a fixed horizontal fiducial mark next
to the mass, each time, to mark the equilibrium position.
Measure the time t for an appropriate number of oscillations.
Determine the period T of the oscillating mass.
1. Identify the techniques that could be used to perform the investigation as accurately as possible.
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2. Repeat the above steps until you have 6 different values of m. Record all your measurements in a table.
Also include in your table values of T2, Ig (m / kg) and Ig (T / s).
Table 2
Graph 2
3. Plot a graph of T on the y axis against m on the x axis.
0.000 0.100 0.200 0.300 0.400 0.500 0.6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f(x) = 1.09534246575342 x + 0.324424657534247
Time Period vs Mass
Mass (kg)
Tim
e Pe
riod
(s)
Time for 10
oscillations, t (s)
m
(kg)t1 t2 tave T (s) T2 (s2) lg m (kg) lg T (s)
0.100 3.86 3.81 3.84 0.384 0.147 -1.000 -0.4160.200 5.79 5.80 5.80 0.580 0.336 -0.699 -0.2370.300 6.83 6.83 6.83 0.683 0.446 -0.523 -0.166
0.400 7.80 7.83 7.82 0.782 0.612 -0.398 -0.1070.500 8.63 8.52 8.58 0.858 0.736 -0.301 -0.0670.550 9.00 9.10 9.05 0.905 0.819 -0.260 -0.043
4. Describe and explain your observations using relevant knowledge and understanding of physics. You
should make reference to your graph.
Describe Explain
Section C: Quantitative analysis
Graph 3
Plot a graph of T2 on the y axis against m on the x axis. Draw a line of best fit.
0.000 0.100 0.200 0.300 0.400 0.500 0.6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
f(x) = 1.45413698630137 x + 0.0191698630136988
T2 vs Mass
Mass (m)
T2 (s2)
1. Determine the gradient of the line.
Gradient= 1.4541
The time period T and the mass m are related by the equation
T=2π √ mk2. Determine the value of k
T2=2π(m/k)
T2*k= 2πm
k=(2πm)/T2
Integral of m=d(m)/d(T2)=k
k=( 1.4541) -1
k = ……………………………..Unit kgs -2
3. Justify the number of significant figures for k.
T 2 is recorded to 3s.f. and m is recorded to 3s.f. so k can only be justified to 3.s.f
Section D: Quantitative analysis – A2 Physics
Graph 4
Plot a graph of lg T on the y axis against lg m on the x axis. Draw a line of best fit.
-1.100 -1.000 -0.900 -0.800 -0.700 -0.600 -0.500 -0.400 -0.300 -0.200
-0.450
-0.400
-0.350
-0.300
-0.250
-0.200
-0.150
-0.100
-0.050
0.000
f(x) = 0.491832848091422 x + 0.0882958843563058
lg T vs lg mlg m (kg)
lg T (s)
1. Determine the gradient of the line.
Gradient = 0.4918
2. Determine the y-intercept of the line.
y-intercept = 0.0883
The time period T and the mass m are related by the power law
T=μmν
where µ and ν are constants for the oscillating mass-spring system.
3. Show that the equation above may also be written as
lg T = lg µ+ ν lg m.
lg T= lg µ + lg mv
lg T= lg µ + v lg m
as log(mv)= v log m and log(ab)= log a + log b
4. Use this relationship and your value for the gradient and the intercept to determine
i. a value for ν
ν = ……………………………..
ii. a value for µ
µ = ……………………………..
The spring constant k is related to µ by the equation
μ=2π√k
5. using your value for µ and the above equations, determine a value for k.
k = ……………………………..
6. Justify the number of significant figures for k.
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Section E: Evaluative
1. Determine the percentage uncertainty in the following quantities for the first row of your data in
table 2;
i. t
ii. T
iii. T2.
2. Determine the percentage difference between the values of k obtained in section A and D.
Percentage difference = ……………………………..
3. From graph 4, determine the percentage uncertainty in the following;
i. gradient
Percentage uncertainty in the gradient = ……………………………..
ii. y-intercept
Percentage uncertainty in the y-intercept = ……………………………..
iii. ν
Percentage uncertainty in ν = ……………………………..
iv. µ
Percentage uncertainty in µ = ……………………………..
v. k.
Percentage uncertainty in k = ……………………………..
4.
Comment upon the following for your investigation;
i. Accuracy
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ii. Reliability
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iii. Validity.
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5. Suggest some of the significant limitations of the experiment and how each of these can be improved.
For one of these limitations discuss the effect it may have on the experimental value of k obtained in
section D.
Limitations Effect on k Possible improvementsGive details of
improvements