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