Simple Harmonic Motion Experiment: Analyzing the Behavior of a Helical Spring | Study notes Engineering Physics

EXPERIMENT B

Keele University Physics/Astrophysics Laboratory

School of Physical and Geographical Sciences Experimental Scripts

Simple Harmonic Motion : The Helical Spring

1. Introduction

A helical spr ing is a simple ex ample of a large cla ss of dynam ical struct ures whose low energy

excited states perform simp le harmonic motion. Exa mples of this type of behav iour are to be found in

virtually a ll areas of physics, both classic al and quantum, rangin g from springs, pendulums and sound

waves to the vibrations of molecules and crystal structures.

In its equilibrium state the forces on a system balance out to zero. When the system is excited out

of equilibrium this balance is broken and there is a restoring force wh ich acts to pu ll the system back

towards the equilibrium st ate. For s imp le h armon ic motion to occ ur the r estor ing for ce m u st be l ine arly

proportional to the displacement of the system aw ay f rom its equilibrium state. Simp le harmon ic motio n

is a periodic motion in which the displacement of the system oscillates about its equilibrium value. In this

experiment on a spring both a spects of this motion are examined (i) the dependenc e of the restoring

force on the springs displacement and (ii) the oscillatory simple harmonic motion itself.

2. Experimental Procedure

i) The Static Deflection (Displacement) of the Spring

a) Suspend your spring vert ic al ly f rom the arm of the retort stand and on it s lo wer end han g a ma ss

of 0.1 kg. You shou ld find that th is loa d is s uff icient to just open al l the turns of the sprin g so that

the spring may be expected to show uniform properties for extensions and loads greater than this.

b) Measure the length

of the spring. Take thi s length to be the distance bet ween the end turns or

some other convenient marker s furthe r apart th an th ese. Repe at th is mea s urement for a w ide range

of higher

values but

DO NOT LET

EXCEED 1.3 m

in order not to exceed the elastic lim it

of the spring. You will need to estimate the error in the lengths

that you measure, what do you

think would be an appropriate estimate ?

For small masses a sprin g is expected to obey Hooke's Law where an incre ase in the force of

extension

ΔF

is proportional to the corresponding increase in length

ΔL

according to the equation.

ΔΔ ΔFgMkL=− =−

(1)

where

is the force constant for yo ur particul ar spring. Plot a graph of

against

and using a least

squares fitting procedure (i.e. the Linefit program described in section 2.5) fit the straight line equation

LL g

kM=+

⎛

⎝

⎜⎞

⎠

⎟

(2)

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Keele University Physics/Astrophysics Laboratory 28

Simple Harmonic Motion : The Helical Spring

Introduction

A helical spring is a simple example of a large class of dynamical structures whose low energy excited states perform simple harmonic motion. Examples of this type of behaviour are to be found in virtually all areas of physics, both classical and quantum, ranging from springs, pendulums and sound waves to the vibrations of molecules and crystal structures. In its equilibrium state the forces on a system balance out to zero. When the system is excited out of equilibrium this balance is broken and there is a restoring force which acts to pull the system back towards the equilibrium state. For simple harmonic motion to occur the restoring force must be linearly proportional to the displacement of the system away from its equilibrium state. Simple harmonic motion is a periodic motion in which the displacement of the system oscillates about its equilibrium value. In this experiment on a spring both aspects of this motion are examined (i) the dependence of the restoring force on the springs displacement and (ii) the oscillatory simple harmonic motion itself.

Experimental Procedure i) The Static Deflection (Displacement) of the Spring a) Suspend your spring vertically from the arm of the retort stand and on its lower end hang a mass M, of 0.1 kg. You should find that this load is sufficient to just open all the turns of the spring so that the spring may be expected to show uniform properties for extensions and loads greater than this. b) Measure the length L of the spring. Take this length to be the distance between the end turns or some other convenient markers further apart than these. Repeat this measurement for a wide range of higher M values but DO NOT LET L EXCEED 1.3 m in order not to exceed the elastic limit of the spring. You will need to estimate the error in the lengths L that you measure, what do you think would be an appropriate estimate? For small masses a spring is expected to obey Hooke's Law where an increase in the force of extension Δ F is proportional to the corresponding increase in length Δ L according to the equation. Δ F = − g Δ M = − k Δ L (1)

where k is the force constant for your particular spring. Plot a graph of L against M and using a least squares fitting procedure (i.e. the Linefit program described in section 2.5) fit the straight line equation

L = L (^) 0 + ⎛⎝⎜ gk ⎞⎠⎟ M (2)

Keele University Physics/Astrophysics Laboratory 29

to the data and hence determine the force constant k and the constant L 0. In order to convert the error in the slope to the error in k you’ll need to use one of the formulae in section (2.2.d.i).

ii) Simple Harmonic Motion of the Spring A simple standard theory, given in any textbook on mechanics, shows that a mass M suspended from a spring with a force constant k, will execute simple harmonic motion when displaced from equilibrium. The periodic time T, i.e. the time to complete one cycle, of this oscillation is given by the equation

Measure the periodic time T for a particular mass M by measuring the time TN for the mass to complete N periods of oscillation (i.e. cycles) and divide this value by N. You should repeat this measurement a number of times in order to be able to estimate the error bar in the value of TN.. You should then repeat this process for a wide range of masses M. In order to use this raw data to get a value for k the spring constant you need to plot a graph of T^2 against M. This means that you must process tour TN values in order to obtain values of T^2 and their error bars. We can do this processing using a spreadsheet in the following way. For the sake of this example we’ll say that we choose N=10 and repeated each measurement of TN 5 times. Thus in A1,B1,..,E1 we enter the 5 values of TN for the first mass M 1 , A2,B2,...,E2 we enter the 5 values of TN for the second mass M 2 and so on down through rows, 3,4,5,etc. In cell F1 we enter the formula =AVERAGE(A1:E1) and in cell G1 we enter the formula =STDEV(A1:E1) and then fill down columns F and G. This calculates the mean and standard deviation of the T 10 values for each mass. In fact we want the error bar for the mean of the T 10 values rather than the standard deviation (see section(2.2.c.i)), so in cell H1 enter the formula =G1/SQRT(5.) and fill down. Now we want the periodic times so in I1 enter =F1/10. and in J1 enter =H1/10. and fill down. The columns I and J are now the periodic times and their error bars. In cell K1 enter =I1I1 and fill down to get the T^2 values, and in cell L1 enter =2.I1*J1 and fill down to get the error bars for T^2 (see

Simple Harmonic Motion Experiment: Analyzing the Behavior of a Helical Spring, Study notes of Engineering Physics

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T = 2 π Mk (3)

T 2 =^42

⎝⎜^

⎠⎟^ (4)