What factors affect the period of a Baby Bouncer?

To examine what factors determine the period of each oscillation:

* How does the Bouncer’s performance depend on the size of the baby?

* What effect has the sort of support on the bounce?

DIAGRAMATIC REPRESENTATION OF THE SYSTEM:

The oscillation pattern of such a mass-spring system can be characterised as a harmonic oscillator.

CONSIDERATION OF THE THINGS WHICH CAN BE CHANGED IN SUCH AN OSCILLATING SYSTEM:

The things which can be changed in such a system that will impact the period of each oscillation are:

* The mass of the baby (load)

* The material which the spring support is composed of

* The length of the spring support

* The stiffness of the spring support

* The strength of the spring support

* The thickness of the spring support

* The amplitude

[Whether the baby bounces up and down in a vertical manner or whether he or she imparts a rotational motion or forward/backward motion will have some impact on the oscillation period].

The above points are called variables and experiments could be conducted to investigate the effect of changes in each variable (whether absolute, such as changing the material of the spring support, or progressive, such as the addition of progressively large masses to the spring support to determine its elasticity.

) In order to perform each series of experiments it would be necessary to keep all of the other components of the oscillating system constant (constants), whilst altering the specific variable to be investigated.

DISCUSSION OF LIKELY IMPACT OF THESE VARIABLES ON THE PERIOD OF OSCILLATION:

Looking at each variable in turn:

* The mass of the baby (load): This is investigated and analysed later on in this report .

In summary, it can be seen from the experimental data that the spring used obeyed Hooke’s Law (discussed in more detail later in the report). The spring used was composed of ductile material (a material which can be stretched).

The likely impact of increasing the load on the period of oscillation is that the oscillation period will increase in duration.

* The material which the spring support is composed of: Usually spring supports are made out of metal or rubber. There is an increasing trend, for safety reasons, for the spring support to be made out of rubber. Employing different materials in the construction of the spring support would have a direct effect on the duration period of the oscillation, for example rubber, has a Young Modulus 0.01 GPa, Lead 18 GPa, Aluminium 70 GPa, Brass 90 – 110 GPa, Copper 130 GPa and steel 210 GPa. The Young Modulus is the ratio of stress to strain resulting from tensile forces, provided Hooke’s Law is obeyed.

* Length of the spring support: Because springs are coiled, they can extend significantly even under a relatively small load. The extension is intimately associated with variables such as strength, thickness, material etc. If the wire the spring is made of was not coiled, it would still be possible to stretch it, but this could take a relatively large force.

In practice, most metals are not particularly elastic and usually can only be stretched by circa 0.1% of their original length. Beyond this they become permanently deformed. However, rubber is not as stiff, and strains of several hundred percent are achievable. The period of oscillation of the Baby Bouncer will be affected by the length of the spring support because the longer the support the longer the period of oscillation.

* The stiffness & the strength of the spring: To change the shape of the spring, a pair of forces is required:

When a spring is squashed, (thus shortening it), the forces are compression forces, but when the spring is stretched the forces are tensile forces.

The terms “stiffness” and “strength” sometimes can be confused. However, “stiffness” describes the spring’s inertia to being extended or compressed, whereas “strength” quantifies how much stress (defined as the load acting per unit of cross-sectional area of the wire) is required to reach the point when the material breaks. The value of stress at this point is called the ultimate tensile stress of the material.

It is interesting to note that ductile materials exhibit plastic behaviour beyond the elastic limit and become permanently deformed. The stiffness of the spring support is likely to affect the oscillation period of the system since an increase in stiffness should result in a decrease in period time due to less extension. The higher the strength of the spring, the more likely that the oscillation period will be lengthened.

* Thickness of the spring: With an increase in thickness it follows that the cross sectional area increases. Hence, it is likely that a thicker spring will result in a shorter oscillation period than a thinner spring, provided that other factors are kept constant.

* The Amplitude: This is the maximum distance that an object moves from its equilibrium position. A simple harmonic oscillator moves back and forth between the two positions of maximum displacement, at x = A and x = – A. I believe that the likely impact of an increase in amplitude is an increase in period time as oscillation will increase.

* There are other factors which could affect the oscillation period of a spring in an extremely minor way, such as air resistance, frictional resistance and temperature.

EXPLANATION OF WHY THE PERIOD OF OSCILLATION MATTERS IN THE BABY BOUNCER, PLUS CONSIDERATION OF WHAT THE EFFECT WOULD BE ON THE BABY’S RIDE IF THE FREQUENCY WAS TOO HIGH, OR TOO LOW:

The period of oscillation in a real Baby Bouncer system is of critical importance. This is because the safety aspects of such a system are of paramount importance since a baby is fragile and highly sensitive to stresses and strains. The brain, internal organs, muscular and skeletal systems of an infant at this stage, are in the process of development, hence must be treated carefully.

Additionally, the objective of using a Baby Bouncer from the parents’ perspective is to aid the baby’s preparation for learning to walk and also, used correctly, the baby exercises its legs muscles and can have great fun. Therefore a sensible period of oscillation is desirable for optimum usage and enjoyment.

If the frequency is too high the effect on the baby’s ride could be dangerous, manifesting itself in the baby being shaken around and possible regurgitating it’s food. Also, a high frequency could result in harm to the baby’s muscles and internal organs. The baby could become scared of the Baby Bouncer and anxious about being placed in it on future occasions. Hence, it is vital that the manufacturers of the Baby Bouncer ensure that the spring support exhibits physical characteristics which ensure that this does not happen.

In addition, if the frequency is too high it could create structural faults within the spring, which could create a dangerous situation and may also prevent the bouncer from being safely used again.

By contrast, if the frequency is too low the baby may become bored and find it less fun. Additionally, the spring could eventually become excessively stretched, reducing the eventual elasticity of the spring.

Thus, it can be seen how important it is for companies which manufacture Baby Bouncing systems to carefully conduct experiments in order to determine the optimal system configuration for the product to be sold successfully. It is not just the aesthetic appeal of a bouncing system which will sell it, but more importantly, the safety profile of the system. In the unfortunate event of such a system’s failure it could lead to significant injury to the infant and possible legal implications. (If such an event did occur, the security of fixing the Baby Bouncer to the door frame would have to be examined, in addition to whether the door frame was in good condition, i.e. had physical integrity and was not rotten etc.).

The scientists / engineer’s working for a Baby Bouncer manufacturer would employ a tensile testing machine capable of producing large compressive and tensile forces to investigate the physical characteristics of the spring support.

The oscillation characteristics of any Baby Bouncer system will respond to the range of variables previously listed. In addition, the joint (limb) kinematics and muscle activation patterns produced by infants who have developed differing bouncing skill levels will have impact on the oscillation characteristics of the system. The relationship between several components of bouncing could be determined (in order to investigate the difference between a “skilled” and a “less-skilled” infant in terms of bouncing ability).

The components which could be investigated could be:

* The oscillation pattern of the mass-spring system which can be characterised as a harmonic oscillator ;

* The infant’s contribution to the bouncing behaviour, which can be characterised in part as a forcing function and in part as a harmonic oscillator ;

* The combination of these two components which corresponds to the output (or the bouncing behaviour).

This would be a relatively simple experiment to conduct by using a load cell (attached to the ceiling) to record the loading of the spring and harness: using two force platforms mounted in the floor beneath the infant’s feet to measure vertical ground reaction forces and placing each infant in a dark full length baby suit with reflective markers placed strategically at several placed on the suit. Then a video camera could be placed at right angles to the sagittal plane to record the results of the experiment. Several surface-mounted electrodes could be placed on each infant’s leg muscles to help determine the pattern of their specific interjoint co-ordination.

Experiments such as this have been performed, the results showing that there was a trend from a chaotic to an organised pattern of interjoint co-ordination as the level of “bouncing skill” increased. Infants that were classified as moderately-skilled and skilled, bounced at one of two distinct frequencies, being 1.5 times and 2 times the resonant frequency of the spring respectively. Similarly, the baby contributions and kinematics of the lower limbs were distinctly different for the two frequencies of bouncing. This suggests that one group of infants, in the experiments conducted employed a spring-like control mechanism for lower limb movement whilst the others incorporated a point-specific (focussed, position-specific) control of forces into the mechanism for lower limb movement.

INDIVIDUAL EXPERIMENT – INVESTIGATING THE IMPACT OF ONE OF THE PREVIOUSLY DISCUSSED VARIABLES:

AIM:

To investigate the impact of mass on the period of oscillation in a Baby Bouncer

HYPOTHESIS:

I predict that a steady increase in mass will result in a steady increase in the period of oscillation, from my general and scientific knowledge.

It was apparent, from observation of my younger brother when he was a baby, that he was able to bounce very quickly in his Baby Bouncer. The period of oscillation looked to be shorter than the oscillation period when he was six months older, when he had gained considerable weight (mass). Although his leg muscles may have increased in power and size, I believe the predominant reason as to why the period of oscillation on his Baby Bouncer was longer, at an older age, was due to his increase in mass.

Hooke’s Law is a well known law which states that the ‘Extension (x) produced in an object is proportional to the load applied (F), provided that the elastic limit is not exceeded’. This means that an increase in mass will result in an increase in extension length of the spring. Accordingly, the period of oscillation will increase because the spring has to extend further, due to the increased extension. It also means that a decrease in mass results in a decrease in extension, therefore a reduced period of oscillation, due to the spring having to extend less. This can be illustrated below:

Hooke’s Law may be written as F = k x, where k = F / x and is called the spring constant (otherwise known as the spring’s stiffness). This is the force per unit extension, (or the force needed to extend the spring by one metre in length). The spring constant can be calculated from a load-extension graph, such as the one below:

The gradient of the section A to B gives the spring constant, thus the spring constant (k) is 1 / gradient.

Beyond point B the graph is no longer a straight line, indicating that the spring is not behaving according to Hooke’s Law. The reason for this is that the spring has probably become permanently deformed and stretched beyond its elastic limit. However, in my experiment, I believe that the spring will not reach its elastic limit as the maximum mass I am using is not adequate to deform the spring.

I predict that doubling the mass applied will result in doubling the extension of the spring.

Mass is proportional to force, meaning an increase in mass results in an increase in force, (and vice versa). This therefore means that in this system, acceleration must be constant in order to obey Newton’s Second Law of Motion, which states that ‘Force is equal to the mass of the object multiplied by its acceleration’. The direction of the acceleration is always in the same direction as the force. As the spring lowers, it accelerates because of the force pulling the mass down – gravitational potential energy and as it rises, its undergoes negative acceleration due to the force acting on it – kinetic energy. The total energy of the system during oscillation, (sum of its potential and kinetic energy) remains constant if there is no damping, (the process whereby oscillations die down due to a loss of energy). Therefore acceleration remains constant in the system.

Using the equation,

It is visible that an increase in mass will result in an increase in period time (when using the same spring), as the spring constant remains the same, (the spring constant of the spring I will use can be calculated using a load extension graph, explained earlier). Pie, (?) is a constant number, (equal to 22/7), and therefore remains constant. This must mean that mass proves to be proportional to the period time, which is illustrated in the graph below:

PILOT INVESTIGATION:

Before conducting this experiment, a preliminary (or pilot) investigation was conducted in order to check for any discrepancies with my initial plan, and to observe the trend that should be expected during the actual investigation.

The spring size had to be decided upon. Initially, an investigation was conducted with a very long, narrow spring. However, the extension proved to be too great, making it harder to measure each oscillation and increasing the space required for the experiment. I wanted to experiment to be as confined as possible, which would allow others more space to operate and give myself less space to look after. Therefore a 2.1cm, fatter spring was used and it proved very successful as the extension with a large mass was small enough to be easily measured. This spring will be used in this investigation.

The clamp which the spring was attached to, was stable when the smaller masses, (100g and 200g) were applied. However, when the large masses, such as 500 grams were applied to the spring, the clamp stand proved to be unstable and amost toppled off the desk. To combat this problem, I decided to place books on the base of the clamp stand, giving it adequate weight to remain stable. The trouble with the clamp stand is illustrated below:

In order for spring oscillation to commence, an initial force must be applied. Originally, a 2cm downwards pull was applied to the spring. However, when using the small masses, the extension was very small, and therefore hard to measure. As the initial force applied is proportional to the extension, I realised a larger force needed to be applied. Therefore a 4cm force was applied, which gave an ample oscillation for times to be recorded from, hence this force being used in this investigation.

METHOD:

Apparatus required:

* Clamp stand

* Spring, (2.1cm long and 1.5cm wide)

* Weights, (5 individual 100g weights)

* Weight holder

* Stop clock

* Ruler

Once all the listed apparatus has been obtained and safety goggles have been put on, the apparatus should be set up as illustrated below:

Diagram of apparatus:

Basic method:

Once the apparatus has been set up as above,

* The initial downward force of a 4cm extension will be applied to the spring. This will be a constant additional force due to the minor damping present.

* The time for ten oscillations will be recorded. Each oscillation will be measured when the mass passes the set mid-point on the ruler, (which is the position where the spring naturally hangs once mass has been applied, with no additional force). The mid-point will be different for each mass tested.

* The Spring extension will be recorded.

* This procedure will be carried out three times for each mass tested.

* The different masses tested will be 100g, 200g, 300g, 400g and 500g. Therefore the procedure will be carried out fifteen times in total, providing there are no anomalous results which would require repeating.

* The results will then be clearly displayed and analysed.

FAIR TEST:

When conducting this experiment, many factors should be considered to make it as fair as possible.

The input variable in this experiment is the mass applied to the spring. Its effect on the period of oscillation will be analysed, and therefore is the output variable.

There will also be controlled variables, or constants. The same spring will be used for each experiment, which is vital as changing the spring will change the stiffness, (spring constant) and therefore alter the period of oscillation.

For each attempt, the same initial additional force of 4cm will be applied vertically. This is of paramount importance, because if different initial forces were applied, the spring would obtain different amounts of energy, therefore affecting the period of oscillation for each attempt, which would make the test unfair.

The same person will apply the initial additional energy to the spring as different people may judge the distance differently, which would affect the force applied and therefore affect the oscillation period. Consistency of experimental technique is important in order to conduct the experiment in a fair and accurate way.

RELIABILITY:

Reliability and repeatability are crucial in order for any experiment to succeed. This experiment will be kept as reliable as possible by employing a few simple procedures:

The time for ten oscillations will be measured instead of measuring one only. Not only is this far easier, but it enables an average time for one oscillation to be calculated. The oscillation period for each mass will be recorded three times, therefore giving three averages – i.e. for each mass, one oscillation will be measured thirty times. The averages will be put together in order to discover a reliable period of oscillation for each mass used.

Human error will be aimed to be kept as minimal as it can be, by being as accurate and consistent as possible in experimental technique.

SAFETY:

Few safety measures can be made in this experiment, however there are some. The spring will be firmly secured by the clamp stand to prevent it from becoming loose and coming off the stand. This would result in the weights falling on the floor (or someone’s foot). The spring may fly across the room. Safety goggles will be worn just in case the spring does become loose and heads towards one’s eyes. All other standard laboratory procedures and safety standards will be strictly adhered to.

RESULTS:

Mass (g)

Time 1 (s)

Time 2 (s)

Time 3 (s)

Average time for 10 oscillations (s)

Average time for one oscillation (s)

Extension of the spring (cm)

100

5.97

5.88

6.16

6.003

0.600

3.1

200

7.00

7.18

8.43

7.537

0.754

7.1

300

10.50

9.45

10.03

9.993

0.999

11.0

400

12.03

11.59

11.92

11.847

1.185

15.1

500

14.42

14.52

14.25

14.397

1.440

19.2

GENERAL OBSERVATIONS:

* The mass extended the spring according to my prediction

* There was sometimes a pendulum-like motion of the mass. This was particularly true for the smaller masses which have a smaller extension. However, this posed no problem for data collection as results were seen as anomalous and the test was repeated.

* The effect of the pendulum-like motion was due to the spring being extended at an angle, and the force being applied in a direction other than vertical.

In order for the time taken for one oscillation to be graphically represented, the times will have to be made larger, using index notation, shown below:

Mass (g)

Average time taken for one oscillation (s)

Time taken for one oscillation x 102

100

0.600

60.0

200

0.754

75.4

300

0.999

99.9

400

1.185

118.5

500

1.440

144.0

ANALYSIS:

The actual experimental results have been expressed graphically in terms of three scatter graphs:

(i) Period of one oscillation plotted against the mass applied.

(ii) Oscillation period squared against mass applied.

(iii) Load-extension graph

Careful consideration and analysis of these graphically displayed results suggests that:

(for graph (i) ), my prediction was correct in suggesting that an increase in mass would result in an increased period of oscillation. This can be clearly seen from the positive linear correlation between the mass and the oscillation period. It is interesting to observe that the increasing intervals in oscillation time are reasonably constant, (circa 0.2 seconds) for every 100 gram increase in mass applied.

(for graph (ii) ), it can be seen that there is a positive linear relationship between the period for one oscillation squared and an increase in mass. This can be explained by the equation:

where, T = period time (seconds),

? = 22 / 7 (remains constant),

k = spring constant (stiffness, which also remains constant)

m = mass (grams)

(for graph (iii) ), it was clear that there appeared to be a perfect correlation between an increase in mass and an increase in extension of the spring (being circa 3cm), (positive linear correlation). Thus, it can be seen that Hooke’s Law was obeyed. The spring did not reach its elastic limit as predicted because the maximum mass used for the experiment was not sufficient to exceed the spring’s elastic limit.

Given that Hooke’s Law has been obeyed, and F = k x, it can be seen that k = F / x, k being the spring constant, otherwise known as the spring’s stiffness. This is the force per unit extension (or the force required to extend the spring by 1 metre in length).

The spring constant can be calculated from a load-extension graph, such as Graph(iii). The gradient of the line of best fit gives the spring constant.

Thus, the spring constant (k) = 1 / gradient

In the case of Graph (iii), k = 1 / (? y � ? x)

= 1 / (12cm � 300g)

= 25

Springs composed of different materials will have a range of different spring constants.

If the line in Graph (iii) was extended significantly, at some point the straight line would break down. The reason for this would be that the spring’s elastic limit would be exceeded and the spring would become permanently deformed.

Looking at the load-extension graph, the strain energy stored can be calculated. It is equal to the area under the load-extension graph.

An alternative way of looking at this is that the strain energy equals 1/2 Fx, (and is expressed in Joules).

Strain energy is the energy in the solid which has been deformed. If a spring support is strained elastically (and the elastic limit is not exceeded), the energy can be recovered. However, if the spring has been plastically deformed, some of the work done has gone into moving atoms passed one another and this energy cannot be recovered (i.e. the metal of the spring warms up slightly).

The amount of work done during the extension of a spring is:

WORK DONE = FORCE X DISTANCE MOVED,(in the direction of the force).

In addition to all of the above, it is important to consider other appropriate theoretical points which are of significance to the understanding of such a physical system as a Baby Bouncer. An appreciation and understanding of stress in such a system is important. The stress on the spring support is defined as “The load acting per unit of cross-sectional area of the wire”.

Thus, STRESS = LOAD (F) � CROSS-SECTIONAL AREA

The strain is the extension per unit length produced by tensile or compressive forces, i.e. “it is the fractional increase in the original length of the spring”.

Thus, STRAIN = EXTENSION � ORIGINAL LENGTH

N.B. Strain is a ratio – i.e. it has no units, but both ‘extension’ and ‘original length’ must be expressed in the same units as the measurement, (e.g. metres).

Strain is sometimes expressed as a percentage.

The stiffness of the spring material that is undergoing experimental stretching can be calculated by the ratio of stress to strain. This is called the Young Modulus of the material:

YOUNG MODULUS = STRESS  STRAIN

(Young modulus units are usually Pascals (Pa) or Nm -2)

Hence, the Young Modulus is the ratio of stress to strain resulting from tensile forces, provided that Hooke’s Law is obeyed.

The Young Modulus of a particular material making up the spring support of a Baby Bouncer ‘describes’ its degree of stiffness when the material is acting in an elastic way. Hooke’s Law is obeyed until a discontinuity occurs in the physical structure of the spring support and the elastic limit of the material is reached, (thus it approaches breaking point).This did not occur in this experiment.

It is imperative to ensure in a Baby Bouncer that the elastic limit of the spring support is never reached (for safety reasons, primarily).

The elastic limit is the point (just after the limit of proportionality) beyond which the spring support would cease to demonstrate physical elasticity (i.e. in the sense that it does not return to its original shape and size when the distorting force [in this case, the Baby’s weight] is removed).

The point, just after the elastic limit, at which a distorting force causes a major change in the material of the spring is termed the yield point.

In a ductile material, (i.e. a material which can be stretched), the internal structure alters because the intermolecular bonds between the molecular layers break and the layers flow over one another. This change is termed plastic deformation (the material becomes plastic). It continues as the force is increased and the material eventually breaks. [A brittle material, by contrast, will break at its yield point].