F+N+mg = 0
The above equation represents the forces at equilibrium.
F = friction force (Newton)
N = normal contact force
M = being the mass
G = the gravitational pull i.e. 9.8mg/s2
F = mg sin α
N = mg cos α
But as the object starts to slide F = μ N
Substitute F and N:
mg sin α = μmg cos α
μ = mg sin α
mg cos α
μ = tan α
This also illustrates that mass becomes irrelevant to friction, thus I will not be taking mass into account in my investigation.
As illustrated below the angle has an influence on acceleration.
mg sinθ -μN = ma
mg sinθ -μ(mg cosθ) = ma
g(sinθ-μ cosθ) = a
Acceleration at which the child travels will eventually fall.
The following formula( Newton’s Second Law of Motion ) is used to find the new acceleration:
ma = - μmg
again mass does not affect acceleration as show :
ma = - μmg
a = - μg
as I have established, μ is 0.35 and gravity (g) is 9.8 ms-2
Assumptions:
To set up this mathematical model of the slide I must simplify the real life situation in order to be able to use the mathematical principles.
- Assumption: Treat the child as a particle.
This is because the position of the child on the slide will have an impact on the velocity the child will travel down the slide.
E.g. - the child holding on to the sides of the slide
-
the position of the child’s leg i.e. placing their feet on the slide→ the shoes/ trainers worn are usually made out of material of high friction e.g. rubber. Which will increase the co-efficient of friction thus decrease velocity.
Both of these will act as a resistance force and because it varies with each
child whether they do this, which means that there is no fixed value, which
can be calculated. Thus the assumption to treat the children as particles,
where the mass will be focused on one point will be made.
This will to prevent complications and make calculations simpler.
- Assumption: Gravity is the only force taken into consideration and to ignore any external factors, such as air resistance.
Factors such as air resistance, will affect the velocity at which the child travels when he/she slides down. This resistance will vary and will require specialised equipment to calculate. Therefore by ignoring the effects of air resistance on a particle (the child) that is moving at an inclined to the horizon a simplified mathematical model for the particle motion can be obtained.
-
Assumption: That the change in direction at point B will happen instantly, therefore no loss in speed.
Change in direction, for example if the slide levels off at the end it will mean a change in acceleration, where there will be a loss in speed
- Assumption: the child sits at top of the slide thus having a fixed initial velocity of 0i+ 0j.
Initial velocity will vary from child to child as they approach the slide, this will in turn affect the speed of which the child travels down the slide. For example, if the child pushed at the top of the slide, which will increase its velocity.
- Assumption: all the children will wear the material of clothing that I have chosen as part of my preliminary test.
The material of clothing will have an affect on the co-efficient of friction, which varies with the type of material. This will in turn affect the acceleration of which the child travels down the slide. So a constant μ value will be used through this investigation.
- Assumption: There will be no other factors that will affect friction, such as footprints or mud on the surface of the slide.
This will increase friction and as a result decrease velocity, and hence the time of the journey. Due to the fact that it will be very difficult to calculate such friction as it will not have a constant value, but would vary. Therefore, I have decided to not take it into account. This will make my investigation simpler to model, thus calculate.
Before proceeding in my design I must make a few specification:
Firstly the age group the slide is aimed for. This will allow me to make decisions about the slide, such as the velocity to suit the age group that will be using it.
I have decided to design a slide for children of aged 5 years old to 8 years old.
After going to research in local playgrounds I have come to decide that the length of my slide will be 3.5 metres, which appear to be the average length for slides.
The material used for the slide, which I will later investigate and test into that determine friction.
Variables and constants
The constants in my investigation are:
- Gravity (g) = 9.8 m/s2
- Length of slide (S) = 3.5 metres
- Co-efficient of friction (μ) = to be calculated
The variables in my investigation are:
-
The angle at which the slide is horizontal from (θ)
- Velocity (V)
- Acceleration (a)
Investigation into materials
In this experiment I will investigate into different materials that could be used for the surface of the slide against clothing material wore by the children and analyse how the nature of the material affects the co-efficient of friction.
Having looked at slides from local parks and playgrounds, I have decided to base my testing on three materials; plastic, wood and metal, which seemed to be the three most common. From this test, I will collect data on each material of their friction. This will be against material clothing worn by children. For this I have researched into clothing worn by children ages 5 to 8 yrs old, through visiting children clothes store and found out that the three most commonly worn materials were denim, cotton and polyester, which I will use to test.
The purpose of this test will allow me to look at the relationship of friction between materials and find the co-efficient of friction, which tells us when the child will start to slide.
EQUIPMENT:
-
Material to represent the slide → Wood
→ Plastic
→ Metal
-
Clothing material → Denim
→ Cotton
→ Polyester
-
Ruler → use to measure the height and length
-
Textbooks → use as the mass of the child
I will collect 3 flat surface boards (representing the slide) of the material I have selected to test. Firstly, I will measure the length of each board; this will help with calculating the angle, as it would not be practical to measure the angle physically, furthermore it will not be accurate.
Textbooks will be wrapped around by the three different material clothing, which will be representing the child.
One end of the board will be placed against the wall to prevent movements.
The textbooks wrapped in clothing will then be placed on the opposite side. Initially, the board will be horizontal and gradually the board from one end will be lifted, ensuring that when lifting both sides is being lifted at the same level.
The board is lifted until the point where the textbooks start to slide; at this point the height is recorded. The diagram below shows how the experiment was set up.
Accuracy and reliability
This will be repeated three times for each material, to produce reliable results, where the average height will then be calculated. Also for accuracy the length s and height will be measured to the nearest 0.5cm, where u will be able to visually read, from the measuring ruler. Furthermore, the same textbook was used for each of the test, hence the surface areas remains constant, making the results reliable. As a greater surface area will increase friction, thus affect the point where the textbook began to slide.
Table 1 – wood
Against wood denim required the smallest angle until friction became limiting. As a result it produced the lowest co-efficient of friction value (μ). Polyester was not far from denim, it seems as though it had the same effect on wood. Polyester required an angle of 0.2° greater than denim’s in order for it to start to slide.
Table 2 – plastic
With plastic, polyester produced the lowest co-efficient of friction value. On the other hand denim produced the highest co-efficient of friction with plastic, although it produced the lowest co-efficient of friction with wood.
Table 3 – metal
Metal produced the similar results as plastic, where polyester has the highest
co-efficient of friction.
Average of co-efficient
Due to the fact that children wear a mixture of these types of clothing, I have come to a decision to take the average co-efficient of friction:
Wood = (0.49+0.61+0.50)/ 3
= 0.53 (2 d.p)
Plastic = (0.40+0.37+0.27) / 3
=0.35 (2 d.p)
Metal = (0.67+0.57+0.31) / 3
= 0.52 (2.d.p)
Validation of experiment:
For my investigation I will take the co-efficient value of 0.35, which will remain constant through out my calculations, this is because the lower the co-efficient, the smaller the angle inclined to the horizontal is required, in order for the object begins to slide. For safety reasons, I have chosen this co-efficient.
This co-efficient is valid to use, due to the fact that the experiment produced reliable and accurate results and I use a valid method for the test.
Firstly the use of the textbooks, although it is not a true representative of the child, i.e. it is much lighter in weight than a child. However, as I established before mass does affect the co-efficient of friction, so it will not affect acceleration.
Suitable materials were used to work out the co-efficient of friction, as the materials I used to test are the same materials used on the surface of slides in children’s playground, having researched into them. Also the clothing materials are common material worn by children. Due to these reasons my co-efficient of friction is a valid value to use.
Analysis of results
From the results, it is clear that wood is the material with the highest co-efficient of friction, followed by metal and then plastic.
With the clothing Denim is has the highest co-efficient of friction out of the three materials I tested. Nevertheless, cotton has proven to have a very high co-efficient, being extremely close to denim’s co-efficient of friction with polyester having the lowest co-efficient of friction.
Not only I am I considering the co-efficient friction, but I am also going to have a look at its practicality.
Firstly, wood is not waterproof unless vanished and will produce splinters in the long run, which will become hazardous to the children, thus become a safety issue.
Similarly metal in not waterproof and will rust, even though the material is high in strength.
I have decided to choose the material with the lowest co-efficient of friction, this will mean that the angle of the slide from the horizontal will not be greatly large in order for the child to be able to slide down. Angle corresponds with height of the slide, therefore having a smaller angle means a lower height and so will be safer for the children to use. In this case plastic seems to be the most favourable with an average co-efficient of 0.35. Plastic also is waterproof and will not produce splinters of any kind. Furthermore, many slides are made from plastic according to my earlier research.
Now that I have established the co-efficient of friction (which it will remain constant for the rest of my investigation), it will be used for further calculation to help me find acceleration and thus velocity.
To obtain the acceleration I must work out the angle of the slide from the horizontal.
However, the angle must agree on the following:
α > tan –1 0.35
Where the angle must be greater than the angle of co-efficient in order for the child to overcome friction and so be able to slide.
From graph one it is clear that the angle and acceleration has a positive correlation, as the angle increases so does acceleration as the force to overcome friction comes greater. From the gradient I have worked out that the acceleration increases by 0.18 ms-2 for every degree that increase.
The decision of the angle that will be used, thus acceleration will again be on the basis of safety. A point of no friction i.e. at high angle acceleration may become dangerous; not only because of the speed but the height the slide must be in order to achieve that angle. Which explains the choice of my acceleration of 2.7 ms-2 ( 2.d.p), which has an angle of 34 degrees.
With the use of the following formula:
V2= U2 + 2aS
Where V= velocity
U= initial
a=acceleration
S = distance
I will be able to calculate the velocity given; U = 0
a = 2.7 (ms-2)
S = 3.5 metres
V2 = 0 + 2(2.7 X 3.5)
V2 = 18.9
V = 4.3ms-1
Time of journey
V = U + at
4.3 = 0 + 2.7 t
t = 4.3 /2.7
t = 1.6 seconds
The slide will level off at the end and the child will project into a sand pit as shown below:
However, I need to establish where the child will land if the ramp is 1 metre long. Therefore, I will be able to find where to position the sand pit.
Therefore:
a= 0.35 x -9.8
a= - 3.4 m/s2
The acceleration is negative which illustrates that acceleration has decreased due the resistance of frictional forces.
Now that the new acceleration is know, the new velocity can be calculated:
V2= U2 + 2aS
Where :
U= 4.3 m/s2
A =-3.4 m/s2
S = 1 metre
V2 = 4.32 + 2 (-3.4) (1)
V = 3.4 m/s
When looking at projectiles motion in two dimensions the x direction and y direction are involved, allowing us to the position of landing.
Firslty initals must be considered:
initial velocity = ( 3.4i +0j )
initial position = ( 0i + 0.05j )
By integrating and with the use of the intials, the position vector formula can be obtained:
a = 0i – 9.8j
v = 3.4i – 9.8tj
r = 3.4ti – (4.9 t2 + 0.5)j
In order to find the time it takes for the child to project and land in the sand pit the y dimension is taken into account and will equal to zero( as the child will be on the ground in the sand pit, thus will have no height in the y- direction).
r j = 0
-4.9t2 + 0.5 = 0
t2 = -0.5 / -4.9
t = 0.32secs
Now that the time of the projectile is known, it can be substituted into the x dimension to find the distance where the child lands.
r i = 3.4 (0.32)
= 1.10metres
But to ensure that the child does land on the sand pit the sand pit must be at least twice as long where the child lands as a safety precaution.
Evaluation
During my investigation the two most important factors that I have considered is safety and excitement, which is the basis of the design of the slide.
The slide has a height of---, which is a suitable height, as it is not too high that would make it dangerous if the child was to fall down.
The velocity of –and acceleration of 2.6 m/s2 seems to be reasonable, it is at a safe speed, yet still fast enough for it to be fun. Furthermore, the slide in general has aspects that would be appealing to children, such as the run off leading to a little projection into a sand pit.
Validation of model
During my investigation I had made assumptions, to make designing my slide more simple and easy, when using mathematical calculations. However, these assumptions will have an affect on my design and how realistic my design is.
Firstly, my assumptions on treating each child as a round spherical particle as they travel down the slide. The assumption that the child is a particle means that the mass is concentrated on one point, which is the centre of mass. However, this is not true in reality, as the mass of the child is distributed according to surface area, which could easily be increased with any means of contact, including the child’s hands or shoes on the slide. This will in turn mean that an increase in surface area, which is in contact with the slide, will increase friction. As a result acceleration will decrease, so the journey down the slide will be slower and the child may not complete the whole journey of the slide and so the time of journey will be increase.
Therefore, the angle chosen for this slide may be smaller then the ones in reality, as a greater frictional force would need to be overcome.
One-way of solving this problem is to obtain the maximum and minimum coefficient of friction, which the mean value from both will then be used for further calculations. This way each child would be able to slide down with safety being considered.
I made the assumption every child using the slide will either wear denim, polyester or cotton as I used the average co-efficient of those material when designing my slide. For example if a child was to wear clothing of lower co-efficient of friction in comparison to the co-efficient friction that I used will result in velocity being higher than what I have calculated (4.3m/s2), therefore affecting safety. The projection of the child into the sand pit will be at a further distance.
In addition if the child was pushed at the top of the slide then, the forward force will be increased and so speed will increase.
I did not consider air resistance in my investigation. However, in real life situation this will have an affect. Air resistance acts as a frictional force, thus will slow the speed of which the child travels down the speed. Furthermore it will affect the excitement the child experiences.
Seasons are a factor to consider, as it has an effect on the friction of the slide. When it rains or snow the friction of the slide decreases, thus acceleration will increase and may become dangerous to the child as he/she will slide at a greater rate. This in turn will affect velocity, thus the run off, which then may need extending in order to be safe. As weather varies a great deal it becomes a limitation.
In my investigation I didn’t take this factor into account, which would affect my final design. However, if the slide was use for an indoor activity, then surrounding conditions will be constant, thus seasons no longer becomes an issue.
Given more time I would further investigation into the mean height of children with the age group the slide is designed for. This extended investigation will be used to design a slide of an appropriate size, which will make the slide more realistic. I would have tested more materials.
However, the time of the journey of the slide may appear fairly short. Given more time I would investigate into the time journeys of different angles.
Conclusion
In general, I found that it was difficult to implement the two criteria of making the slide safe yet exercising safety.
It was hard to achieve the greatest amount of safety when you need to make the slide exciting as well.
Overall my findings from my experiment demonstrate that plastic and polyester has the lowest coefficient, which means that a smaller angle is required from the horizontal, for the child to over come the frictional forces. As the angle however, the acceleration increases. Looking at the equation: V2= U2 + 2aS, it is clear that velocity (V) corresponds to acceleration (a), as acceleration increase in value so will velocity. As a result the child will experience a shorter journey time.
Also when increasing force of the direction the child will travel down the slide will increase acceleration, thus acceleration is directly proportional to the resultant force applied.
F = ma
Where m is a constant.
