Variables
The coefficient of friction (µ=f/R), the height of the slide, the length of the slide and the value of ø (the angle of elevation) are the variables that need to be determined.
The following formula can be used to work out The coefficient of friction (µ).
f 0 -WSinØ 0
0 R -WCosØ 0
Working out The coefficient of friction (µ).
The above formula can be rearranged to give the formulas:
The values obtained for F and R can then be substituted into the following formula
µ=f/R. This equals µ= which equals µ=TanØ.
To obtain a value for µ I will go to my local park and measure the vertical height and horizontal length of the slide. The slide I will measure will be made out of aluminum, which is the material I will be using. I will be using this metal because it has a high strength to weight ratio, its light and also it is corrosion resistant. I will then use the values I obtain to obtain a value for the angle using basic trigonometry.
The height will be the opposite and the length will be the adjacent. Therefore the angle will equal Tan-1(Opposite/Adjacent). I will use my local park slide to determine a value for µ because I assume that the park slide will have taken into consideration the different values of µ when different materials are on the slide and therefore they must have picked a suitable value for µ (most likely to be the lowest value).
Results obtained form Local Park
Using these results I calculated the coefficient of friction to be 0.32 (2dp). I will use this as my value for µ for my model.
Selecting a suitable speed
When selecting a suitable speed, considerations such as weather it will be a safe speed for a child to travel at and will the speed be enjoyable for the child need to be made. On average a person can walk about 4 miles in an hour. In ms-1 this equals
=4/(5/8)= 6.4kmh-1
=6.4/(60x60)= 2/1125kms-1
=(2/1125)x1000= 1.78ms-1.
This would be a safe speed for a young child. If an adult is supervising the child, the adult can take then to the top of the slide and let the go and still have enough time to walk round to the bottom and meet the child before the child lands at the bottom. Although this speed seems as if it is safe it may not be enjoyable for the child. This is because older children may find going at walking speed boring. The average running speed is about 12 miles in an hour. In ms-1 this equals
=12/(5/8)= 19.2kmh-1
=19.2/(60x60)= 2/375kms-1
=(2/375)x1000= 5.33ms-1.
This seems as if it would be a fun speed to travel at but it is questionable weather it is a safe speed. As safety has to take priority over enjoyment a compromise speed between 1.78ms-1 and 5.33ms-1 needs to be used. Therefore I have decided to make the final velocity of my slide 4ms-1.
Selecting a height
The slide needs to be fairly high to help the child reach the speed set. Also the slide needs to be at a height where the child will feel safe and not scared. (i.e. A slide 5m High would be scary for a child to climb). Most children’s slides you see are just above head height. My height is approximately 5 foot 11 inches, which is about 1.80m. 0.30m greater then this would give a height of 2.10m, which seems like a safe and high enough height. Therefore the height I will be using in my final design will be 2.10m
Calculating the slide length
The length of the slide can be calculated using the following formula provided that the final velocity is known and also that the height of the slide is known.
g x H x L x (SinØ – 0.3CosØ) =V
V=4ms-1 9.8 x 2.10xL(SinØ – 0.3CosØ) = 4
=42 = 20.58L(SinØ – 0.3CosØ)
16
20.58(SinØ – 0.3CosØ)
(SinØ=Opposite/Adjacent (2.1/L) & CosØ=Adjacent/Hypotenuse(√((L2 2.12)/L)
0.777453838
2.1/L – 0.32(√((L2 2.12)/L)
=0.777453838= L((2.1/L)– 0.32(√((L2 2.12)/L)
=0.777453838= 2.1– 0.32L √L2 2.12
L
=0.777453838= 2.1 – 0.32√(L2 – 2.12)
= 0.777453838 – 2.1
-0.32
=
0.777453838 – 2.1
-0.32
= L = 4.64. Therefore the length of the slide = 4.64m
This graph shows that a slide with height 2.1m has a final speed of 4ms-1 when Ø=26.91° and the slide length equals 4.64m.(also see calculations above).
Calculating The Run Off Area.
To calculate the run off area the run off area the acceleration (Value is negative because the child is slowing down) of the child needs to be found. This can be done using the following formulas:
-f 0 0 a
0 R -Mg 0
F=-Ma F=µR
R=Mg µ=0.32
0.32Mg = -aM
(This is why the mass of the child is not needed because the masses cancel out of the equation)
∴a=-0.32g
Now that the acceleration has been worked out it can be substituted into the following formula to work out the run off area.
V2 = U2 + 2as V=Final Velocity U=Initial Velocity
a=Acceleration s=Displacement
=02=42 + 2x0.32gxs
=-16=-6.272s
∴s=2.55m
Final Design
