Physics of the esd lower school playground
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Physics of the ESD Lower School Playground. Grant Fitts. Spin Wheel. Radius = 1.17 m 1 rev = 2.6 seconds (at max velocity). Just to hold on…. V = 2(1.17) π / 2.6 V t =2.83 m/s Mv 2 /r = F 81.81(2.83 2 )/1.17 = F F = 560 Newtons. Spiral Slide. Height = 1.77 m Straight for .9 m

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Spin wheel
Spin Wheel

  • Radius = 1.17 m

  • 1 rev = 2.6 seconds (at max velocity)


Just to hold on
Just to hold on…

  • V = 2(1.17)π / 2.6

  • Vt =2.83 m/s

  • Mv2/r = F

  • 81.81(2.832)/1.17 = F

  • F = 560 Newtons


Spiral slide
Spiral Slide

  • Height = 1.77 m

  • Straight for .9 m

  • Falls .46 m to ground

  • Ball travels 1.3 m during fall


How fast it should be going

(.5) mv2 = mgh

V = √(2gh)

= √(2 x 9.8 x 1.77)

V = 5.89 m/s

theoretical

.46 = .5 (a) t2

.46 = .5 (9.8)t2

T2 =

√(.46/ (9.8 x .5 ))

T =.306

V = 1.3 / .306

V = 4.24 m/s

How fast it should be going


Why the difference

4.24 = 5.89 - 2a∆x

4.24 = 5.89 -2a(.9)

A =.917 m/s2

Ff = uN

.917(m) = uN

U =(.145)(9.8)/.917

µ= 1.55

4.24 -5.89 / 5.89 x 100 = %error

= 28.01 %

4.24 / 5.89 = eff.

Efficiency= 71.98%

Why the difference?


Straight bar hold
Straight Bar Hold

  • 4.35 meters long

    -.96 seconds to travel

  • 2.6 before-hand

    -ran in .4 seconds


Velocity and efficiency

2.6 /.4 = Vi

Vi = 6.5 m/s

4.35/ .96 = Vf

Vf (average) = 4.53

4.53 = 6.5 - 2a(2.175)

A = -.453 m/s2

81.81(.453) = u(801.74)

µ = .046

Velocity retained (%)

Vf = 6.5 - 2 (.453)(4.35)

Vf = 2.56

2.56 / 6.5 = effic.

Efficiency = 39.4%

Velocity and efficiency


Racing slide
Racing Slide

  • Height = 1.2 m

  • Falls .4 m

  • Travels 1 m in fall


Velocities

.5 (mv2) = mgh

.5v2 = 9.8(1.2)

V = 4.85 m/s

Theoretical

------------------------------

.4 = .5(9.8)t2

T = .286 seconds

1 = .286v

V = 3.497 m/s

actual

3.497 - 4.85 / 4.85 x 100 = %error

27.9 % error

velocities


Swing
Swing!

  • Radius = 1.86 m


To go in a circle
To Go in a Circle…

  • R = 1.86 m

  • Mg = mv2/r

  • 9.8 = v2/1.86

  • Vf = 4.27 m/s

  • .5m(4.272 - vi2) =-mgh

  • .5(18.23 - vi2) = -9.8(3.72)

  • Vi = 9.55 m/s running start


Pendulum status
Pendulum status

  • Times of full swing (from start) in seconds

    2.5, 2.7, 2.8, 2.7, 2.8,2.8,2.9,2.9,3.0,3.1

    Not quite a pendulum because the chain will lose tension when the swing breaches 90 degrees and then drops straight down, jerking forward and losing angular velocity.


Hand rail
Hand Rail

  • ∆h = .7 m

  • Length = 2.45 m

  • 2.1 seconds to slide


Angles and acceleration

2.45 = .5(a)t2

T = 2.1

2.45 = .5a(2.12)

A = 1.11 m/s2

1.11 = 9.8sinø

Ø = 6.5 degrees.

sin-1(.7 / 2.45) = ø

Ø=16.6 degrees

gsinø = gx

2.45 = .5(2.8) t2

T = 1.322 seconds

Angles and acceleration


The difference is friction

1.11(m) = 2.8(m) -uNcosø

1.69 = u(g)cosø

1.69 = u(9.8)cos16.6

µ = .18

The difference is friction


Big stairs to gym
Big Stairs to Gym

  • 35 steps

    -each .18 meters down

    1.4 seconds to fall total height


How far down
How far down

  • Depth = 35(.18)

  • = 6.3meters down

  • Vf2 = 2a∆x

  • V = √(2(9.8)(6.3))

  • V = 11.11m/s for a falling ball


Tricycle racing hill
Tricycle Racing Hill

  • Straight section at bottom = 3.52 m

  • Time after pedaling down hill = 1.9 seconds

  • Time on section without pedaling = 1.1 seconds

  • Hill = 14.75 meters


The hill
The Hill

  • 3.52 / 1.1 = 3.2 m/s

  • 3.22 = 2a∆x

  • 3.22 = 2a(14.75)

  • A= .347 m/s2

  • Gsinø - ugcosø = .347

  • 9.8sinø -(.4)(9.8)cosø= .347

  • Ø = 23.68 degrees

  • Sin23.68= h/ 14.75

  • H = 5.9 m

.5v2 = 9.8(5.9)

V = 10.75

3.2 - 10.75 / 10.75 x 100 = %error

70.2% error


Racing tip
Racing tip…

  • 3.52 / 1.9 = 1.85 m/s

  • 1.852 = 2a(14.75)

  • A = .116

  • 3.2(x) = 1.85

  • X = .578

  • Attempting to pedal cuts your speed to 57.8% of what it would be.

  • http://www.facebook.com/profile.php?id=533654442&ref=profile


Mothers day out slide
Mothers’ Day Out Slide

  • H = 2.17 m

  • Time = 3.2 seconds

  • Track length = 4 m


Differences in velocities

gh = .5v2

9.8(2.17) = .5v2

V = 6.52 (theoretical)

4 = .5a(3.22)

A = .78 m/s2

Vf2 = 2(.78)4

Vf = 2.498 m/s

2.498 - 6.52 / 6.52 x 100 = %error

61.69% error

Differences in velocities


Sources
sources

  • http://cielovistahoa.com/images/playground.jpg


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