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Vertical Suspension
BowlingBall
PIPE
ROPE
LiftVertical Up
GravityVertical Down
There are Two Forces acting on the Ball
There are no Horizontal Forces
Fh = 0There is no Vertical Acceleration, consequently
Flift + Fgravity = 0
Horizontal MovementThere are Four Forces acting on the Box of Bowling Balls
Accelerating to the Right
GravityThe Force of Gravity on an Object is commonly known as Weight.
Normal ForceThe Force the supporting surface exerts on the object is known as the Normal Force.
PullFriction
PullFriction
Gravity
Normal Force
In order for the Box to Accelerate, the pull force must exceed the frictional force, causing a net horizontal force.
Ff + Fp = Fh
There is no Vertical Acceleration, consequently
Fg + Fn = 0
We’ll come back to friction in a bit…
Friction
Gravity
Normal Force
Movement on an Inclined PlaneThere are Three Forces acting on this Box of Bowling Balls
Axis of Motion
Box sliding down a rake
Friction
Gravity
Normal Force
Axis of Motion
Box sliding down a rake
Tilting the Frame of Reference
Gravity
Axis of Motion
Vertical Down
HorizontalTo the Left
Angle of the InclineOff the Horizontal (
Box sliding down a rakeBreaking the Force down into Vertical and Horizontal Components
Trigonometric Funtions
Fv = -(Fg * cos
Fh = -(Fg * sin
Fg = Weight
“-” indicates direction
Box sliding down a rake
Friction
Gravity
Normal Force
Axis of Motion
Fgv
Fgh
Box sliding down a rake
Friction
Gravity
Normal Force
Axis of Motion
Fgv
Fgh
FghFriction
Fgv
Normal Force
The Box is Accelerating Horizontally.Consequently, there is a Net Horizontal Force.
Fgh + Ff = Fh
There is no Vertical Acceleration,so…
Fgv + Fn = 0
Assigning Values to the Terms
Gravity = Weight
Normal Force = Weight * cos
How to Calculate Friction.
What we know so far…
What’s Next…
When Gravity is the only other Vertical Force
Three Kinds of Friction
Static
Kinetic
RollingPulling Force
If Ff < Fp, then Acceleration
If Ff = Fp, then Constant Velocity
Once moving, if Ff > Fp, then Deceleration
How to Find the Frictional Force
Ff = Fn * is theCoefficient of Friction
Coefficients of Static and Kinetic Friction are unitless numbers that quantify how well two surfaces will slide
against one another.
These Coefficients are specific to the material combination NOT any individual material.
IE. Rubber on Asphalt: s = .6, k = .4
s kRubber on Asphalt .6 .4Steel on Ice .1 .05Steel on Steel (dry) .6 .5Steel on Steel (greased) .1 .05Rope on Wood .5 .3Teflon on Steel .04 .04Teflon on Teflon .04 .04Shoes on Ice .1 .05Rubber Soled Shoes on Wood .9 .7Leather Soled Shoes onWood .3 .2Climbing Boots on Rock 1.0 .8
s kOn UHMWLauan w/grain .25 .23Lauan against grain .31 .27Poplar .25 .22Oak .27 .23Steel .21 .18
On MDFUHMW .26 .22MDF .45 .36Pine .40 .28
On Glazed LauanUHMW .29 .22
Values for Theatre, w/ Uncertainty
Say the Box and its contents weigh 50 lbs. The base of the box is UHMW and the surface the box is sitting on is MDF.
Ff = Fn * s
Find Fn
Fn = Fg * cos 0
Fn = 50 lbs * 1 = 50 lbss of UHMW on MDF is .26
So…
Ff = 50 lbs * .26
Ff = 13 lbs
k of UHMW on MDF is .22
So to keep the box moving…
Ff = 50 lbs * .22
Ff = 11 lbs
A Note on Surface AreaExcept in extreme circumstances,
the Surface Area of the materials does NOT affect the Friction between the materials.
More surface to bond,Less Pressure
between surfaces.
Less surface to bond,More Pressure
between surfaces.
Advantage of less surface is cancelled by the increased pressure.
Rolling Friction v. Static and Kinetic Friction
Similar to Static and Kinetic Friction:
Ff = Fn * r
Different from Static and Kinetic Friction:
Rolling Friction is a property of an individual caster. The wheel and bearing materials, diameter of the wheel, and type of
construction define the COF for any given caster. As long as the wheel doesn’t slip on the surface, the surface material is
negligible in determining the r.
More Casters does not dramatically effect overall Friction
Expressed as a Coefficient: r
Where to find Coefficients of Frictionfor your application
Online
Manufacturer’s Specs
Measure it YourselfCOF are affected by minor variances in materials and manufacturing, dust, dirt, impurities, age and condition
of surfaces.
940 lbs
s for UHMW on glazed lauan = .29
Define the Forces and Draw the FBD
Gravity
Normal Force
DriveFriction
Gravity = Weight = -940 lbs
Normal = Gravity = 940 lbs
Frictions = Normal * s 940 lbs * .29 = -272.6 lbs
Drives = ma + Frictions
32 ft over 4 s, accelerated in the wings over 4 s
Frictionk = Normal * k 940 lbs * .27 = -253.8 lbs
k for UHMW on glazed lauan = .27m = Fg/g = 940 lbs/32.2 ft/s2 = 29.2 slugs
UHMW on Glazed Lauan
Drivek = ma + Frictionk
29.2 slugs * 2 ft/s2 + 272.6 lbs = 331.0 lbs
29.2 slugs * 0 ft/s2 + 253.8 lbs = 253.8 lbs
940 lbs
Describe the motion
Max Velocity = d/t = 32 ft/s / 4 s = 8 ft/s
32 ft over 4 s, accelerated in the wings over 4 s
UHMW on Glazed Lauan
Acceleration = v1-v0 / t = 8 ft/s – 0 ft/s / 4 s = 2 ft/s2
Deceleration = v1-v0 / t = 0 ft/s – 8 ft/s / 4 s = -2 ft/s2
Velocity in ft/s
Time in s
940 lbs
r for a generic wagon on decent casters = .05
What if it were on wheels?
Gravity
Normal Force
DriveFriction
Gravity = Weight = -940 lbs
Normal = Gravity = 940 lbs
Frictions = Normal * r 940 lbs * .05 = -47 lbs
Drive0 = ma + Friction = 105.4 lbs
32 ft over 4 s, accelerated in the wings over 4 s
Drive1 = ma + Friction = 47 lbs
On Wheels
HP = F * ft / s
550
Weight = 940 lbs
UHMW on glazed lauan = .29
Define the Forces
Gravity
Normal Force
PullFriction
Gravity = Weight = 940 lbs
Normal = Gravity = 940 lbs
Friction = Normal * s 940 lbs * .29 = 272.6 lbs
Pull = Friction = 272.6 lbs
32 ft over 4 s, accelerated in the wings over 4 s
Calculate the HP
HP = 141 lbs * 32 ft / 4 s
550
HP =
272.6 lbs * 8 ft/s550 * .08
HP = 2.05 lb ft/s
32 ft over 4 s
2.05 HP will generate 141 lbsof Pull Force at this velocity
Static Friction = 141 lbs
Pull Force > Static Friction to Accelerate Object
Weight = 940 lbs
UHMW on Waxed Wood, s = .15
How much HP do we need to accelerate the box?
32 ft over 4 s
Fn = m * a
Weight = 940 lbs UHMW on Waxed Wood, s = .15
Force = Mass * Acceleration
(Ff + Fn) * v550
HP =
(Ff + lbs/32 * a) * v550
HP =
(141 lbs + 940 lbs/32 ft/sec2 * 8 ft/s2) * 8 ft/s550
HP =
a = 8 ft/sec2
HP = 2.48 lbft/s
32 ft over 4 s
HP by friction
HP by acceleration
Weight = 940 lbs
UHMW on Waxed Wood, s = .15
2.04
2.48
2.72
3.30
Design Factor: Do not use more than 75% of name plate rating to overcome friction or if you
will accelerate for more than 5 sec.
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