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.