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Mechanical Advantage of Inclined Planes: Stairs, Ramps, Wedges, and Screws, Study notes of Analytical Geometry and Calculus

University of Puget Sound (UPS)Analytical Geometry and Calculus

An in-depth exploration of inclined planes, including stairs, ramps, wedges, and screws. Learn how to calculate their mechanical advantage using the given formulas and examples. Understand how these simple machines make lifting and moving objects easier.

Typology: Study notes

2021/2022

Uploaded on 09/27/2022

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InclinedPlane
Readthefollowingarticleoninclinedplaneshighlightingimportantinformationwithinthearticle.
Youprobablyarethankfulthatgoingfromthefirstfloortothesecondfloor
hereatGenoaMiddleSchoolwehavestairsandnotaladder.Ifyoudidn’t
know,stairsareasimplemachinereferredtoasaninclinedplane.Inclined
planesarerampsthatarealwaysstationary.Theyworkbyspreadingaweight
outoveralongerdistance.Likeallsimplemachines,theyalsohavea
mechanicaladvantage.Tocalculatethemechanicaladvantage(MA)ofan
inclinedplane(stairs),youdividethediagonaldistancefromthefrontofastep
uptothenextfrontstepbytheoverallheightofthestep.
Arampiswhatmostpeoplethinkofwhentheyhearthewordsinclinedplane.Itisjustlikestairsinthatitisan
inclinedplanethatisstationary(notmoving).It’scalledamachinebecauseitallowsyoutoraiseanobject
fromoneleveltoanotherusinglesseffortthanifyousimplylifteditstraightup.Sincerampsandstairsare
identical,tocalculatetheMA,dividethelengthoftherampbytheheightoftheramp.
Wherearampisstationary,awedgeontheotherhandisaninclinedplanethat
moveswhenitisused.Forexample,ifweweresplittingalog,awedgeisagreattool
thatforceswoodapartsplittingitintosmallerpieces.Wedgesarealsousedtostop
thingssuchasdoors.Byplacingawedgeunderthedoor,itworksjustlikeanaxe
(exceptforthesplittingpart).Inthiscasethough,itstopsthedoorfrommoving.The
MAofawedgeisfoundbydividingthelengthofthewedgebythethicknessatthe
widestend.Thewedgeisoneoftheleastefficientsimplemachinesbecausesomuch
effortisuseduptryingtopushitthroughsomethinglikethepieceofwoodabove.
Ascrewisalsoaninclinedplane.Itcanbethoughtofasarampwrappedarounda
cylinder.Asyourotatethescrewonefullturn,thescrewmovesforwardadistanceequaltothesizeofthe
thread.Thisdistanceisusuallysmall.Sinceyouapplyeffortoveramuchlongerdistance(screwshavereal
longrampswrappedaroundthem),screwshavealarge
ly
mechanicaladvantage.
Createagraphicwebforinclinedplanes:
Inclined
Plane
pf3
pf4

Partial preview of the text

Download Mechanical Advantage of Inclined Planes: Stairs, Ramps, Wedges, and Screws and more Study notes Analytical Geometry and Calculus in PDF only on Docsity!

Inclined Plane

Read the following article on inclined planes highlighting important information within the article.

You probably are thankful that going from the first floor to the second floor here at Genoa Middle School we have stairs and not a ladder. If you didn’t know, stairs are a simple machine referred to as an inclined plane. Inclined planes are ramps that are always stationary. They work by spreading a weight out over a longer distance. Like all simple machines, they also have a mechanical advantage. To calculate the mechanical advantage (MA) of an inclined plane (stairs), you divide the diagonal distance from the front of a step up to the next front step by the overall height of the step.

A ramp is what most people think of when they hear the words inclined plane. It is just like stairs in that it is an inclined plane that is stationary (not moving). It’s called a machine because it allows you to raise an object from one level to another using less effort than if you simply lifted it straight up. Since ramps and stairs are identical, to calculate the MA, divide the length of the ramp by the height of the ramp.

Where a ramp is stationary, a wedge on the other hand is an inclined plane that moves when it is used. For example, if we were splitting a log, a wedge is a great tool that forces wood apart splitting it into smaller pieces. Wedges are also used to stop things such as doors. By placing a wedge under the door, it works just like an axe (except for the splitting part). In this case though, it stops the door from moving. The MA of a wedge is found by dividing the length of the wedge by the thickness at the widest end. The wedge is one of the least efficient simple machines because so much effort is used up trying to push it through something like the piece of wood above.

A screw is also an inclined plane. It can be thought of as a ramp wrapped around a cylinder. As you rotate the screw one full turn, the screw moves forward a distance equal to the size of the thread. This distance is usually small. Since you apply effort over a much longer distance (screws have real long ramps wrapped around them), screws have a large

ly mechanical advantage.

Create a graphic web for inclined planes:

Inclined

Plane

How does a Ramp work as a simple machine?

  1. Inside the box is a 1 kg weight.
  2. Lift the box and weight 0.2 m off the ground using a spring scale and record the weight in the table.
  3. Calculate the PE of lifting the box.

Weight of the box lifted straight up PE of lifting the box 0.2 m

  1. The ramp has a bar height of 0.2 m.
  2. Measure the length of the wooden board in centimeters using a meter stick and record in the data table.
  3. Calculate the mechanical advantage by dividing the length of the ramp by the height of the bar.

Length of the ramp in centimeters Height of bar 20 cm Mechanical Advantage

  1. Using the PE and the mechanical advantage, predict how much work will be needed to slide the box up the ramp. ____________________________________
  2. Based on your answer to question 7, will the weight of the box sliding up the ramp be lower, the same, or higher (circle)?
  3. Slowly drag the box up the ramp by the hook attached to the side of the box and the other to the spring scale. Record the weight on the spring scale in the data table below.

Weight of the box being pulled up the ramp

  1. How did your answer in question 8 compare to what really happened?? Did the box have less weight?
  1. Using a metric ruler, measure the height H of one step to the closest centimeter. Record this information in the data collection table below.

Genoa Stairs

Body Weight

Diagonal distance between steps

Height of one step

  1. For a flight of stairs, you find the mechanical advantage by dividing the distance you travel diagonally up the step by the height of the step:

Mechanical advantage = diagonal distance between steps = _____________________ Height of one step

  1. Without the help of a machine, the weight your legs would need to produce to get you from the 1 st^ floor of Genoa to the 2 nd^ floor is equal to your body weight. With the stairs, the weight needed is lower than your body weight. This is what makes stairs so great!

Body Weight = ________________ pounds Mechanical Advantage

MA for an Inclined Plane

If you haven’t noticed, all of the different inclined planes have the same formula for calculating the mechanical advantage, and it relies on one simple shape, the triangle.

  1. What is the formula for finding the mechanical advantage of an inclined plane, ramp, wedge, and screw?

MA =