Inclined Plane

Claimed by Catherine Grey Spring 2017

The Main Idea

We typically know an inclined plane as a flat surface that is higher on one end aka a big triangle. Inclined planes are commonly used to move objects to a higher or lower place. These slopes help to lessen the force needed to move an object but do require the object to move a greater distance, the hypotenuse of the triangular plane. Some examples of inclined planes include ramps, stairs, wedges (such as a door stopper), or even mountains which people sled down. To solve an equation with an inclined plane forces may be taken into account or the energy principle if the object is moving or has a rotation that is on the inclined plane.

A Mathematical Model

This figure illustrates most of the forces that could act on an inclined plane. Depending on the question not all will be present, specifically friction.

Variables:

θ = Angle of the plane to the horizontal
g = Acceleration due to gravity
m = Mass of object
N = Normal force (perpendicular to the plane)
F_f = frictional force of the inclined plane (sometimes it is omitted on test problems)
mgSinθ = A force parallel to the plane (mgSinθ > Ff the body slides down the plane)
mgCosθ = A force acting into the plane (opposite to N)
F_netparallel = The net parallel force acting on the object
F_{netperpendicular} = The net perpendicular force acting on the object

Equations:

[math]\displaystyle{ \ F_{net} = F_{net//} + F_{netperp} }[/math]

[math]\displaystyle{ \ F_{net//} = F_f - mgSinθ }[/math]

[math]\displaystyle{ \ F_{netperp} = N -mgCosθ }[/math]

All energy equations are also present, especially when rotation is included on an inclined plane.

A Computational Model

Using GlowScript or Vpython we are able to model a block on an inclined plane moving upward with an initial velocity then pulled downward as it is being affected by gravity.

Code

from __future__ import division from visual import * from visual.graph import *

1. 1. SETUP ELEMENTS FOR GRAPHING, SIMULATION, VISUALIZATION, TIMING
------------------------------------------------------------------------

Set window title

scene.title = "Incline Plane"

Make scene background black

scene.background = color.black scene.center = (0.7, 1, 0) # location at which the camera looks

Define scene objects (units are in meters)

1.4-m long inclined plane whose center is at 0.7 m

inclinedPlane = box(pos = vector(0.6, 0, 0), size = (1.2, 0.02, 0.2), color = color.green, opacity = 0.3)

20-cm long cart on the inclined plane

cart = box(size = (0.2, 0.06, 0.06), color = color.blue)

Set up trail to mark the cart's trajectory

trail = curve(color = color.yellow, radius = 0.01) # units are in meters

SETUP PARAMETERS AND INITIAL CONDITIONS

Define parameters

cart.m = 0.5 # mass of cart in kg

initial position of the cart in(x, y, z) form, units are in meters
cart is positioned on the inclined plane at the far left end

cart.pos = vector(0, 0.04, 0.08)

cart.v = vector(0, 0, 0) # initial velocity of car in (vx, vy, vz) form, units are m/s

angle of inclined plane relative to the horizontal

theta = 25.0 * (pi / 180.0)

rotate the cart and the inclined plane based on the specified angle (counterclockwise)

inclinedPlane.rotate(angle = theta, origin = (0, 0, 0), axis = (0,0,1)) cart.rotate(angle = theta, origin = (0, 0, 0), axis = (0,0,1))

set the initial velocity up the ramp; units are m/s

cart.v = norm(inclinedPlane.axis) cart.v.mag = 3

g = 9.8 # acceleration due to gravity; units are m/s/s

mu = 0.20 # coefficient of friction between cart and plane

Define time parameters

t = 0 # starting time deltat = 0.0005 # time step units are s

print "initial cart position (m): ", cart.pos

CALCULATION LOOP; perform physics updates and drawing

while cart.pos.y > 0.02 :

   rate(1000)    
   Fnet = norm(inclinedPlane.axis)
   
   Fnet.mag = -(cart.m * g * sin(theta))
   
   if cart.v.y > 0:
       Fnet.mag += (mu * cart.m * g * cos(theta))
   else:
       Fnet.mag -= (mu * cart.m * g * cos(theta))
   
   cart.v = cart.v + (Fnet/cart.m * deltat)

   cart.pos = cart.pos + cart.v * deltat

   trail.append(pos = cart.pos)
  
   t = t + deltat
       
OUTPUT

Print the final time and the cart's final position

print "final time (s): ", t print "final cart position (m): ", cart.pos print "final cart velocity (m/s): ", cart.v print "final cart speed (m/s): ", mag(cart.v) print "final cart acceleration (m/s/s): ", (mag(Fnet) / cart.m)

Examples

The easiest example of an object on an inclined plane is removing all forces, but gravity. Since gravity is the only force acting on the object then the computations are easier. To make inclined plane problems harder, adding more forces or calculating for factors other than net force can be included such as finding the acceleration or time it takes for the block to go from the top to the bottom of an inclined plane. As well adding in rotation, with a ball rolling down the plane, would demonstrate other difficulties.

Easy

Medium

Hard

Technical Usage

Terminology

Let's imagine there is a right triangle. The side opposite to the right angle is a Slant. The side on the bottom is Run. The side vertical to the bottom is Rise.

Slope: A slope brings a mechanical advantage to the incline plane.

[math]\displaystyle{ \theta = \tan^{-1} \bigg( \frac {\text{Rise}}{\text{Run}} \bigg) \, }[/math]

Mechanical Advantage

Fw is a gravitational force that applies on the plane
Fi is a force exerted on the object and parallel to the plane

[math]\displaystyle{ \mathrm{MA} = \frac{F_w}{F_i}. \, }[/math]

if the inclined plane is frictionless,

[math]\displaystyle{ \text{MA} = \frac{F_w}{F_i} = \frac {1}{\sin \theta} \, }[/math] (in this case, [math]\displaystyle{ \sin \theta = \frac {\text{Rise}}{\text{Length}} \, }[/math])

if the inclined plane has a friction

[math]\displaystyle{ \mathrm{MA} = \frac {F_w}{F_i} = \frac {\cos \phi} { \sin (\theta - \phi ) } \, }[/math] (in this case, [math]\displaystyle{ \phi = \tan^{-1} \mu \, }[/math])

[math]\displaystyle{ \theta \lt \phi\, }[/math]: Downhill applied force is needed.

[math]\displaystyle{ \theta = \phi\, }[/math]: Infinite mechanical advantage.

[math]\displaystyle{ \theta \gt \phi\, }[/math]: The mechanical advantage is positive. Uphill force is needed.

History

An inclined plane is one of the six classical simplified machines defined by Renaissance scientists. They have been used for thousands of years to move large objects to a specific distance, such as with the potential creation Stonehenge or the formation of the Egyptian Pyramids. In the second century B.C., Archimedes began theorizing some of the properties associated with an inclined plane including its various applications. Then in the first century A.D., Hero of Alexandria went beyond Archimedes initial theories to further explain the functionality of an inclined plane. These theories then proved to be a reality as the years went on and there were more advancements and functions applied to inclined planes.

Connectedness

Every day people experience some sort of inclined plane. From driving up a hill on the way to work to walking down a ramp to get to class (because of all of the construction we always have) an inclined plane is a vital piece of our lives. What I find the most interesting about inclined planes is how people use them and physics for fun without even knowing it. Such as going down a slide at a park or skiing down a snowy mountain, the physics and forces behind an inclined plane bring enjoyment to many people's lives every day.

An inclined plane relates to mechanical engineering because MEs will always be working with various systems including some form of an inclined plane. Such as pulling an object across a ramp or the mechanics behind a water slide at an amusement park, inclined planes are everywhere. The best advice is to become comfortable with how they work and how to calculate the net force acting on the object sitting or moving on the inclined plane.

References

Cole, Matthew (2008). Explore science, 2nd Ed. Pearson Education. https://books.google.com/books?id=RhuciGEQ1G8C&pg=PA178#v=onepage&q&f=false

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/incline.html