Talk:PlanetPhysics/Frictionless Inclined Plane

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: frictionless inclined plane %%% Primary Category Code: 45.50.Dd %%% Filename: FrictionlessInclinedPlane.tex %%% Version: 6 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\begin{document}

The inclined plane is a common example of \htmladdnormallink{Newton's laws of motion}{http://planetphysics.us/encyclopedia/Newtons3rdLaw.html}. It was used in Galileo's experiment to calculate the \htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html} due to gravity and has been used by students for centuries to explore the laws of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. Here we will examine a block sliding down a frictionless inclined plane as shown below. The y axis is perpendicular to the incline and the x axis is parallel to incline.

\begin{center} \vspace{20 pt} \includegraphics[scale=.85]{InclinePlane.eps} \vspace{20 pt}

\label{Figure 1} \end{center}

The first thing to do is draw a free body \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/Commutativity.html} to describe the external \htmladdnormallink{forces}{http://planetphysics.us/encyclopedia/Thrust.html} acting on our \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and the coordinate \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} for the problem.

\begin{center} \vspace{20 pt} \includegraphics[scale=.85]{FreeBodyDiagram.eps} \vspace{20 pt}

\label{Figure 2} \end{center}

Applying Newton's 2nd law to both x and y

$$ \sum \vec{F} = m \vec{a} $$

{\bf y-dir} \\

First note that the block is stationary in the y direction, so we know that the acceleration is zero.

$$ \sum F_y = 0 $$

The two forces in the y direction are due to the normal force applied by the incline and the force due to gravity.

$$ \sum F_y = N - mg \cos \theta = 0 $$

{\bf x-dir} \\

With the block sliding down the frictionless incline, we get

$$ \sum F_x = mg \sin \theta = m a_x $$

There are several quantities of interest, but let us start with the basics of acceleration, \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} and displacement. For acceleration we see that it is constant in the x direction

\begin{equation} a_x = g \sin \theta \end{equation}

Next, let us calculate the velocity at a given displacement. As usual, we integrate acceleration

$$ \frac{dv_x}{dt} = g \sin \theta $$ $$ \int_0^{v_x} dv_x = \int_0^t g \sin \theta dt $$

and carrying out the integration gives us the velocity of the block as a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of time as it slides down the incline.

\begin{equation} v_x(t) = g \, t \sin \theta \end{equation}

Next, integrate again to get displacement down the incline.

$$ \frac{dx}{dt} = g \, t \sin \theta $$ $$ \int_0^{x} dx = \int_0^t g \, t \sin \theta dt $$

Carrying out the integration gives us the displacement of the block as a function of time as it slides down the incline.

\begin{equation} x(t) = \frac{1}{2} g \, t^2 \sin \theta \end{equation}

Now we can focus on other interesting quantities such as the velocity as a function of displacement. To get this solve (3) for t and plug it into (2)

\begin{equation} t = \sqrt{\frac{2x}{g \sin \theta}} \end{equation}

$$ v_x(x) = g \, \sqrt{\frac{2x}{g \sin \theta}} \sin \theta $$

putting all the terms under the \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} root gives

\begin{equation} v_x(x) =\sqrt{2\, g \,x \,\sin \theta} \end{equation}

A fun and easy experiment is to measure the acceleration of gravity by rolling objects down an incline. All you have to do is solve equation (4) for g

$$ g = \frac{2x}{t^2 \sin \theta} $$

and then just measure $\theta$, $x$ and $t$. There is really nothing like carrying out experiments that helped shape physics, so go forth and measure.

{\bf \htmladdnormallink{equilibrium}{http://planetphysics.us/encyclopedia/InertialSystemOfCoordinates.html}.}

We already saw that the forces perpindicular to the plane in the y direction are in equilibrium

$$ N = mg cos \theta. $$

For the x direction parallel to the plane, one must apply a force $F_a$ in the opposite direction (up the plane) to counter the force due to gravity. This equilibrium condition would mean our forces in the x direction would cancel so

$$ F_a + mg sin \theta = 0 $$ $$ F_a = -mg sin \theta. $$

\end{document}