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 +====== Galileo'​s Theory Of Relativity ======
  
 +[[books:​dialogues_concerning_two_new_sciences_by_galileo_galilei|]]
 +
 +<​html>​
 +Imagine a person inside a ship which is sailing on a perfectly smooth ​
 +lake at constant speed. This passeneger is in the ship's windowless ​
 +hull and, despite it being a fine day, is engaged in doing mechanical ​
 +experiments (such as studying the behavior of pendula and the 
 +trajectories of falling bodies). A simple question one can ask of this 
 +researcher is whether she can determine that the ship is moving (with 
 +respect to the lake shore) <​em>​without going on deck or looking out a 
 +porthole</​em>​. </p>
 +<p>
 +Since the ship is moving at constant speed and direction she will not <​em>​feel</​em>​
 + the motion of the ship. This is the same situation as when flying on a 
 +plane: one cannot tell, without looking out one of the windows, that 
 +the plane is moving once it reaches cruising altitutde (at which point 
 +the plane is flying at constant speed and direction). Still one might 
 +wonder whether the experiments being done in the ship's hull will give 
 +some indication of the its motion. Based on his experiments Galileo ​
 +concluded that this is in fact impossible: all mechanical experiments ​
 +done inside a ship moving at constant speed in a constant direction ​
 +would give precisely the same results as similar experiments done on 
 +shore. </p>
 +<p>
 +The conclusion is that one observer in a house by the shore and another ​
 +in the ship will not be able to determine that the ship is moving by 
 +comparing the results of experiments done inside the house and ship. In 
 +order to determine motion these observers must look at each other. It 
 +is important important to note that this is true <​em>​only</​em>​ if the 
 +ship is sailing at constant speed and direction, should it speed up, 
 +slow down or turn the researcher inside <​em>​can</​em>​ tell that the ship 
 +is moving. For example, if the ship turns you can see all things ​
 +hanging from the roof (such as a lamp) tilting with respect to the 
 +floor </p>
 +<p>
 +Generalizing these observations Galileo postulated his <​b>​relativity ​
 +hypothesis:</​b>​ </p>
 +<p>
 +</p>
 +<​center>​
 +<table cols="​1"​ width="​400">​
 +    <​tbody><​tr>​
 +        <​td><​em>​any two observers moving at constant speed and 
 +            direction with respect to one another will obtain the same results for 
 +            all mechanical experiments </​em></​td>​
 +    </tr>
 +</​tbody></​table>​
 +</​center>​
 +<p>
 +(it is understood that the apparatuses they use for these experiments ​
 +move with them). </p>
 +<p>
 +In pursuing these ideas Galileo used the scientific method (Sec. <a href="​node6.html#​sec-sci.method">​1.2.1</​a>​):​ he derived consequences ​
 +of this hypothesis and determined whether they agree with the 
 +predictions. </p>
 +<p>
 +This idea has a very important consequence:​ <​em><​font color="#​00ff00">​velocity ​
 +is not absolute</​font></​em>​. This means that velocity can only be 
 +measured in reference to some object(s), and that the result of this 
 +measurment changes if we decide to measure the velocity with respect to 
 +a diferent refernce point(s). Imagine an observer traveling inside a 
 +windowless spaceship moving away from the sun at constant velocity. ​
 +Galileo asserted that there are no mechanical experiments that can be 
 +made inside the rocket that will tell the occupants that the rocket is 
 +moving . The question ``are we moving''​ has no meaning unless we 
 +specify a reference frame (``are we moving with respect to that star''​ <​em>​is</​em>​
 + ​meaningful). This fact, formulated in the 1600's remains very true 
 +today and is one of the cornerstones of Einstein'​s theories of 
 +relativity. </p>
 +<p>
 +</p>
 +<p>
 +<br>
 +</p>
 +<p>
 +</p>
 +<p>
 +<br>
 +<br>
 +</​p><​div align="​CENTER">​ <a href="​https://​www.ganino.com/​_media/​timg143.gif"​ name="​tex2htmlfigure1904"><​img src="​https://​www.ganino.com/​_media/​timg143.gif"​ alt="​\begin{figure} \framebox [6 in][r]{\parbox[r]{5.5 in}{\scriptsize \bigskip{\em ...  ...f experience. Think of it.\bigskip} \parbox{.2in}{\hspace{.1in}}} {}\end{figure}"></​a></​div>​
 + <​br>​
 +<​p></​p>​
 +<p>
 +</p>
 +<p>
 +<br>
 +</p>
 +<p>
 +</p>
 +<p>
 +<br>
 +A concept associated with these ideas is the one of a ``frame of 
 +reference''​. We intuitively know that the position of a small body 
 +relative to a reference point is determined by three numbers. Indeed ​
 +consider three long rods at 90<​sup><​i>​o</​i></​sup>​ from one another, the 
 +position of an object is uniquely determined by the distance along each 
 +of the corresponding three directions one must travel in order to get 
 +from the point where the rods join to the object (Fig. <a href="​node47.html#​fig:​f2">​4.1</​a>​) </p>
 +<p>
 +<br>
 +</​p><​div align="​CENTER"><​a name="​1833">&​nbsp;</​a>​ <​p></​p>​
 +<​table>​
 +    <​caption><​strong>​Figure 4.1:</​strong>​ A frame of reference. <a name="​fig:​f2">&​nbsp;</​a></​caption>​
 +    <​tbody><​tr>​
 +        <​td><​img src="​https://​www.ganino.com/​_media/​img144.gif"​ alt="​\begin{figure} \centerline{ \vbox to 3 truein{\epsfysize=6 truein\epsfbox[0 0 612 792]{4.galileo/​f2.ps}} }\end{figure}"​ sgi_src="/​usr/​people/​wudka/​public_html/​Physics7/​Notes_www/​img144.gif"​ height="​313"​ width="​314"></​td>​
 +    </tr>
 +</​tbody></​table>​
 +<p>
 +</​p></​div><​br>​
 +<​p></​p>​
 +<p>
 +Thus anyone can determine positions and, if he/she carries clocks, ​
 +motion of particles accurately by using these rods and good clocks. ​
 +This set of rods and clock is called a <​i>​reference frame</​i>​. In 
 +short: <font color="#​00ff00">​a reference frame determines the where and 
 +when of anything with respect to a reference point.</​font>​ </p>
 +<p>
 +A prediction of Galileo'​s principle of relativity is that free objects ​
 +will move in straight lines at constant speed. A free object does not 
 +suffer form interactions from other bodies or agencies, so if it is at 
 +one time at rest in some reference frame, it will remain at rest 
 +forever in this frame. Now, imagine observing the body form another ​
 +reference frame moving at constant speed and direction with respect to 
 +the first. In this second frame the free body is seen to move at 
 +constant speed and (opposite) direction. Still nothing has been done to 
 +the body itself, we are merely looking at it from another reference ​
 +frame. So, in one frame the body is stationary, in another frame it 
 +moves at constant speed and direction. On the other hand if the body is 
 +influenced by something or other it will change its motion by speeding ​
 +up, slowing down or turning. In this case either speed or direction are 
 +not constant as observed in <​em>​any</​em>​ reference frame. From these 
 +arguments Galileo concluded that free bodies are uniquely characterized ​
 +by moving at constant speed (which might be zero) and direction. </p>
 +<p>
 +An interesting sideline about Galilean relativity is the following. Up 
 +to that time the perennial question was, what kept a body moving? ​
 +Galileo realized that this was the <​i>​wrong question</​i>,​ since uniform ​
 +motion in a straight line is not an absolute concept. The right 
 +question is, what keeps a body from moving uniformly in a straight ​
 +line? The answer to that is ``forces''​ (which are defined by these 
 +statements). This illustrates a big problem in physics, we have at our 
 +disposal all the answers (Nature is before us), but only when the right 
 +questions are asked the regularity of the answers before us becomes ​
 +apparent. Einstein was able to ask a different set of questions and 
 +this lead to perhaps the most beautiful insights into the workings of 
 +Nature that have been obtained. </p>
 +<p>
 +Galilean relativity predicts that free motion is in a straight line at 
 +constant speed. This important conclusion cannot be accepted without ​
 +experimental evidence. Though everyday experience seems to contradict ​
 +this conclusion (for example, if we kick a ball, it will eventually ​
 +stop), Galileo realized that this is due to the fact that in such 
 +motions the objects are <​em>​not</​em>​ left alone: they are affected by 
 +friction. He then performed a series of experiments in which he 
 +determined that frictionless motion would indeed be in a straight line 
 +at constant speed. Consider a ball rolling in a smooth bowl (Fig. <a href="​node47.html#​fig:​f1">​4.2</​a>​). </p>
 +<p>
 +<br>
 +</​p><​div align="​CENTER"><​a name="​1835">&​nbsp;</​a>​ <​p></​p>​
 +<​table>​
 +    <​caption><​strong>​Figure 4.2:</​strong>​ Illustration of Galileo'​s ​
 +    experiments with friction <a name="​fig:​f1">&​nbsp;</​a></​caption>​
 +    <​tbody><​tr>​
 +        <​td><​img src="​https://​www.ganino.com/​_media/​img145.gif"​ alt="​\begin{figure} \centerline{ \vbox to 2 truein{\epsfysize=8 truein\epsfbox[0 -70 612 722]{4.galileo/​f1.ps}} }\end{figure}"></​td>​
 +    </tr>
 +</​tbody></​table>​
 +<p>
 +</​p></​div><​br>​
 +<​p></​p>​
 +<p>
 +The ball rolls from it's release point to the opposite end and back to 
 +a certain place slightly below the initial point. As the surfaces of 
 +the bowl and ball are made smoother and smoother the ball returns to a 
 +point closer and closer to the initial one. In the limit of zero 
 +friction, he concluded, the ball would endlessly go back and forth in 
 +this bowl. </p>
 +<p>
 +Following this reasoning and ``abstracting away''​ frictional effects he 
 +concluded that </p>
 +<p>
 +</p>
 +<​center>​
 +<table cols="​1"​ width="​400">​
 +    <​tbody><​tr>​
 +        <​td><​em><​font color="#​00ff00">​Free horizontal motion is 
 +            constant in speed and direction.</​font><​font color="​green">​ </​font></​em></​td>​
 +    </tr>
 +</​tbody></​table>​
 +</​center>​
 +<p>
 +This directly contradicts the Aristotelian philosophy which claimed ​
 +that </p>
 +<ul>
 +    <li>
 +    all objects on Earth, being imperfect, will naturally slow down, 
 +    </​li><​li>​
 +    that in a vacuum infinite speeds would ensue, ​
 +    </​li><​li>​
 +    and that perfect celestial bodies must move in circles. ​
 +</​li></​ul>​
 +<p>
 +In fact objects on Earth slow down due to friction, an object at rest 
 +would stay at rest even if in vacuum, and celestial bodies, as anything ​
 +else, move in a straight line at constant speed or remain at rest 
 +unless acted by forces. </p>
 +</​html>​
galileo_theory_relativity.txt ยท Last modified: 2020/05/17 00:10 by admin