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Energy Conservation Example
A mass hangs on a string forming a pendulum.  
The length of the string is L = 1.5m  
The mass is pulled to one side until the string makes an angle of 30 to the vertical, as shown.  
The mass is released from this position.  
Use the conservation of energy to calculate the speed of the mass at the instant that it passes through the equilibrium position (ie when the string is vertical).  
First we need to find how far the mass falls, vertically.  
     
   
   
It should be clear from the diagram that  
   
   
this gives h = 0.232m  
   
   
   
   
As the mass falls, GPE is converted to KE.  
The principle of conservation of energy tells us that the GPE lost is equal to the KE gained.  
Therefore we have  
 
We notice immediately that (not surprisingly) we don't need to know the mass of the falling body.  
From this we have  
 
which gives v = 2.13ms-1  
   
Notice that the concept of energy and energy conservation turns an apparently complicated question into a very simple one.  
   
By the way, gravity is not the only force acting on the mass. there is, of course, the force due to the string.  
Why does this force not have any effect on the speed of the mass ? (answer below...)  
   
   
   
   
   
   
   
   
   
   
   
It always acts at 90 to the direction of motion of the mass.  
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