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Mechanical Oscillations
A mechanical oscillation is a periodic conversion of energy from potential energy to kinetic energy to potential energy etc.  
If the oscillation is "damped" then some energy is also converted to other forms (usually thermal energy and/or sound) during each of these "cycles" of PE to KE.  
In what follows we will consider undamped oscillations.  
   
If a body on which all forces are initially in equilibrium, is to oscillate, it must first be displaced from this equilibrium position.  
It must then be acted on by a force which is always pulling it back towards the equilibrium position.  
This force is called the restoring force.  
The simplest type of oscillation is called simple harmonic motion, s.h.m.  
   
Definition of S.H.M.  
If a body moves such that its acceleration is directly proportional to its displacement from a fixed point and is always directed towards that point, then the motion is s.h.m.  
   
The fixed point is the equilibrium position referred to above.  
   
Written mathematically  
 
The magnitude of the constant is therefore  
 
but, using Newton's second law, we can write this as  
 
where m is the mass of the oscillating body and F is the restoring force.  
   
This shows that the actual value of the constant for a given oscillation depends on  
1. the restoring force per unit displacement (F/x) and  
2. the mass of the oscillating body.  
   
Three other terms need to be defined:  
The amplitude, r, of an oscillation is the maximum displacement from the equilibrium position.  
   
The red arrows on the diagram describe one complete oscillation.  
   
The frequency, f is the number of oscillations per unit time (usually per second).  
   
The Relation Between S.H.M. and Circular Motion  
If a body moving at constant speed in a circular path is observed from a distant point (in the plane of the motion) it appears to be oscillating.  
It will be shown later that it not only appears to be oscillating but that the oscillation is s.h.m.  
   
Similarly, if we set up apparatus as shown below, it will be noticed that the shadow of the pendulum bob appears to be oscillating (with s.h.m.) when the pendulum itself is either oscillating or moving in a circular path at constant speed... you see no difference in the motion of the shadow (as long as the amplitude of the oscillation is small).  
   
See here for an animation showing the relation between s.h.m. and circular motion.   
   
These observations suggest that, for any s.h.m. we can find a corresponding circular motion.  
   
When we say that a circular motion corresponds to an s.h.m. we mean:  
1. The radius of the circle is equal to the amplitude of the s.h.m. (hence "r" for amplitude)  
2. The time period of the circular motion is equal to that of the s.h.m.  
   
For a given s.h.m. the corresponding circular motion is usually called the auxiliary circular motion.  
   
This comparison between circular motion and s.h.m. leads to a convenient way of finding the value of the constant of proportionality between acceleration and displacement, for a given s.h.m., mentioned above.  
   
Finding the Constant of Proportionality for a Given S.H.M.  
   
  A body oscillates between point A and B around the equilibrium position O as shown.   
   
At the instant shown, it has displacement, x, and we assume that it experiences a restoring force, the magnitude of which is proportional to x, so the motion is simple harmonic.   
   
     
This diagram shows an auxiliary circular motion for this s.h.m.   
Consider the point p to be like the shadow of point p'.    
   
Point p', moving in a circular path at constant speed will have acceleration given by   
   
see here for proof    
   
The point p always follows point p' but its motion is restricted to the line A-B so, at any instant, its acceleration is equal to the component of the acceleration of p along this direction.  
Therefore, the acceleration of p is   
   
However, looking at the diagram, we see that   
   
This gives us   
   
Remembering that a and x must always be in the opposite sense, we include a negative sign to finally obtain   
   
We have therefore shown that the constant of proportionality between acceleration and displacement for a body moving with s.h.m. is simply equal to the square of the angular speed of the axillary circular motion  
Furthermore, since   
   
we see that we have an easy way to find the value of this constant in any given oscillation... just measure the time period  
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