Pendulums, springs, and natural frequencies

I remember while preparing for the JEE I came across a simple spring mass system and was quite surprised by the fact that it should have a natural frequency at which it must vibrate when left to its own devices. An analogous but more intuitive system is a regular clock pendulum. The pendulum completes one cycle in exactly a second, and, therefore, has a natural frequency of 1 Hz. It used to fascinate me that the pendulum should do just that without any external intervention (There is some external intervention to make up for energy dissipations from air viscosity, friction etc. but we'll neglect those 'non-ideal' effects for now.)

So  I was thinking about the simple spring mass system the other day and was quite pleased with the fact that one can explain the 'complicated concept' of natural frequencies in terms of purely physical intuition. Think of a spring which is attached to a wall at one end and a block which is free to move back and forth at the other end. The spring is made up of a material whose weight is much smaller than the weight of the block (assumption of massless spring) and the whole assembly is placed on a very slippery surface (assumption of frictionless surface.) Let's call the initial location of the block which corresponds to zero stretching in the spring,  'S'. Now we take the block and pull it so that the spring stretches by say 5 cm. and leave it. If one has ever held a spring in hand he knows that it takes effort to stretch it. In fact, it becomes progressively more difficult to stretch the spring to larger lengths. The same it true with compressing it - so the spring has a tendency to force itself to an unstretched position. Therefore, in our spring block system, the spring pulls the block back and keeps pulling it till the spring reaches its unstretched position. But by this time the block has attained a velocity so it doesn't stop and starts compressing the spring. The spring tries to stop this compression but it takes a certain amount of time before the spring is able to bring the block to a stop. By that time the block has already compressed the spring by a certain distance and that distance is exactly equal to the initial stretching of the spring (5 cm. in this case.) So our initial stretch has been completely converted into an equivalent compression. Now that the block is at rest again, the spring starts pulling it back until it crosses 'S', starts stretching the spring due to its speed, and stops when the spring is stretched by 5 cm. This whole cycle of stretching-compression-stretching takes a certain amount of time and now we ask ourselves a question, is it possible for this time to be any less than what it is? We also put the constraint, for the time being, that the amplitude of vibration (5 cm. in this case) has to be the same. For that to be the case, the block would have to travel at a higher speed on average and stop faster. This also means that it will have to be traveling at a higher speed when it crosses 'S'. But if that is the case, the spring will not be able to stop it within 5 cm. If the spring wants to stop it within 5 cm., it will have to be a stiffer spring but that is not allowed in our current thought experiment. Now let's relax the constraint that the amplitude of vibration has to be 5 cm. But that is again problematic because now the block has to travel a larger distance to complete one cycle. So even though it may be traveling at a higher speed on average, it will still take the same amount of time due to the larger distance it has to cover in each cycle.

This brings us to the conclusion that as long as we do not change the spring and the mass, we cannot change the time the system takes to complete one cycle - and this precisely is the natural frequency of the system - a constant for this simple system! In fact, as argued above, it is possible to complete the cycle in a faster time (higher frequency) if the spring is stiffer. It's even possible to do it with the same spring if we decrease the weight of the block - because it's easier to stop lighter objects than heavier objects. So the frequency of our system (number of cycles in a second) seems to increase with increasing stiffness and decreasing mass. Well, that's about how much we can deduce without mathematics! The exact relation is frequency=C*{k/m}^.5, where 'k' is the stiffness of the spring and 'm' is the mass of the block (C is a constant.)

The pendulum is a very analogous system where the effect of the spring is replaced by the pull of gravity. In fact, a lot of systems in the real world are walking this tightrope where there exists a certain force which wants to pull them back to a rest position. In a more complicated way, they all display preferences for certain 'frequencies'. They all want to complete their cycles in a certain time. It's all very drab and academic when a spring-mass system does that but it's all so artistic and cultured when a violin does the exact same thing!