Why don't we generally notice the effects of special relativity in our daily lives? Be specific.
The short answer is that relativistic effects are generically of order (v/c)^2, where v is some typical speed of the system you are looking at and c is the speed of light. In our daily lives the typical speeds are much lower than the speed of light (which is about 3*10^8 m/s), so relativistic effects are very small.
Put differently and slightly more rigorously, suppose you do an experiment e.g. you measure the period of a pendulum. If you calculate theoretically what you should find using special relativity and using classical mechanics, then the fractional difference of the two predictions is:
[Prediction of Relativity - Prediction of Classical Mechanics]/Prediction of Classical Mechanics =
some constant times (v/c)^2
for v a typical velocity of the system. So, for a pendulum swinging at an average velocity of 1 m/s, you would expect the fractional difference to be of order 10^(-17) and even for very fast moving objects, say 1 km/s, the relativistic correction is only of order 10^(-11).
But why are relativistic effects only of order (v/c)^2 and not of order v/c? This is because relativity does not merely say that the speed of light is some finite large number but also ...
We show that relativistic effects are generically of order (v/c)^2. We also point out that there are relativistic effects that are not small in our daily lives. We conclude by explaining the classical limit c --> infinity.