Why isn't a pendulum clock considered a perpetual machine? Why do people say that a perpetual machine is one that runs for eternity? Wouldn't a machine that runs taking no energy from outside but sometimes needs to be restarted, such as pendulum clock, be considered a perpetual machine? It just needs to be restarted one or two times a month. Given that physicists love approximations, wouldn't restarting a machine two times a month be higher order smaller term that we could throw away and call the pendulum clock a perpetual machine?
 A: There is no such thing as a physical quantity that is small or large in an absolute sense. Quantities can only be small or large compared to other quantities. Furthermore, comparisons must be between quantities with the same physical dimensions (which can be unitless).
For example, sometimes a velocity is much less than the speed of light: $v \ll c$, or equivalently $v/c \ll 1$. Sometimes we say "$v$ is small" but the understanding is that we mean to say "$v$ is small compared to $c$."
In your case, you want to throw out a quantity. Either it's the time between rewindings $T \approx 1\ \mathrm{month}$ (which you say is long) or the frequency of rewindings $f \approx 4\times10^{-7}\ \mathrm{Hz}$ (which you say is small). But you haven't specified what the quantity is large or small compared to.
Finally, whenever a quantity is properly neglected, it can be viewed in the paradigm $x + \epsilon \approx x$, where $\lvert \epsilon \rvert \ll \lvert x \rvert$. You can replace one number by another if they differ by small amount (compared to the actual value). But you can never just replace a number by $0$ in vacuo. For example, if your formula is $P = 3\pi (\epsilon/x)^2$, then even if $\lvert \epsilon \rvert \ll \lvert x \rvert$, $P \approx 0$ can be a very bad approximation. If the scale of the problem is set by $Q = 16 (\epsilon/x)^4$, then $P$ is in fact a large quantity, and you wouldn't be justified in saying $Q + P \approx Q + 0 = Q$.

As for perpetual motion itself, we define it to mean you don't inject energy ever. One month is short compared to "infinite time," and in fact any finite length of time is "infinitely short" compared to $\infty$. As far as infinity is concerned, restarting your clock once a month is no different from restarting it once every nanosecond.
A: It is not considered a perpetual motion machine for two reasons:

*

*a month is not perpetuity


*the pendulum clock does take energy from outside- when you say 'restarted' what you mean is resupplied with energy from outside.
As for your opening line and your final suggestion, the entire point of the concept of a perpetual motion machine is to show that nothing can run forever without being resupplied with energy- to illustrate a principle. Making approximations that blur the principle you are trying to highlight would be rather pointless.
A: I have an old pendulum clock that needs winding every week.  Winding involves raising a block of iron weighing a few kilograms through a distance of about a metre.  In no way is this a perpetual motion machine.
