Q7.11 Thinking about the pendulum (T)
- Write down the equation for the period of the SHM of a simple pendulum; check that it makes sense.
- Estimate the value of the amplitude θm below which the period of a real pendulum will agree with the SHM prediction to within 1%.
- Show with a graph how you would expect the period of a real pendulum to depend on the amplitude.
- A pendulum is located in a lift which accelerates uniformly downwards with acceleration a=g/10. What is the effect on its period?
Hint
Reveal

Hint
It is helpful to refer back to Q7.2. In the second bit think of an extreme case (assume the bob is suspended from a stiff string, or rod). In the last bit remember The Principle of Equivalence.Solution
Reveal

Solution
Perhaps this is the right moment to consider a question which may have occurred to you as you work though (I jest not) your lecture notes:
"Do I have to memorize this equation?"
I mean this as a general question about any equation, not just the one at issue here.
The answer is: it depends on the equation. That needs elaboration!
There are certain key equations which you should always be able to ‘write down’ without any assistance. Some examples (not an exhaustive list!):
- The constant acceleration equations
- Newton’s 2nd law
- The gravitational force law
- The generic equation of SHM
- The Taylor Series expansion (surprised?)
These are core equations which are applicable in such a wide range of contexts that they must be known in this demanding sense.
There are other equations which are worth remembering because, though less general or fundamental, they are still handy: Some examples:
- The period (or frequency) of a simple pendulum
- The period (or frequency) of the mass spring system
- The speed of a wave on a string
- The TSE for sine and cosine
In both cases you should realise that ‘remembering’ or ‘memorizing’ should not be a blind process (such as you might adopt of you had to memorize bits of a telephone directory, or the like): you will always find that you can recall better what you understand; and you should always be able to check it (and refine it if needbe) using the rules that all equations have to satisfy.
There are also many equations which you should not feel you have to ‘carry around’ with you. In this category are all the equations that appear as intermediate steps in arguments (you might have to recover them, but you don’t have to remember them); and equations that, while important, can be established easily from one of the ones you do have at your finger tips (Example: the equation for
for a particle exhibiting SHM, which you should get by differentiating the equation for x, which you should know.)
With that as motivation we write down the results for the period (and angular frequency) of the simple pendulum:
We ’check it makes sense’ (or ’explore’) this equation in the usual ways. We check the units:as it should be to match the units of the period T.Then we check the way the result varies with the physical parameters (in the extreme, special cases):
- Switching off g sends the period to infinity, which is fine since with no g there is no restoring force and a displaced pendulum just stays where it is.
- Decreasing L decreases the period: you know this is right from painful experiences in laboratory classes.
- The answer to this bit is shorter: we’ve done the work. The result for the period of the pendulum rests upon the TSE approximation sinθ≃θ. In Q7.2 you should have found that this approximation is good to 1% provided θ does not exceed about 0.24 radians or 14∘. The amplitude must satisfy this condition.
Let’s build this up in bits. We know that for all values of θm less than the estimate we have just made the period should be given by the SHM result, which is independent of θm. Thus , for small θm a graph of Ttrue (the period of a real pendulum) versus θm should be flat, coinciding with the SHM result.
As we increase θm we must expect the result to differ from the SHM prediction. We get some clue how this departure kicks in from the form of the next term in the TSE for sinθ (Q7.2)
The correction term pulls the value of the sine down below the sinθ≃θ approximation. This suggests the restoring force is less than this approximation implies; we would thus expect the period to begin to rise above the SHM result.
We can come to the same conclusion on physical rather than mathematical grounds. Think of a very extreme case: make the amplitude θm=π so we start the pendulum upside down. (Of course we’d better think of the string as being a rigid light rod! We can do so without compromising any of the assumptions we made in establishing the SHM result for T). The upside down pendulum is actually in a position of (unstable) equilibrium. If started off (exactly) there it will stay there and the period of oscillation (about the point of stable equilibrium) will be infinite. If we start it off just a little away from there it will gradually (but with increasing speed) move further away; the closer we start it off to the vertical the longer it will take to move away from it.
The implication: we must expect the period to diverge to plus infinity as θ approaches pi. The sketch puts these ideas together. The picture it suggests is essentially correct, as one can show by solving the EOM without using the sinθ≃θ approximation. - A pendulum is located in a lift which accelerates uniformly downwards with acceleration a=g/10. What is the effect on its period?
![\[ T=2\pi \sqrt{\frac{L}{g_{\mbox{eff}}}} \hspace{0.5cm}\mbox{\rm where}\hspace{0.5cm} g_{\mbox{eff}} =g-a = \frac{9g}{10} \]](mastermathpng-4.png)