Problems
Problem-solving is a skill you have to learn, as opposed to one you can be taught, and the only way to learn it is to practise. You should aim to do all the examples on both the example sheets, whether they are set as assessed homework or not.
Note that I stole all the examples in Sheet 1 from an exam which allows 15 minutes per question. They aren't all equally difficult, so some of them may genuinely take more than 15 minutes, but if you are still stuck after half an hour you need to ask for help!
Click on the titles below to download each problems sheet as a pdf file. Click here to jump down to sheet 2.
These are general examples using a range of physical principles. All the equations you need are on the Toolkit sheet. To see a couple of examples of how to apply our four-step strategy to problems like these, look at pages 9-15 of Lecture 1.
Some basic hints, applicable to all questions
- It is usually best to start with a clearly labelled diagram.
The diagram does not have to be of high artistic quality, but it should be
large enough to label clearly and to add things like force arrows as you go
on. It may be necessary to draw several diagrams (e.g. before action
starts, during action, and after action stops).
- Think about conserved quantities. Momentum is always
conserved (allowing for the effect of forces, of course). Total
energy is always conserved, but kinetic energy may not be: not all
collisions are elastic. Unless told otherwise, you should assume that
strings don't stretch, so the speed and acceleration of the two ends of the
string must be equal in magnitude (though not necessarily in direction, of
course).
- If what's happening is not immediately obvious, try starting with
special values and working back to the general case.
-
Consider example 2 in lecture
1, and suppose you couldn't see what would happen when you dropped the small
weight. Well, what would happen if the large weight were very large indeed
(e.g., the rope was bolted to the floor)? Obviously, mass m would
come to a complete stop once the rope became taut. What would happen if
the "large" weight were actually very small – for example, a feather?
Obviously, mass m would just keep on falling, trailing the loose
rope. So what happens in between those two cases? Mass m
presumably neither stops nor carries on unaffected – it must do
something in between, i.e. carry on falling, but more slowly.
- Look for clues in the wording of the question. For example,
if the question carefully specifies that an angle is small, it means that the
solution is likely to use the small angle approximations for sine and/or
cosine; if it specifies that collisions are elastic, it means that you need
to use conservation of kinetic energy; if it talks about "rate", you
may very well need to differentiate something (derivatives are rates of
change).
- Although we have been known to provide unnecessary information
as a red herring, you should be suspicious if your answer appears not to
use something that your physical intuition tells you should be important
(for example, if your answer to a question on heat flow through a window
doesn't seem to depend on the thickness of the window, it's probably
wrong – experience tells us that heat flow does depend on
the thickness of the barrier). Conversely, if your answer seems to
require you to know information that has not been provided, and which you
couldn't reasonably be expected to guess, then the chances are you are
not using the right physics.
Most of these are exercises in dimensional analysis. There are several examples in Lecture 2. Note that you can apply
dimensional analysis to a problem even when you have only a very hazy idea of
the underlying physics.
Scaling problems are closely related to dimensional analysis problems. If you
have concluded that y ∝ x2, then if you double x you multiply y by 4, regardless of the actual values of x and y.
Simple example of a scaling problem
- Five men take five days to dig five holes in the road. How many holes can two men dig in two days?
- It's very tempting to say "two", isn't it? But think of scaling:
- Obviously, the number of holes that can be dug is proportional to the
number of men doing the digging – 10 men could dig 10 holes in
five days.
- The number of holes that can be dug is also proportional to the time
spent digging: if the five men spend another five days they can dig
another five holes, so five men can dig 10 holes in 10 days.
- Therefore, using (1), one man takes five days to dig one hole. Using
(2), one man digs 1/5 of a hole in one day (presumably this means a
hole the same size, but only 1/5 as deep!).
Therefore, two men in two days can dig 2x2x1/5 = 4/5 of a hole.
Hints for dimensional analysis problems
- Remember the basic dimensions: mass,
length, time, charge
and temperature. (Electric current can
be used instead of charge – the SI base unit is actually current, but
charge is usually easier to work with, at least in my opinion.)
[There are two other base units in SI: the mole (amount of substance) and
the candela (luminous intensity). The
candela is a horrible
unit, very seldom used in physics, whose dimensions can anyway be expressed
in terms of more common units such as watts. The mole (the amount of any
given substance containing Avogadro's number of elementary units of that
substance) essentially has the same dimensions as mass. So you aren't going
to need either of those in dimensional analysis problems.]
- To work out units for more complicated ("derived") quantities, use any
equation that suits: for example,
kinetic energy = ½mv2 will give you the dimensions
of energy as [M] [L]2 [T]-2; so will gravitational
potential energy = mgh, E = mc2, or
any other equation for energy.
- Reducing all quantities to the five base dimensions will always work, but
is not always necessary: for example, if you have force on one side of
your equation and energy on the other, it may well be enough to recall
that a joule is a newton-metre, i.e. [E] = [F] [L] where [E] stands for
"dimensions of energy" and [F] for "dimensions of force".
- It is rather easy to make mistakes in dimensional analysis problems, since
they typically ask you to solve a set of simultaneous equations, and
misplacing a minus sign will mess this up completely. Therefore, it is
essential to check that your final result makes physical sense.
- If working directly with dimensions does not come naturally to you, using
the SI base units will also work: you can reduce your quantities to kg, m,
s, C and K. But, whatever you do, don't mix the two –
otherwise you will sooner or later mix up m (metres; length) with [M] (mass;
kilograms), leading to inevitable disaster.
