Standard form consists of a number between 1 and 10 multiplied by a power of 10. For big numbers and very small numbers standard form is very useful.
The following number is shown in standard form:
3.28 × 105
= 3.28 × 100 000 = 328 000
Consider this number:
We find that there are 18 digits after the first digit, so we can write the number in standard form as:
4.505 × 1018
For fractions we count how far back the first digit is from the decimal point:
In this case it is six places from the decimal point, so it is:
3.42 × 10-6
A negative power of ten (negative index) means that the number is a fraction, i.e. between 0 and 1.
Convert the following numbers into standard form:
86; 381; 45300; 1 500 000 000; 0.03; 0.00045; 0.0000000782
There is no hard and fast rule as to when to use standard form in an answer. Basically if your calculator presents an answer in standard form, then use it. I generally use standard form for:
When doing a conversion from one unit to another, for example from millimetres to metres, I consider it perfectly acceptable to write:
15 mm = 15 × 10-3 m
Using a Calculator
A scientific calculator is an essential tool in Physics, just like a chisel is to a cabinet-maker. A calculator geared just to money is fine for an accounts clerk, but quite useless to a physicist. All physics exams assume you have a calculator, and you should always bring a calculator to every lesson. They are not expensive, so there is no excuse for not having one.
The calculator should be able to handle:
angles in degrees and radians;
natural logarithms and logarithms to the base 10.
Most scientific calculators have this and much more.
There are no hard and fast rules as to what calculator you should buy:
I am assuming that you know the basic functions of your calculator, but I need to draw your attention to a couple of points:
Do NOT write 2.317. This means "2.31 to the power 7" (= 351). This will get a mark deduction because it's an arithmetical error.
Consider this calculation:
Vrms = 13.6
Your calculator will give the answer as Vrms = 9.6166526 V
There is no reason at all in A-level Physics to write any answer to any more than 3 significant figures. Three significant figures is claiming accuracy to about one part in 1000. Blindly writing your calculator answer is claiming that you can be accurate to one part in 100 million, which is absurd. Before calculators were common, slide rules were used extensively. They would give answers to two significant figures, three at a push.
The examination mark schemes give answers that are no more than 2 significant figures. So our answer becomes:
Vrms = 9.62 V (3 s.f.)
Vrms = 9.6 V (2 s.f.)
Do any rounding up or down at the end of a calculation. If you do any rounding up or down in the middle, you could end up with rounding errors.
Many questions tell you to write your answer to an appropriate number of significant figures. The rule is:
The answer should be to the same number of significant figures as the quantity with the lowest number of significant figures.
For a question that quotes these data:
h = 6.63 × 10-34 Js
e = 1.6 × 10 -19 C
m = 9.11 × 10 -31 kg
V = 1564 V
The answer will be to no more than 2 significant figures, as the quantity e is to 2 significant figures.
Some other tips on use of calculators:
On most calculators the number is keyed in before the function (sin, cos, log)
Take one step at a time and write intermediate results.
It is easy to make a mistake such as pressing the × key rather than the ÷ key. It is a good idea to do the calculation again as a check.
As you get more experienced, you will get a feel for what is a reasonable answer. 1000 N is a reasonable force that a car would use to accelerate; 2 × 10-10 N is most certainly not.
Use your calculator to do the
following calculations. Write your answers to no more than three