Page 53 - Physics Coursebook 2015 (A level)

Chapter 3: Dynamics – explaining motion
WORKED EXAMPLE
3 It is suggested that the time T for one oscillation of a swinging pendulum is given by the equation
T 2 = 4π2(l/g) where l is the length of the pendulum and g is the acceleration due to gravity. Show that this equation is homogeneous.
For the equation to be homogeneous, the term on the left-hand side must have the same base units as all the terms on the right-hand side.
Step1 ThebaseunitoftimeTiss.Thebaseunitof the left-hand side of the equation is therefore s2.
Step2 Thebaseunitoflism.Thebaseunitsofgare m s−2. Therefore the base unit of the right-hand side is
m = s2. (Notice that the constant 4π2 has no (m s−2)
units.)
Since the base units on the left-hand side of the equation are the same as those on the right, the equation is homogeneous.
Prefixes
Each unit in the SI system can have multiples and sub- multiples to avoid using very high or low numbers. For example 1 millimetre (mm) is one thousandth of a metre and 1 micrometre (μm) is one millionth of a metre.
The prefix comes before the unit. In the unit mm, the first m is the prefix milli and the second m is the unit metre. You will need to recognise a number of prefixes for the A/AS course, as shown in Table 3.3.
Multiples Sub-multiples
QUESTIONS
7 Find the area of one page of this book in cm2 and then convert your value to m2.
8 Write down in powers of ten the values of the following quantities:
a 60pA
b 500MW
c 20 000 mm
WORKED EXAMPLE
4
The density of water is 1.0 g cm−3. Calculate this value in kg m−3.
Step1 Findtheconversionsfortheunits: 1g=1×10−3kg
1cm3 =1×10−6m3
Step2 Usetheseinthevalueforthedensityofwater: 1.0gcm−3= 1.0×1×10−3
1 × 10−6
= 1.0 × 103 kg m−3
Multiple Prefix Symbol
Multiple Prefix Symbol
10−1 deci d
10−2 centi c
The pull of gravity
Now we need to consider some specific forces – such as weight and friction.
When Isaac Newton was confined to his rural home
to avoid the plague which was rampant in other parts of England, he is said to have noticed an apple fall to the ground. From this, he developed his theory of gravity which relates the motion of falling objects here on Earth to the motion of the Moon around the Earth, and the planets around the Sun.
The force which caused the apple to accelerate was the pull of the Earth’s gravity. Another name for this force is the weight of the apple. The force is shown as an arrow, pulling vertically downwards on the apple (Figure 3.4). It is usual
to show the arrow coming from the centre of the apple –
its centre of gravity. The centre of gravity of an object is defined as the point where its entire weight appears to act.
103
106
109
1012
1015
Table 3.3
kilo k
mega M
giga G
tera T
peta P
10−3 mill
10−6 micro
10−9 nano
m
μ
n
41
10−12 pico p
Multiples and sub-multiples.
You must take care when using prefixes.
■■ Squaring or cubing prefixes – for example: 1cm=10−2m
so1cm2 =(10−2m)2 =10−4m2 and1cm3 =(10−2m)3 =10−6m3.
■■ Writing units – for example, you must leave a small space between each unit when writing a speed such as 3 m s−1, because if you write it as 3 ms−1 it would mean
3 millisecond−1.
weight = mg
Figure 3.4 The weight of an object is a force caused by the Earth’s gravity. It acts vertically down on the object.