Throughout physics, we calculate, measure and use many quantities. All quantities consist of a value and a unit. In physics, we mostly use units from the SI system. These units are all defined with extreme care, and for a good reason. In science and engineering, every measurement must be made on the same basis, so that measurements obtained in different laboratories can be compared. This is important for commercial reasons, too. Suppose an engineering firm in Taiwan is asked to produce a small part for the engine of a car that is to be assembled in India. The dimensions are given in millimetres and the part must be made with an accuracy of a tiny fraction of a millimetre. All concerned must know that the part will fit correctly – it would not be acceptable to use a different millimetre scale in Taiwan and India.

Base units, derived units

The metre, kilogram and second are three of the seven SI base units. These are defined with great precision so that every standards laboratory can reproduce them correctly.
Other units, such as units of speed ($m\,{s^{ - 2}}$) and acceleration ($m\,{s^{ - 2}}$) are known as derived units because they are combinations of base units. Some derived units, such as the newton and the joule, have special names that are more convenient to use than giving them in terms of base units. The definition of the newton will show you how this works.

Defining the newton

Isaac Newton (1642–1727) played a significant part in developing the scientific idea of force. Building on Galileo’s earlier thinking, he explained the relationship between force, mass and acceleration, which we now write as $F = ma$. For this reason, the SI unit of force is named after him.
We can use the equation F = ma to define the newton (N).
One newton is the force that will give a $1 kg$ mass an acceleration of $1\,m\,{s^{ - 2}}$ in the direction of the force.
$1N = 1kg \times 1\,m\,{s^{ - 2}}\,or\,1N = 1kg\,m\,{s^{ - 2}}$

The seven base units

In mechanics (the study of forces and motion), the units we use are based on three base units: the metre, kilogram and second. As we move into studying electricity, we will need to add another base unit, the ampere. Heat requires another base unit, the kelvin (the unit of temperature).
Table 3.4 shows the seven base units of the SI system. Remember that all other units can be derived from these seven. The equations that relate them are the equations that you will learn as you go along (just as $F = ma$ relates the newton to the kilogram, metre and second). The unit of luminous intensity is not part of the AS 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 AS & A Level courses, as shown in Table 3.5.
You must take care when using prefixes. A Level courses.

Other SI units

Using only seven base units means that only this number of quantities have to be defined with great precision. It would be confusing if more units were also defined. For example, if the density of water were defined as exactly $1\,g\,c{m^{ - 3}}$, then $1000\,c{m^3}$ of a sample of water would have a mass of exactly $1 kg$.
However, it is unlikely that the mass of this volume of water would equal exactly the mass of the standard kilogram.
All other units can be derived from the base units. This is done using the definition of the quantity. For example, speed is defined as $\frac{{dis\tan ce}}{{time}}$ , and so the base units of speed in the SI system are $m\,{s^{ - 1}}$.
Since the defining equation for force is $F = ma$, the base units for force are $kg\,m\,{s^{ - 2}}$.
Equations that relate different quantities must have the same base units on each side of the equation. Ifthis does not happen the equation must be wrong.
When each term in an equation has the same base units the equation is said to be 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 AS & A Level courses, as shown in Table 3.5.
You must take care when using prefixes.

Squaring or cubing prefixes

For example:

$1\,cm\, = \,{10^{ - 2}}m$
so $1\,c{m^2}\, = \,{({10^{ - 2}}m)^2} = {10^{ - 4}}{m^2}$
and $1\,c{m^3} = {({10^{ - 2}}m)^3} = {10^{ - 6}}{m^3}$

Writing units

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 $3\,m\,{s^{ - 1}}$ it would mean $3\,milli\sec ond{\,^{ - 1}}$.

EXAM-STYLE QUESTIONS

 

1) Which list contains only SI base units? [1]
A: ampere, kelvin, gram
B: kilogram, metre, newton
C: newton, second, ampere
D: second, kelvin, kilogram

2) The speed v of a wave travelling a wire is given by the equation $v = {\left( {\frac{{Tl}}{m}} \right)^n}$
where T is the tension in the wire that has mass m and length l.
In order for the equation to be homogenous, what is the value of n? [1]
A: $\frac{1}{2}$
B: 1
C: 2
D: 4

3) When a golfer hits a ball his club is in contact with the ball for about $0.000 50 s$ and the ball leaves the club with a speed of $70\,m\,{s^{ - 1}}$. The mass of the ball is $46g$
a: Determine the mean accelerating force. [4]
b: What mass, resting on the ball, would exert the same force as in part a? [2]
[Total: 6]

4) The mass of a spacecraft is $70 kg$. As the spacecraft takes off from the Moon, the upwards force on the spacecraft caused by the engines is $500 N$. The acceleration of free fall on the Moon is $1.6\,N\,k{g^{ - 1}}$.
Determine:
a: the weight of the spacecraft on the Moon [2]
b: the resultant force on the spacecraft [2]
c: the acceleration of the spacecraft. [2]
[Total: 6]

5) A metal ball is dropped into a tall cylinder of oil. The ball initially accelerates but soon reaches a terminal velocity.
a: By considering the forces on the metal ball bearing, explain why it first accelerates but then reaches terminal velocity. [3]
b: State how you would show that the metal ball reaches terminal velocity.
Suggest one cause of random errors in your readings. [4]
[Total: 7]

6) Determine the speed in $m\,{s^{ - 1}}$ of an object that travels:
a: $3.0 μm$ in $5.0 ms$ [2]
b: $6.0 km$ in $3.0 Ms$ [2]
c: $8.0 pm$ in $4.0 ns$. [2]
[Total: 6]

7) This diagram shows a man who is just supporting the weight of a box. Two of the forces acting are shown in the diagram. According to Newton’s third law, each of these forces is paired with another force.

For a the weight of the box, and b the force of the ground on the man, state:
i- the body that the other force acts upon [2]
ii- the direction of the other force [2]
iii- the type of force involved. [2]
[Total: 6]

8) A car starts to move along a straight, level road. For the first $10 s$, the driver maintains a constant acceleration of $1.5\,m\,{s^{ - 2}}$. The mass of the car is $1.1 \times {10^3}\,kg$.
a: Calculate the driving force provided by the wheels, when:
i- the force opposing motion is negligible [1]
ii- the total force opposing the motion of the car is $600 N$. [1]
b: Calculate the distance travelled by the car in the first $10 s$. [2]
[Total: 4]

9) These are the speed–time graphs for two falling balls:

a: Determine the terminal velocity of the plastic ball. [1]
b: Both balls are of the same size and shape but the metal ball has a greater mass.
Explain, in terms of Newton’s laws of motion and the forces involved, why the plastic ball reaches a constant velocity but the metal ball does not. [3]
c: Explain why both balls have the same initial acceleration. [2]
[Total: 6]

10) A car of mass $1200 kg$ accelerates from rest to a speed of $8.0\,m\,{s^{ - 1}}$ in a time of $2.0 s$.
a: Calculate the forward driving force acting on the car while it is accelerating. Assume that, at low speeds, all frictional forces are negligible. [2]
b: At high speeds the resistive frictional force F produced by air on a body moving with velocity v is given by the equation $F = b{v^2}$, where b is a constant.
i- Derive the base units of force in the SI system. [1]
ii- Determine the base units of b in the SI system. [1]
iii- The car continues with the same forward driving force and accelerates until it reaches a top speed of $50\,m\,{s^{ - 1}}$. At this speed the resistive force is given by the equation $F = b{v^2}$. Determine the value of b for the car. [2]
iv- Use your value for b in iii and the driving force calculated in part a to calculate the acceleration of the car when the speed is $30\,m\,{s^{ - 1}}$. [2]
v- Sketch a graph showing how the value of F varies with v over the range 0 to $50\,m\,{s^{ - 1}}$. Use your graph to describe what happens to the acceleration of the car during this time. [2]
[Total: 10]

11) a: Explain what is meant by the mass of a body and the weight of a body. [3]
b: State and explain one situation in which the weight of a body changes while its mass remains constant. [2]
c: State the difference between the base units of mass and weight in the SI system. [2]
[Total: 7]

12) a: State Newton’s second law of motion in terms of acceleration. [2]
b: When you jump from a wall on to the ground, it is advisable to bend your knees on landing.
i- State how bending your knees affects the time it takes to stop when hitting the ground. [1]
ii- Using Newton’s second law of motion, explain why it is sensible to bend your knees. [2]
[Total: 5]