CXC Physics Notes Section A SI Base Units The SI (Système International d'Unités) is a globally agreed system of units, with seven base units. Formally agreed by the 11th General Conference on Weights and Measures (CGPM) in 1960, the SI is at the centre of all modern science and technology. The definition and realisation of the base and derived units is an active research topic for meteorologists with more precise methods being introduced as they become available. There are two classes of units in the SI: base units and derived units. The base units provide the reference used to define all the measurement units of the system, whilst the derived units are products of base units and are used as measures of derived quantities: The seven SI base units, which comprise: The ampere (A) - unit of measurement of electric current The kilogram (kg) - unit of measurement of mass The metre (m) - unit of measurement of length The second (s) - unit of measurement of time The kelvin (K) - unit of measurement of thermodynamic temperature The mole (mol) - unit of measurement of amount of substance The candela (cd) - unit of measurement of luminous intensity Other units of physics, (joules, pascals etc.) are derived from these base units. It is therefore important to define the derived quantity properly as this will give the answer to a sharp minded physicist on what the derived unit is. Example, work is defined as force that acts on a body and displaces it from the point of application to the direction of the force. What does this mean exactly? Force = ma (mass x acceleration); unit – kg m/s2 Displacement = s (displacement is different from distance, the s is not seconds); unit – m By the definition, it basically states that work is force acting on a body displacing it in a particular direction. Therefore the SI derived unit of work is the combination of force x displacement which equates to: kg m/s2 x m = kg m2/s2
SI Derived Units Derived units are units which may be expressed in terms of base units by means of mathematical symbols of multiplication and division. Certain derived units have been given special names and symbols, and these special names and symbols may themselves be used in combination with the SI and other derived units to express the units of other quantities. NOTE: ALL these derived quantities when defined WILL give you the SI derived units, there is absolutely NO quantity that DOES NOT follow this basic principle of physics Examples of SI derived units expressed in terms of base units Derived Quantity
SI derived unit Name
Symbol
area
square metre
m2
volume
cubic metre
m3
speed, velocity
metre per second
m/s
acceleration
metre per second squared
m/s2
wavenumber
1 per metre
m-1
density, mass density
kilogram per cubic metre
kg/m3
specific volume
cubic metre per kilogram
m3/kg
current density
ampere per square metre
A/m2
magnetic field strength
ampere per metre
A/m
concentration (of amount of substance)
mole per cubic metre
mol/m3
luminance
candela per square metre
cd/m2
refractive index
(the number) one
1(a)
(a) The symbol '1' is generally omitted in combination with a numerical value.
SI Prefixes SI prefixes are used to form decimal multiples and sub-multiples of SI units. They should be used to avoid very large or very small numeric values. The prefix attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit.
Multiplying Factor
SI Prefix
Scientific Notation
1 000 000 000 000 000 000 000 000 yotta (Y)
1024
1 000 000 000 000 000 000 000
zetta (Z)
1021
1 000 000 000 000 000 000
exa (E)
1018
1 000 000 000 000 000
peta (P)
1015
1 000 000 000 000
tera (T)
1012
1 000 000 000
giga (G)
109
1 000 000
mega (M)
106
1 000
kilo (k)
103
0.001
milli (m)
10-3
0.000 001
micro (µ)
10-6
0.000 000 001
nano (n)
10-9
0.000 000 000 001
pico (p)
10-12
0.000 000 000 000 001
femto (f)
10-15
0.000 000 000 000 000 001
atto (a)
10-18
0.000 000 000 000 000 000 001
zepto (z)
10-21
0.000 000 000 000 000 000 000 001 yocto (y)
10-24
Vectors A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories, vectors and scalars. A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces that occur in two dimensions. Examples of vector quantities that have been previously discussed include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. For example, suppose your teacher tells you "A bag of gold is located outside the classroom. To find it, displace yourself 20 meters." This statement may provide yourself enough information to pique your interest; yet, there is not enough information included in the statement to find the bag of gold. The displacement required to find the bag of gold has not been fully described. On the other hand, suppose your teacher tells you "A bag of gold is located outside the classroom. To find it, displace yourself from the centre of the classroom door 20 meters in a direction 30 degrees to the west of north." This statement now provides a complete description of the displacement vector – it lists both magnitude (20 meters) and direction (30 degrees to the west of north) relative to a reference or starting position (the centre of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed.
Representing Vectors Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object. Such diagrams are commonly called as free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram. a scale is clearly listed a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail. the magnitude and direction of the vector is clearly labelled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).
Vector Additions A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). In forces as the example, the net force experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. That is the net force was the result (or resultant) of adding up all the force vectors. Observe the following summations of two force vectors:
These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces). Sample applications are shown in the diagram below.
In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below. There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are: the Pythagorean theorem and trigonometric methods
the head-to-tail method using a scaled vector diagram
The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the
hypotenuse of a right triangle.
To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. This problem asks to determine the result of adding two displacement vectors that are at right angles to each other. The result (or resultant) of walking 11 km north and 11 km east is a vector directed north-east as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant (i.e., the hypotenuse of the right triangle).
The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km.
Force A force is a push or pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. Forces only exist as a result of an interaction.
Contact versus Action-at-a-Distance Forces For simplicity sake, all forces (interactions) between objects can be placed into two broad categories: contact forces, and forces resulting from action-at-a-distance Contact forces are those types of forces that result when the two interacting objects are perceived to be physically contacting each other. Examples of contact forces include frictional forces, tensional forces, normal forces, air resistance forces, and applied forces. Action-at-a-distance forces are those types of forces that result even when the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite their physical separation. Examples of action-at-a-distance forces include gravitational forces. For example, the sun and planets exert a gravitational pullon each other despite their large spatial separation. Even when your feet leave the earth and you are no longer in physical contact with the earth, there is a gravitational pull between you and the Earth. Electric forces are action-at-a-distance forces. For example, the protons in the nucleus of an atom and the electrons outside the nucleus experience an electrical pull towards each other despite their small spatial separation. And magnetic forces are action-at-a-distance forces. For example, two magnets can exert a magnetic pull on each other even when separated by a distance of a few centimetres. Examples of contact and action-at-distance forces are listed in the table below. Contact Forces Frictional Force Tension Force Normal Force Air Resistance Force Applied Force Spring Force
Action-at-a-Distance Forces Gravitational Force Electrical Force Magnetic Force
Type of Force (and Symbol)
Description of Force
Applied Force
An applied force is a force that is applied to an object by a person or another object. If a person is pushing a desk across the room, then there is an applied force acting upon the object. The applied force is the force exerted on the desk by the person.
Fapp
Gravity Force
The force of gravity is the force with which the earth, moon, or other massively large object attracts another object towards itself. By definition, this is the weight of the object. All objects upon earth experience a force of gravity that is directed "downward" towards the centre of the earth. The force of gravity on earth is always equal to the weight of the object as found by the equation:
(also known as Weight)
Fgrav = m * g where g = 9.8 N/kg (on Earth) and m = mass (in kg) (Caution: do not confuse weight with mass.)
Fgrav Normal Force
Fnorm
Friction Force
Ffrict
The normal force is the support force exerted upon an object that is in contact with another stable object. For example, if a book is resting upon a surface, then the surface is exerting an upward force upon the book in order to support the weight of the book. On occasions, a normal force is exerted horizontally between two objects that are in contact with each other. For instance, if a person leans against a wall, the wall pushes horizontally on the person. The friction force is the force exerted by a surface as an object moves across it or makes an effort to move across it. There are at least two types of friction force - sliding and static friction. Thought it is not always the case, the friction force often opposes the motion of an object. For example, if a book slides across the surface of a desk, then the desk exerts a friction force in the opposite direction of its motion. Friction results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. The maximum amount of friction force that a surface can exert upon an object can be calculated using the formula below: Ffrict = µ • Fnorm The friction force is discussed in more detail later on this page.
Air Resistance Force
Fair Tension Force
The air resistance is a special type of frictional force that acts upon objects as they travel through the air. The force of air resistance is often observed to oppose the motion of an object. This force will frequently be neglected due to its negligible magnitude (and due to the fact that it is mathematically difficult to predict its value). It is most noticeable for objects that travel at high speeds (e.g., a skydiver or a downhill skier) or for objects with large surface areas. Air resistance will be discussed in more detail in Lesson 3. The tension force is the force that is transmitted through a string, rope, cable or wire when it is pulled tight by forces acting from opposite ends. The tension force is directed along the length of the wire and pulls equally on the objects on the opposite ends of the wire.
Ftens Spring Force
Fspring
The spring force is the force exerted by a compressed or stretched spring upon any object that is attached to it. An object that compresses or stretches a spring is always acted upon by a force that restores the object to its rest or equilibrium position. For most springs (specifically, for those that are said to obey "Hooke's Law"), the magnitude of the force is directly proportional to the amount of stretch or compression of the spring.
Confusion of Mass and Weight A few further comments should be added about the single force that is a source of much confusion to many students of physics – the force of gravity. As mentioned above, the force of gravity acting upon an object is sometimes referred to as the weight of the object. Many students of physics confuse weight with mass. The mass of an object refers to the amount of matter that is contained by the object; the weight of an object is the force of gravity acting upon that object. Mass is related to how much stuff is there and weight is related to the pull of the Earth (or any other planet) upon that stuff. The mass of an object (measured in kg) will be the same no matter where in the universe that object is located. Mass is never altered by location, the pull of gravity, speed or even the existence of other forces. For example, a 2-kg object will have a mass of 2 kg whether it is located on Earth, the moon, or Jupiter; its mass will be 2 kg whether it is moving or not (at least for purposes of our study); and its mass will be 2 kg whether it is being pushed upon or not. On the other hand, the weight of an object (measured in Newton) will vary according to where in the universe the object is. Weight depends upon which planet is exerting the force and the distance the object is from the planet. Weight, being equivalent to the force of gravity, is dependent upon the value of g – the gravitational field strength. On earth's surface g is 9.8 N/kg (often approximated as 10 N/kg). On the moon's surface, g is 1.7 N/kg. Go to another planet, and there will be another g value. Furthermore, the g value is inversely proportional to the distance from the centre of the planet. So if we were to measure g at a distance of 400 km above the earth's surface, then we would find the g value to be less than 9.8 N/kg. (The nature of the force of gravity will be discussed in more detail in a later unit of The Physics Classroom.) Always be cautious of the distinction between mass and weight. It is the source of much confusion for many students of physics.
The Newton Force is a quantity that is measured using the standard metric unit known as the Newton. A Newton is abbreviated by an "N." To say "10.0 N" means 10.0 Newton of force. One Newton is the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s. Thus, the following unit equivalence can be stated: 1 Newton = 1 kg • m/s2
Force is a Vector Quantity A force is a vector quantity. As learned in an earlier unit, a vector quantity is a quantity that has both magnitude and direction. To fully describe the force acting upon an object, you must describe both the magnitude (size or numerical value) and the direction. Thus, 10 Newton is not a full description of the force acting upon an object. In contrast, 10 Newton, downward is a complete description of the force acting upon an object; both the magnitude (10 Newton) and the direction (downward) are given. Because a force is a vector that has a direction, it is common to represent forces using diagrams in which a force is represented by an arrow. Such vector diagrams were introduced in an earlier unit and are used throughout the study of physics. The size of the arrow is reflective of the magnitude of the force and the direction of the arrow reveals the direction that the force is acting. Furthermore, because forces are vectors, the effect of an individual force upon an object is often cancelled by the effect of another force. For example, the effect of a 20-Newton upward force acting upon a book is cancelled by the effect of a 20-
Newton downward force acting upon the book. In such instances, it is said that the two individual forces balance each other; there would be no unbalanced force acting upon the book.
Other situations could be imagined in which two of the individual vector forces cancel each other ("balance"), yet a third individual force exists that is not balanced by another force. For example, imagine a book sliding across the rough surface of a table from left to right. The downward force of gravity and the upward force of the table supporting the book act in opposite directions and thus balance each other. However, the force of friction acts leftwards, and there is no rightward force to balance it. In this case, an unbalanced force acts upon the book to change its state of motion.
Newtons First Law Isaac Newton (a 17th century scientist) put forth a variety of laws that explain why objects move (or don't move) as they do. These three laws have become known as Newton's three laws of motion. The focus of Lesson 1 is Newton's first law of motion – sometimes referred to as the law of inertia. Newton's first law of motion is often stated as An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Two Clauses and a Condition There are two clauses or parts to this statement – one that predicts the behaviour of stationary objects and the other that predicts the behaviour of moving objects. The two parts are summarized in the following diagram.
The behaviour of all objects can be described by saying that objects tend to "keep on doing what they're doing" (unless acted upon by an unbalanced force). If at rest, they will continue in this same state of rest. If in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East). If in motion with a leftward velocity of 2 m/s, they will continue in this same state of motion (2 m/s, left). The state of motion of an object is maintained as long as the object is not acted upon by an unbalanced force. All objects resist changes in their state of motion – they tend to "keep on doing what they're doing." There is an important condition that must be met in order for the first law to be applicable to any given motion. The condition is described by the phrase "... unless acted upon by an unbalanced force." As the long as the forces are not unbalanced – that is, as long as the forces are balanced – the first law of motion applies. This concept of a balanced versus and unbalanced force will be discussed in more detail later in Lesson 1. Suppose that you filled a baking dish to the rim with water and walked around an oval track making an
attempt to complete a lap in the least amount of time. The water would have a tendency to spill from the container during specific locations on the track. In general the water spilled when: the container was at rest and you attempted to move it the container was in motion and you attempted to stop it the container was moving in one direction and you attempted to change its direction. The water spills whenever the state of motion of the container is changed. The water resisted this change in its own state of motion. The water tended to "keep on doing what it was doing." The container was moved from rest to a high speed at the starting line; the water remained at rest and spilled onto the table. The container was stopped near the finish line; the water kept moving and spilled over container's leading edge. The container was forced to move in a different direction to make it around a curve; the water kept moving in the same direction and spilled over its edge. The behaviour of the water during the lap around the track can be explained by Newton's first law of motion.
Everyday Applications of Newton's First Law There are many applications of Newton's first law of motion. Consider some of your experiences in an auto mobile. Have you ever observed the behaviour of coffee in a coffee cup filled to the rim while starting a car from rest or while bringing a car to rest from a state of motion? Coffee "keeps on doing what it is doing." When you accelerate a car from rest, the road provides an unbalanced force on the spinning wheels to push the car forward; yet the coffee (that was at rest) wants to stay at rest. While the car accelerates forward, the coffee remains in the same position; subsequently, the car accelerates out from under the coffee and the coffee spills in your lap. On the other hand, when braking from a state of motion the coffee continues forward with the same speed and in the same direction, ultimately hitting the wind-shield or the dash. Coffee in motion stays in motion. Have you ever experienced inertia (resisting changes in your state of motion) in an auto mobile while it is braking to a stop? The force of the road on the locked wheels provides the unbalanced force to change the car's state of motion, yet there is no unbalanced force to change your own state of motion. Thus, you continue in motion, sliding along the seat in forward motion. A person in motion stays in motion with the same speed and in the same direction... unless acted upon by the unbalanced force of a seat belt. Yes! Seat belts are used to provide safety for passengers whose motion is governed by Newton's laws. The seat belt provides the unbalanced force that brings you from a state of motion to a state of rest. Perhaps you could speculate what would occur when no seat belt is used.
