Chapter3 (Force and Motion) Summary
This chapter contains the most important concept that you have to grasp, so study hard to understand the content.
Newton's first law of motion (牛頓運動第一定律 ) states that
Every object remains in a state of rest or of uniform speed along a straight line unless acted on by an external
unbalanced (non-zero) net force.
任何物體均保持它的靜止或勻速直線運動狀態，除非它受到外加的淨力而迫使它改變這種狀態。
Note the following:
1. Net force ( 淨力 ) means resultant force ( 合力 ) which is the vector sum of all forces acting on the object.
2. The net force may be zero because (1) there may be no forces acting on the object (e.g. in outer space) or (2)
the forces cancel (or balance) each other.
3. When there is no net force, the actual state of motion of the body depends on its initial state.
If it is initially at rest, it will be continually at rest.
If it is initially moving, it will continue to move with constant velocity (uniform motion in a straight line).
Newton's first law is sometimes expressed in the following Chinese wordings:
若沒有外加的淨力，則靜者恆靜，動者恆動。
4. Newton's first law reveals (揭示) that objects tend to maintain their states of motion. The ability of an object to
resist changes in its state of motion is called inertia ( 慣性 ). The mass of an object gives a measure of its
inertia. The more massive an object, the harder it is to change its motion.
Newton's first law also gives a definition of force.
Force is that which changes the state of rest or uniform motion of an object.
力是改變物體靜止狀態或等速直線運動狀態的一種作用。
Examples of inertia

In a train moving with uniform velocity, a stone is thrown up vertically. Even it is in the air, it has a horizontal
velocity same as that of the train. So it will drop back to the position of the train where it has been thrown.

When a space-ship flies in the space, free from the attraction of any planet, it will travel in a straight line
forever without any waste of fuel.

When a bus turns round a corner, we feel like being thrown away from the bus. It is because we tend to move
in the original direction.

When a car starts (or stops), we tend to move backwards (or forwards).

As soon as a stone is dropped from an ascending helicopter, it has the same upward velocity as the helicopter.
Friction
Whenever an object slides over another, friction arises which always acts in the opposite direction to the motion. In
other words, friction always opposes ( 阻止 ) the motion.
Friction is caused because no two surfaces are ever perfectly smooth. Close examination using a microscope
reveals all kinds of irregularities on the surface. The surface irregularities interlock, bump into each other and
motion is thus hindered.
Try to understand the following : A moving object will stop because friction opposes the motion. We cannot
completely eliminate (消去) friction, but when friction is greatly reduced (e.g. both surfaces in contact are very
smooth), the object will travel a much longer distance before coming to rest. By logical reasoning, if friction is
thoroughly eliminated, a moving object will move forever at its original constant speed. Thus force is not necessary
to keep an object in motion. In fact, you should have the concept that an object traveling at constant velocity is as
natural as it is at rest.
Methods of reducing friction
1.
2.
3.
4.
Place the object on a thin layer of polystyrene beads. (To reduce contact area)
Rolling the object instead of sliding it on a surface. (To reduce contact area)
Use lubricants such as oil or grease to separate the rubbing surfaces.
Separate the contact surfaces by air --- producing an air cushion.
Near friction-free conditions have to be artificially produced on earth, but spaceships travelling in outer space are
completely free from any external influences such as friction or gravity. A spaceship keeps moving with constant
speed along a straight line even if the rocket motors are turned off. The rocket motors are turned on only when the
spaceship accelerates, decelerates or changes direction.
Newton's second law of motion (牛頓運動第二定律 ) states that
The acceleration of an object is directly proportional to, and in the same direction as, the unbalanced force acting
on it, and inversely proportional to the mass of the object.
In SI unit, the unit of mass is kg, the unit of acceleration is m s-2, the unit of force is newton (N).
Newton's second law in mathematical form:
F = ma
作用於一個物體的外力總和 = 物體的質量

物體的加速度
Note the following :
1. Force is the cause ( 因 ), acceleration is the result or effect ( 果 ), i.e. force produces acceleration. The
direction of acceleration follows the direction of force. The magnitude of acceleration a is given by F/m.
This is an extremely important concept. You should always remember this point.
2. When using Newton's second law, F represents the external net (or unbalanced) force.
3. If F = 0, a = 0. This implies that the object moves with constant velocity (not speed) (including at rest, v = 0).
Thus Newton's first law can be regarded as a special case of second law. Note that a body moving with
constant velocity must be travelling in a straight line while a body moving with constant speed may be moving
uniformly in a circle.
4. We define 1 newton as the force that gives 1 kg mass an acceleration of 1 m s-2. That is
1 N = 1 kg x 1 m s-2
5. In dealing with mechanics problems, the most difficult problem is that students are not able identify all the
forces acting on the object of interest. This requires a lot of practice. Always draw a force diagram (free-body
diagram) showing all the forces acting on the object of interest.
Worked example
Suppose an object of mass 5 kg is travelling to the right with a velocity of 4 m s -1 on a
smooth horizontal surface. A constant force of 10 N now acts on the object to the right, i.e.
in the same direction as the object's motion.
(a) Find the speed of the object after 2 s.
(b) Find the distance travelled by the object after 2 s.
If the force acts to the left, i.e. opposite to the direction of motion of the object.
(c) Find the speed of the object after 2 s.
(d) Find the distance travelled by the object after 2 s.
Solution
(a) Because the force is directed to the right, the object accelerates to the right and therefore the object will move
faster and faster with acceleration a given by
a = F / m = 10 N / 5 kg = 2 m s-1
To find the speed of the object after 2 s, we use the equation of motion v = u + at
required speed v
= u + a t = 4 + 2 (2) = 8 m s-1
(b) Required distance s = u t + ½ a t 2 = 4 (2) + ½ (2) (2)2 = 12 m
(c) In this case, the force is acting to the left, so the acceleration is also directed to the left, in other words, the
acceleration is in opposite direction to the object's initial velocity, therefore the object will move slower and
slower with deceleration a given by
a = F / m = 10 N / 5 kg = 2 m s-1
To find the speed of the object after 2 s, again, we use the equation of motion v = u + at but note that in this
case, we have two directions in one equation : u is directed to the right but a is directed to the left. Remember that
the direction of initial velocity is always taken as positive direction, so a must be negative.
required speed v
= u + a t = 4 + (-2) (2) = 0 m s-1
(d) Required distance s = u t + ½ a t 2 = 4 (2) + ½ (-2) (2)2 = 4 m
Frictional force has a maximum value, called limiting friction. When a force greater than the limiting friction is
applied to the body, the body will accelerate. But when the force applied is less than the limiting friction, there is
simply no motion.
Inertia and mass
Inertia is the resistance of an object to a change in its state of rest or uniform motion in a straight line. It is obvious
that the greater the mass, the greater is the inertia. That means an object of greater mass is much more difficult to
start or stop moving than a smaller mass.
Mass and Weight
The mass of an object can be defined as the 'amount of matter' in it. The mass of an object does not vary from place
to place.
The weight of an object is defined as the gravitational attraction on the object by the earth (or other planets). It is
produced by gravity ( 重力 ) and since it is a force, it is a vector and measured in newton. In the absence of air
resistance, the uniform acceleration of a freely-falling body is produced by the force of gravity only and its value is
g = 10 m s-2. So
The weight of an object of mass m = m a = mg where g = 10 m s-2.
In symbols,
W = mg
The weight of a man of 60 kg = mg = 60 x 10 = 600 N
The value of g varies slightly from place to place on the earth's surface and is totally different on the moon ( g is
smaller on the moon ) and other planets.
Newton's third law of motion states that
For every action force, there is an equal and opposite reaction force.
對每一個作用力而言，都必然存在一個大小相同，方向相反的反作用力。
Alternatively, Newton's third law can also be expressed as follows:
If a body A exerts a force (action) on body B, then B exerts an equal and opposite force (reaction) on A.
若物體A有一力作用於物體B，則B有一力作用於A。這兩個力(作用力與反作用力)大小相同，方向相反。
Note the following :
 It does not matter which force is action and which is reaction. The important point is that action and reaction
exist simultaneously and they are equal in magnitude but opposite in direction.
 Another important point is that action and reaction act on two different bodies (in most cases).
 This law is most confusing. Try to remember the law and apply it to the real situations.
Try to identify the action-reaction pairs in the following case.
A book resting on a table
Vector addition of forces
An object may subject to several forces. Force is a vector because its direction is also very important and has to be
mentioned if we want to completely describe a force. The resultant (or vector sum) of several forces can be found
by 'tip-to-tail' method or parallelogram method as in the case of displacement.
Example 6
Two objects A and B of mass 2 kg and 4 kg respectively are held by two light strings as shown. Find the tensions
in both strings.
A, 2 kg
B, 4 kg
Note that if the string is light, the tension throughout the string is the same everywhere.
In secondary level, the strings used are usually light.
9.10
Example 7
A light inextensible string, passing over a smooth light pulley, has two masses 3 kg and 2 kg attached to its ends.
The masses are now released from rest.
(a) Find the acceleration of the system.
(b) Find the tension in the string.
Solution :
(a) Since the string is light and the pulley is smooth, the tension is the same
throughout the string, say T. The string is inextensible, so the two masses
move with equal acceleration (in magnitude), say a. It is obvious that the 3
kg mass moves downward and the 2 kg mass move upward.
3 kg
Applying Newton's 2nd law to the 3 kg mass, we have
2 kg
Applying Newton's 2nd law to the 2 kg mass, we have
Example 8
Two wooden blocks of mass 6 kg and 4 kg are placed together on a smooth horizontal table. The blocks are
pushed by a horizontal force of 50 N. Find the acceleration of the blocks and the reaction between them.
Solution :
Let a be the acceleration of the blocks
and F be the force between them.
6 kg 4 kg
50 N
6 kg
4 kg
9.11
Example 9
Two trolleys of mass 1 kg and 3 kg are tied together on a horizontal surface with a long string as shown. Assume
that the frictional force is 1 N. When a horizontal force of 10 N acts on the 3 kg mass,
(a) find the acceleration of the 3 kg trolley,
(b) find the tension of the string connecting
10 N
the two trolleys,
1 kg
Example 10
3 kg
Suppose a man of mass 50 kg is inside a lift. The man is standing on a weighing machine (a balance, say). The lift
is initially at rest at the ground floor. Find his apparent weight (i.e. the reading of the balance) in the following four
cases :
(1) The lift starts to move upwards with acceleration 2 m s-2.
(2) The lift is about to reach the destination, say the 6th floor, i.e. it is moving upwards with deceleration 2 m s-2.
(3) The lift starts to move downwards with acceleration 2 m s-2.
(4) The lift is about to reach the ground floor, i.e. it is moving downwards with deceleration 2 m s -2.
Case (1)
Velocity is upward, acceleration is upward (since the lift and the man are accelerating), so net force
must be upward (recall : force produces acceleration), hence R > mg.
Case (2)
Velocity is upward, acceleration is downward (since the lift and the man are decelerating), so net
force must be downward, hence R < mg.
Case (3)
Case (4)
9.12
Velocity is downward, acceleration is downward (since the lift and the man are accelerating), so net
force must be downward, hence R < mg.
Velocity is downward, acceleration is upward (since the lift and the man are decelerating), so net force
must be upward, hence R > mg.
Example 11
In the above problem, again find the apparent weight if
(a) the lift is moving at a constant velocity 3 m s-1.
(b) the cable suddenly breaks, and the lift falls freely.
9.13
9.5
Resolution of forces
Two forces can be replaced by a single resultant force which is the vector sum of these forces. The resultant force
produces the same effect as the two forces. Conversely, it is possible to replace a single force with two forces.
These forces are called the components ( 分力 ) of the original force. The process of splitting a force into its
components is called resolution of force ( 力的分解 ).
In physics problems, we always resolve a force into two components which are perpendicular to each other.
As an example, the single force F is resolved into two components, one in x-direction and the other in y-direction.
y
From geometry,
x-component of F = Fx = F cos 
y-component of F = Fy = F sin 
Fy
F
0
Fx

x
Note the following :
1. The components are smaller in magnitude than the original force .
2. The resultant of the components must, of course, be the original force.
Example 12
A picture of mass m is hung by two strings as shown. In which case is the tension the greatest ?
(1)
(2)
(3)
Example 13
A pendulum has a bob of mass 0.5 kg. It is hung in a bus which is accelerating forwards at 2 m s-2.
(a) Draw a force diagram for the bob, showing all the forces acting on it.
(b) Find the inclination of the string.
(c) Find the tension in the string.
2 m s-2
9.14
When motion along an inclined plane is considered, components parallel to and perpendicular to the incline should
be resolved.
As an example, a block of mass m is on an inclined plane which makes an angle  with the horizontal. After
understanding, try to memorize the following because you will frequently use it.
Component of weight down the incline = m g sin 
Example 14
(a) Find the acceleration of a block of mass 2 kg placed on an inclined
plane of slope 30o. The friction along the plane is assumed to be 2
N.
(b) What happens if the slope is decreased to 5 o ?
9.6
Pressure
The effect of a force depends on the area over which the force acts. High-heel shoes can be rather uncomfortable
because the whole weight of the body is acting over the very small area of the shoes. Platform shoes are more
comfortable because the weight of the body is supported on a much larger area. As mentioned in chapter 6,
Pressure =
Force acting at right angle to an area
area
The unit of pressure is called pascal, abbreviated as 'Pa'.
1 Pa = 1 N / m2
Example 15
A block measuring 0.1 m x 0.2 m x 0.8 m has a mass of 20 kg. What is the maximum and minimum pressure it
can exert on the ground ?
maximum pressure results when the contact area is the smallest.
9.15
Example 16
Cube A exerts a pressure P on the table. What is the pressure exerted by another cube B made of the same material
but with the length of all the sides halved ?
cube A
cube B
Note the following :
 Nails can be driven into a concrete wall because of their sharp points.
 Heavy animals have thick legs to reduce the pressure on the ground.
 Every man can sleep on a bed of nails without getting hurt. The reason is that the total area of contact between
the back and all the nails is approximately the same as when lying on a hard surface, the pressure on the body
is greatly reduced. But one has to be very careful when getting onto and off a bed of nails.
9.16
科學家介紹
牛頓(Issac Newton, 1642-1727)
牛頓在伽里略逝世那年出生於英格蘭一個農民家裏。小時
上學成績一
般，但愛制作機械模型，而且對問題愛尋根究底。1661年
18 歲 時 考 入
劍橋大學"三一"學院。學習踏實認真，三年後被選為優等
生 ， 1665 年
畢業後留校研究。這年六月，劍橋因瘟疫的威脅而停課，
他回家後一
連住了20個月。這20個月的清靜生活使他對在校所研究的
問題有了充
分的思考時間，因而成了他一生中創造力最旺盛的時期。他一生中最重要的科學和數學發
現，如微積分、萬有引力定律、光的色散等在這一時期都已基本上孕育成熟。
1667年牛頓回到劍橋，翌年獲得碩士學位。1669年開始當數學教授，時年26歲。此後在
力學方面的深入研究使他在1687年出版了物理科學中極其偉大的科學著作<<自然哲學的
數學原理>>(Philosophiae Naturalis Principia Mathematica)。在這著作中，
他把伽里略和其他物理學家的一些發現總結起來，並把力作一明確定義，將運動的規律以
數學形式，很有系統地、有條理地和很有邏輯地表達出來。
牛頓在這著作中還提出了萬有引力定律。他用自己發明的微積分(一種極之有用的數學理
論)解釋了開普勒(Kepler)的行星橢圓軌道定律，從而圓滿地解決了行星的運動問題。
至今，用牛頓的運動定律仍可準確預測行星的運動。牛頓定律的出現是人類認識自然界的
一次非常重大的突破。
牛頓運動三大定律形成了經典力學(Classical mechanics)，在以後的二百多年裏幾
乎統領了物理學的各個領域。在實際應用上，牛頓定律仍是許多工程技術，例如機械、土
建、動力等的理論基礎。
牛 頓 在 他 著 作 中 亦 寫 下 了 他 對 絕 對 空 間 (absolute space) 和 絕 對 時 間 (absolute
time)的概念。然而，他這方面的概念並不正確。原來時空是有關連的。現在我們知道，
牛頓定律在低速(指速度遠低於光速)和質量不太小(指質量不接近微觀粒子，如電子)時
是符合實驗結果的。但在高速的情況下，則要使用愛恩思坦(Albert Einstein)的相對
論(relativity)，而在低質量的情況下則必須用量子力學(Quantum mechanics)來處
理。
1689年和1701年牛頓兩次以劍橋大學代表的身份被選入議會。1696年被任命為皇家造幣
廠監督。後更被任命為皇家造幣廠廠長。1705年被女王授予爵士爵位。他終生未婚，晚
年由侄女照顧。1727年3月20日病逝，享年85歲。在他一生的後二十多年裏，他轉而研
究神學，並且寫了一些有關基督教的書籍，是一位十分敬虔的基督徒。
牛頓臨終前的一段遺言，充份表現出這位偉大科學家的謙遜和對追求真理的熱切。他這樣
說 ：「 我 好 像 是 海 濱 上 玩 耍 的 孩 子 ， 時 而 拾 到 幾 塊 瑩 潔 的 石 子 ， 時 而 拾 到 幾 片 美 麗 的 貝
殼，並為之歡欣。那浩瀚的真理海洋仍展現在面前。」