Having described motion in the Kinematics workbook, we now turn to what causes it. A force is a push or a pull between two objects, and every force is the result of an interaction. In this workbook you will learn Newton's three laws of motion, how to draw and analyse free-body diagrams, how to find the resultant of several forces, and the nature of the main contact and field forces you will meet throughout IB Mechanics.
By the end of this workbook you should be able to:
explain that a force is an interaction between two bodies, and state Newton's three laws of motion
draw and interpret free-body diagrams for objects in a range of situations
analyse a free-body diagram to find the resultant force acting on a system, including resolving forces into components
explain the condition for equilibrium and apply it to simple systems
describe the nature and use of the contact forces: normal force, friction, viscous drag and buoyancy
describe the nature and use of the field forces: gravitational, electric and magnetic force
1. Forces as interactions
A force is a push or a pull that one object exerts on another. Forces are measured in newtons (N), and — like displacement, velocity and acceleration — a force is a vector: it has both a size and a direction.
Key idea. A force never exists on its own. It is always the result of an interaction between two objects, so whenever you identify a force you should be able to say which object it acts on, and which object is causing it.
A force can change an object's velocity (speed it up, slow it down, or change its direction), or change its shape. Forces are usually grouped into two families, both covered later in this workbook: contact forces, which need the two objects to be touching (such as friction or the normal force), and field forces, which act between objects even when they are not touching (such as gravity).
Weight and mass
Mass is a measure of the amount of matter in an object; it is a scalar, measured in kilograms, and does not change if the object is moved somewhere else. Weight is the force of gravity acting on an object's mass. It is a vector, measured in newtons, and depends on the local gravitational field strength, g.
weight, Fg = mg (SI unit: N)
Gravitational field strength, g: the force of gravity acting on each kilogram of mass at a particular location, measured in N kg−1. Near the Earth's surface, g = 9.8 N kg−1, which has exactly the same numerical value as the acceleration of free fall, 9.8 m s−2.
Worked example 1.1
An astronaut has a mass of 85 kg. Calculate her weight (a) on the Earth's surface, where g = 9.8 N kg−1, and (b) on the Moon's surface, where g = 1.6 N kg−1.
Answer:
(a) Fg = mg = 85 × 9.8
(b) Fg = mg = 85 × 1.6
Notice that her mass, 85 kg, is exactly the same in both parts — only her weight changes, because g is different on the Moon.
Check your understanding
1Explain the difference between mass and weight, making clear which is a vector and which is a scalar.
Mass is a scalar measuring the amount of matter in an object, in kilograms; it stays the same wherever the object is. Weight is a vector — the force of gravity acting on that mass, Fg = mg, measured in newtons; it depends on the local gravitational field strength, so the same object has a different weight on the Moon or on Mars than it does on Earth, even though its mass is unchanged.
2A rover has a mass of 190 kg. Calculate its weight on the surface of Mars, where the gravitational field strength is 3.7 N kg−1.
Fg = mg = 190 × 3.7
3Explain what would happen to an astronaut's weight, and to her mass, if she travelled to a point in deep space, far from every planet and star.
Her mass would stay exactly the same, since mass does not depend on location. Her weight, however, would fall to zero, because far from any planet or star the gravitational field strength is essentially zero, and Fg = mg = m × 0 = 0. She would still have exactly the same amount of matter (mass) as before, but gravity would no longer be pulling on it.
2. Newton's three laws of motion
Isaac Newton summarised the relationship between force and motion in three laws. Together they explain almost everything about how and why objects move the way they do.
Newton's first law
Newton's first law. An object stays at rest, or moving with constant velocity, unless a resultant (non-zero) force acts on it.
An object obeying this law is said to be in equilibrium. This includes an object at rest (a book resting on a table) and an object moving at a constant velocity in a straight line (a car cruising at a steady speed, where the forward driving force exactly balances the resistive forces acting backwards).
Newton's second law
Newton's second law. The resultant force acting on an object is equal to its mass multiplied by its acceleration.
resultant force, F = ma (SI unit: N = kg m s−2)
Newton (unit): one newton is defined as the resultant force needed to give a mass of 1 kg an acceleration of 1 m s−2.
Worked example 2.1
A car of mass 1450 kg has a resultant forward force of 3800 N acting on it, starting from rest. Calculate (a) its acceleration, and (b) the distance it covers in the first 4.0 s (use the SUVAT equations from the Kinematics workbook).
Answer:
(a) F = ma, rearranged: a = F ÷ m = 3800 ÷ 1450
(b) s = ut + ½at², with u = 0: s = ½ × a × 4.0², using your value of a from part (a)
Newton's third law
Newton's third law. Whenever object A exerts a force on object B, object B exerts an equal and opposite force on object A.
These two forces are called a third-law pair. A third-law pair always: (i) involves the same type of force, (ii) acts along the same line, (iii) is equal in size and opposite in direction, and — most importantly — (iv) acts on two different objects, never on the same object.
Fig. 2.1 A common mistake is to treat the weight and normal force on the same book (left) as a third-law pair. They are not — they act on the same object. The true third-law pair (right) is "book pushes down on table" and "table pushes up on book": the same interaction, viewed from each of the two objects involved.
Common mistakeWeight and normal force are usually not a third-law pair. They often look like one (equal size, opposite direction) but they act on the same object and are usually different types of force (one gravitational, one a contact force). A genuine third-law pair is only ever found by looking at what force A exerts on B, and what B exerts back on A.
Check your understanding
4A skydiver falls at a constant (terminal) velocity. Use Newton's first law to explain what this tells you about the forces acting on her.
Since her velocity is constant, Newton's first law tells us the resultant force acting on her must be zero — she is in equilibrium. This means her weight (acting downwards) must be exactly balanced by the air resistance acting on her (acting upwards), even though both forces are individually non-zero.
5A resultant force of 640 N acts on a shopping trolley of mass 40 kg. Calculate its acceleration.
F = ma, rearranged: a = F ÷ m = 640 ÷ 40
6A swimmer pushes backwards against the water with her arms and moves forwards through the pool. Use Newton's third law to explain why this happens.
The swimmer exerts a backward force on the water (swimmer on water). By Newton's third law, the water must exert an equal and opposite force on the swimmer — a forward force (water on swimmer). This third-law pair acts on two different objects (the water and the swimmer), so it is the force from the water on the swimmer that actually pushes her forwards through the pool.
7Using Fig. 2.1, explain why the weight of the book and the normal force from the table on the book are not a third-law pair, even though they are equal in size and opposite in direction.
Both the weight and the normal force act on the same object — the book. A genuine third-law pair must act on two different objects. The weight is also a gravitational force (Earth pulling on the book), whereas the normal force is a contact force (table pushing on the book) — different types of force. The actual third-law pair here is the book pushing down on the table, and the table pushing up on the book, which involves the same contact interaction seen from each of the two objects.
3. Free-body diagrams
A free-body diagram shows every force acting on a single object, drawn as arrows from a single point representing that object — usually treated as a "point particle" for simplicity. It shows only the forces acting on that one object; forces that object exerts on anything else are never included.
Key idea. To draw a free-body diagram: (1) identify the single object you are considering, (2) list every force acting on it (weight is almost always present; contact forces appear wherever it touches something; field forces appear if it sits in a field), (3) draw each force as an arrow from the object's point, roughly in proportion to its size, and label each one clearly.
Fig. 3.1 Free-body diagram of a block on a rough inclined plane, just before it starts to slide. Here the weight has already been resolved into two components: mg sinα acting down the slope and mg cosα acting into the surface (α is the same angle called θ elsewhere in this workbook). The normal force, N, acts perpendicular to the surface, and friction is shown at its maximum static value, μsN — you will meet this notation in Section 5. Diagram by P. Williamson (Wikimedia Commons).
Free-body diagrams work the same way for any object — a book on a table (weight and normal force only), a pendulum bob swinging on a string (weight and tension), the Moon orbiting the Earth (gravitational force only, since nothing is touching it), or a box being pulled at a constant speed across the floor (weight, normal force, tension in the pulling rope, and friction).
Check your understanding
8List the forces acting on a book resting on a horizontal table. State which of these are contact forces and which are field forces.
Two forces act on the book: its weight (a field force — the gravitational force of the Earth acting on it, no contact needed) and the normal force from the table (a contact force — it only exists because the book and table are touching).
9A skydiver is falling at her terminal (constant) velocity. List the forces acting on her and state how their sizes compare.
Two forces act on her: her weight, acting downwards, and air resistance (a form of viscous drag), acting upwards. Since she is moving at a constant velocity, she is in equilibrium (Newton's first law), so these two forces must be equal in size and opposite in direction — they cancel out to give a zero resultant force.
10Using Fig. 3.1, list the forces and force components shown, and state which are contact forces and which are field forces.
The normal force, N, and the friction force, μsN, are both contact forces, since they only exist because the block is touching the surface of the incline. The remaining two arrows, mg sinα and mg cosα, are not separate forces — they are the two components of a single field force, the block's weight (gravitational), resolved parallel and perpendicular to the slope.
4. Resultant force and equilibrium
When several forces act on an object at once, they combine to give a single overall force called the resultant force — the vector sum of all the individual forces. Because forces are vectors, they must be added using vector addition, not simple arithmetic, unless they happen to act along the same line.
Fig. 4.1 Adding forces A and B "tip-to-tail" gives the resultant force, R, drawn from the start of A to the end of B. Here R ≈ 62 N, at about 23° above the horizontal.
Key idea. An object is in equilibrium when the resultant force acting on it is zero — this is exactly the condition described by Newton's first law. Equilibrium does not mean no forces are acting; it means they all cancel out.
Forces that act at an angle are often easier to deal with by resolving them into two perpendicular components — typically horizontal and vertical, or (for an object on a slope) parallel and perpendicular to the slope. A force F at an angle θ to a chosen direction has components F cosθ along that direction and F sinθ perpendicular to it.
Worked example 4.1
A box of weight 250 N rests on a slope inclined at 30° to the horizontal (as in Fig. 3.1). Calculate the components of its weight (a) parallel to the slope, and (b) perpendicular to the slope.
Answer:
(a) component parallel to the slope = W sinθ = 250 × sin 30°
(b) component perpendicular to the slope = W cosθ = 250 × cos 30°
Live simulation: resultant force builder
Force A always points horizontally. Use the sliders to change the size of A and the size and angle of force B, and watch the resultant, R, update. Try the "show equilibrant" button to see the single extra force that would be needed to bring the object into equilibrium.
Resultant force builder
Resultant, R = 0.0 N
Fig. 4.2 Building a resultant force from two component forces, A and B.
Check your understanding
11Two forces act on a ring: 30 N pulling due north and 40 N pulling due east. Calculate the magnitude of the resultant force.
Since the two forces are at right angles, use Pythagoras' theorem: R = √(30² + 40²)
12A force of 18 N acts at 40° above the horizontal. Calculate its horizontal and vertical components.
horizontal component = F cosθ = 18 × cos 40°
vertical component = F sinθ = 18 × sin 40°
13In the simulation above, set force B to the same size as force A (for example, both 40 N) and its angle to 180°. Describe the resultant force, and state what this tells you about the object's motion.
With B equal in size to A but pointing in exactly the opposite direction, the resultant force falls to zero — the two forces cancel each other out. By Newton's first law, this means the object is in equilibrium: it stays at rest, or continues to move with whatever constant velocity it already had.
14A crate of weight 180 N sits on a slope inclined at 25° to the horizontal. Calculate the component of its weight acting parallel to the slope (this is the component that friction must balance to keep the crate in equilibrium).
component parallel to the slope = W sinθ = 180 × sin 25°
5. Contact forces I: normal force and friction
Contact forces only exist where two objects are touching. This section covers the normal force and friction — two contact forces you will meet in almost every mechanics problem.
Normal force
Normal force, N: the component of the contact force between two surfaces that acts perpendicular to those surfaces. It adjusts automatically to prevent the two surfaces passing through each other.
The word "normal" here means perpendicular, not "usual" — the normal force is always at right angles to the surface, whatever angle that surface happens to be at.
Friction
Friction is the component of the contact force that acts parallel to the surfaces, opposing relative sliding motion (or the tendency to slide) between them. There are two cases:
static friction: f ≤ μsN dynamic (kinetic) friction: f = μdN
Key idea. While an object stays still, static friction adjusts itself to exactly match whatever force is trying to move it, up to a maximum value of μsN. Once the applied force exceeds this maximum, the object starts to slide, and dynamic friction takes over — it stays roughly constant, at μdN, and is normally somewhat smaller than the maximum static value.
Fig. 5.1 As the applied force increases, static friction rises to match it exactly, up to a maximum of μsN. Beyond this point the object slides and friction drops to the roughly constant dynamic value, μdN.
Worth rememberingFriction does not depend on surface area. The coefficient of friction (μs or μd) depends only on the nature of the two surfaces in contact — how rough or smooth they are — and not on how large the contact area is.
Surfaces
Typical μs
rubber tyre on dry road
0.8
steel on steel
0.8
glass on metal
0.6
waxed wood on wet snow
0.1
Worked example 5.1
A football of mass 1.5 kg sits on grass, where μs = 0.7 and μd = 0.3. A player kicks it, giving a horizontal force of 16 N for a brief instant. Show that the ball starts to slide, and calculate its acceleration once it does.
Answer:
normal force, N = weight = mg = 1.5 × 9.8
maximum static friction, fmax = μsN = 0.7 × (1.5 × 9.8)
since the applied force (16 N) is greater than fmax, the ball starts to slide, so friction becomes dynamic: f = μdN = 0.3 × (1.5 × 9.8)
resultant force = applied force − f
acceleration, a = resultant force ÷ mass
Live simulation: friction explorer
Drag the applied force slider from zero upwards and watch how the friction force responds. Notice how friction matches the applied force exactly while the block stays still, then settles to a lower, constant value once it starts to slide.
Friction explorer
μs = 0.6, μd = 0.4 state: static
Fig. 5.2 A block on a rough surface. Friction self-adjusts to match the applied force while the block is static, up to a maximum of μsN; once it slides, friction drops to μdN and the block accelerates.
Check your understanding
15Explain the difference between static and dynamic (kinetic) friction.
Static friction acts when there is no relative sliding between the two surfaces; it adjusts itself to exactly match the force trying to cause motion, up to a maximum value of μsN. Dynamic (kinetic) friction acts once the surfaces are actually sliding past each other; it has a roughly constant value of μdN, which is normally somewhat smaller than the maximum static value.
16A crate of mass 12 kg rests on a floor where μs = 0.45. Calculate the maximum static friction force before the crate begins to slide.
normal force, N = mg = 12 × 9.8
fmax = μsN = 0.45 × (12 × 9.8)
17Using Fig. 5.1, describe what happens to the friction force as the applied force is increased from zero, right up to the point where the object begins to slide, and beyond.
At first, the friction force rises to match the applied force exactly (f = F), keeping the object in equilibrium, up to a maximum value of μsN. Once the applied force exceeds this maximum, the object begins to slide, and the friction force drops to the roughly constant dynamic value, μdN, which stays approximately the same even as the applied force is increased further.
18Why will an object at rest, when a pushing force increases to just above the maximum static friction, always accelerate?
Once the pushing force rises just above the maximum static friction, μsN, the object starts to slide, so friction immediately drops to the lower dynamic value, μdN (since μd is normally smaller than μs). At that instant the pushing force is still just above μsN, which is itself above μdN, so the pushing force now exceeds the (now smaller) friction force by a clear margin. This leaves a non-zero resultant force acting on the object, so by Newton's second law it must accelerate — it can never simply move off at a new constant velocity, because the sudden drop in friction always leaves an unbalanced resultant at the moment sliding begins.
19A student says that a wide car tyre grips the road better than a narrow one because it has a larger contact area with the road. Explain why this reasoning is not correct in terms of the coefficient of friction.
The coefficient of friction depends only on the nature of the two surfaces in contact (how rough or smooth the tyre and road are), not on the size of the contact area. A wider tyre does not, by itself, increase the maximum friction force available, since fmax = μsN depends on the normal force and the coefficient of friction, not the contact area.
6. Contact forces II: drag and buoyancy
Viscous drag
Viscous drag is the resistive force a fluid (liquid or gas) exerts on an object moving through it, opposing the object's motion. For a small sphere moving slowly through a fluid, it is given by Stokes' law:
Fd = 6πηrv
where η is the fluid's viscosity (unit Pa s), r is the sphere's radius, and v is its speed. A more viscous fluid produces a larger drag force at the same speed.
Fluid
Viscosity, η / Pa s
heavy oil
0.7
light oil
0.1
water
1 × 10−3
air
1.8 × 10−5
Terminal velocity: as a falling object speeds up, drag increases (since drag depends on speed). Eventually drag becomes large enough to balance the object's weight (and any buoyancy force), so the resultant force falls to zero and the object continues at a constant, maximum speed — its terminal velocity.
Buoyancy
Any object in a fluid experiences an upward buoyancy force (upthrust), because the pressure of the fluid pushing up on the bottom of the object is greater than the pressure pushing down on its top. This is described by Archimedes' principle:
Fb = ρVg
where ρ is the density of the fluid, and V is the volume of fluid displaced by the object (which equals the object's submerged volume). In words: the buoyant force equals the weight of fluid displaced.
Fig. 6.1 A floating block: the buoyant force, Fb, equals the weight of water displaced by the submerged part of the block. The block floats in equilibrium when Fb = W.
Worked example 6.2
A block of wood has a volume of 34 cm3 (34 × 10−6 m3) and a mass of 29 g (0.029 kg). The density of water is 1000 kg m−3. Calculate (a) the block's weight, and (b) the maximum possible buoyant force on it (if it were fully submerged). Use your answers to explain whether the block floats or sinks.
Answer:
(a) W = mg = 0.029 × 9.8
(b) Fb(max) = ρVg = 1000 × (34 × 10−6) × 9.8
Since Fb(max) (part b) is greater than W (part a), the block can displace enough water to balance its weight before it is fully submerged, so it floats, sitting only partly under the water.
Explore how the density of an object compared with the density of the fluid it is placed in determines whether it floats or sinks, and how the buoyant force changes as more of the object becomes submerged.
Interactive simulation — open the online version of this workbook to launch it.
20An object of volume 60 cm3 (60 × 10−6 m3) is fully submerged in water of density 1000 kg m−3. Calculate the buoyant force acting on it.
Fb = ρVg = 1000 × (60 × 10−6) × 9.8
21Two identical steel ball bearings are dropped, one into light oil and one into heavy oil. Explain, in terms of viscous drag, why the ball reaches a lower terminal velocity in the heavy oil.
Heavy oil has a much larger viscosity, η, than light oil. Since the drag force Fd = 6πηrv is directly proportional to viscosity, the ball in heavy oil experiences a larger drag force at any given speed. This means the drag force builds up to balance the ball's weight at a lower speed in heavy oil than in light oil, so its terminal velocity — reached once drag balances weight — is lower.
7. Field forces
Unlike contact forces, field forces act between objects even when they are not touching — they act "at a distance", through a field that surrounds the object producing the force.
Key idea. Contact forces need the two objects to be physically touching (friction, normal force, tension, drag, buoyancy). Field forces do not — they act through a field that extends through space around the object producing it (gravitational, electric and magnetic force).
Gravitational force
Every object with mass creates a gravitational field around it, and exerts a gravitational (attractive) force on every other mass within that field. Near the Earth's surface, this is simply the weight of an object, Fg = mg, which you met in Section 1.
Electric force
Any charged object creates an electric field around it, which exerts a force on other charged objects within that field. Like charges repel each other; unlike charges attract. You will study the electric force in detail, including how to calculate it, later in the course.
Magnetic force
Magnets, and electric currents, create a magnetic field around them, which exerts a force on other magnets, magnetic materials, or moving charges within that field. Like the electric force, you will meet the details of the magnetic force in a later topic.
Contact forces (need touching)
Field forces (act at a distance)
normal force, friction, viscous drag, buoyancy
gravitational, electric, magnetic
Check your understanding
22Classify each of the following as a contact force or a field force: (a) tension in a rope, (b) the attraction between a charged balloon and a wall, (c) friction between a tyre and the road, (d) the pull between two bar magnets.
(a) tension — contact force. (b) attraction between a charged balloon and a wall — field force (electric). (c) friction — contact force. (d) pull between two magnets — field force (magnetic).
23Two students sitting near each other in class exert a gravitational force on one another, yet neither of them notices it. Explain why, comparing this force with the weight each student experiences due to the Earth.
Every mass does produce a gravitational field and attracts every other mass, so the students do exert a tiny gravitational force on each other. However, this force depends on the masses involved, and a person's mass is minute compared with the Earth's enormous mass. The gravitational force between the two students is therefore far too small to notice, while the gravitational force from the whole Earth (their weight) is large enough to feel clearly.
24State which field force is most likely responsible for each situation: (a) a compass needle turning to point near a wire carrying a current, (b) an apple falling from a tree.
(a) the magnetic force — a current-carrying wire produces a magnetic field, which exerts a force on the compass needle. (b) the gravitational force — the Earth's gravitational field pulls the apple downwards.
Glossary
Force
A push or a pull that one object exerts on another; a vector, measured in newtons (N), and always the result of an interaction between two bodies.
Mass and weight
Mass is the amount of matter in an object (a scalar, in kg, unchanged by location). Weight is the gravitational force acting on that mass, Fg = mg (a vector, in N), which does depend on location.
Gravitational field strength, g
The gravitational force acting on each kilogram of mass at a location, in N kg−1; near the Earth's surface, g = 9.8 N kg−1.
Equilibrium
The state of an object when the resultant force acting on it is zero, so (by Newton's first law) it stays at rest or moves with constant velocity.
Newton's third-law pair
Two forces of the same type, equal in size, opposite in direction, acting along the same line, on two different objects — never on the same object.
Free-body diagram
A diagram showing every force acting on a single object, drawn as labelled arrows from a point representing that object.
Resultant force
The single overall force equivalent to all the individual forces acting on an object, found by adding them as vectors.
Normal force
The component of the contact force between two surfaces that acts perpendicular to those surfaces.
Friction
The component of the contact force that opposes relative sliding between two surfaces. Static friction (f ≤ μsN) adjusts to match the applied force up to a maximum before sliding begins; dynamic friction (f = μdN) acts once sliding starts, and is roughly constant.
Viscous drag
The resistive force a fluid exerts on an object moving through it, opposing its motion; for a small sphere, given by Stokes' law, Fd = 6πηrv.
Terminal velocity
The constant, maximum speed reached by an object falling through a fluid, once the drag force (and any buoyancy) has increased enough to balance its weight.
Buoyancy
The upward force exerted by a fluid on an object within it, equal to the weight of fluid displaced (Archimedes' principle), Fb = ρVg.
Contact force and field force
A contact force (e.g. normal force, friction) only exists where two objects are touching. A field force (e.g. gravitational, electric, magnetic) acts between objects at a distance, through a field.
