Full chapter
Dynamics
All 4 topics and the revision summary on one page.
01
Forces on one body
Make a guess. You are not marked.
A car goes round a bend at a constant speed. Is there a resultant force on it?
Show the answer
Yes, towards the inside of the bend, because its direction is changing.
A change of direction is a change of velocity, which needs a resultant force. Friction from the road on the tyres provides it, pointing towards the inside of the bend.
Draw the forces on the chosen body, then combine them to find the resultant force.
A force comes from an interaction, such as a string pulling or a surface pushing. Review force types and weight if you need help naming the forces.
A free-body diagram shows the forces acting on one selected body. Represent the body with a dot or a simple outline. Draw labelled arrows from it: the arrowhead gives direction, and the length can represent magnitude using a chosen scale.
- Choose the body: for example, the book, rather than the book and table together.
- Identify its interactions: Earth attracts the book, and the table pushes on it.
- Draw only forces on that body: the book's force on the table belongs on a different diagram.
- Check the directions and labels: weight is downwards; the normal contact force is perpendicular to the table.
Forces on a stationary book
Selected body: book
These forces balance on the same body.
The resultant is the single force equal to the vector sum of all the forces on the body. Along one line, forces in the same direction add; forces in opposite directions subtract. Keep their directions in the answer.
Balanced forces do not require the object to be at rest
Forces are balanced when their resultant is zero. The body's acceleration is then zero, so its velocity stays constant. A body already at rest remains at rest; a moving body continues at the same speed in the same direction.
This is Newton's first law: a body stays at rest, or keeps moving at constant velocity, unless a resultant force acts on it. Your syllabus does not ask you to state Newton's laws, but school mark schemes often name them.
Balanced forces on a moving trolley
Selected body: 2.0 kg trolley
The trolley is moving right along a horizontal track.
Zero resultant: constant velocity. The trolley need not be at rest.
The trolley has forces acting on it, but their resultant is zero. The string's forward pull balances the resistance. If resistance were absent, a forward pull would no longer be needed to maintain the same velocity.
An unbalanced force changes velocity
A non-zero resultant force produces acceleration in its direction. Since velocity includes direction, several changes are possible:
- An object at rest can start moving.
- A resultant in the direction of motion can increase its speed.
- A resultant opposite to its motion can reduce its speed.
- A resultant can turn its motion. An object following a curved path has changing velocity even if its speed stays constant.
For example, increasing the trolley's string pull to 7 N while resistance remains 3 N gives a resultant of 4 N forwards. Its velocity now changes. We calculate how quickly it changes in resultant force and acceleration.
Do not add a "force of motion". Motion is what the body does; each force must come from an interaction. If a separate velocity arrow helps, label it as velocity and keep it distinct from the force diagram.
The trolley's support equals its weight here because the pull is horizontal and there is no vertical acceleration. Do not assume support always equals weight when other vertical forces or vertical acceleration are present.
Check your understanding
A trolley travels along a straight level track at a constant 2 m/s. What must be true about the forces on it?
Show the answer
The resultant force is zero; individual forces may still act.
Constant speed in a fixed direction means constant velocity and zero acceleration. The forces balance. Weight and support may act, and any forward pull may balance resistance.
02
Pairs of forces
Make a guess. You are not marked.
A rocket engine pushes exhaust gases out backwards. What pushes the rocket forwards?
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The gases push forwards on the rocket, the third-law partner force.
The rocket pushes the gases backwards, so the gases push the rocket forwards with an equal force. This works in space, where there is no air.
When two bodies interact, each exerts a force on the other. The two forces have equal magnitudes and opposite directions.
A free-body diagram contains forces on one body. An interaction pair spans two bodies, so its members belong on different body diagrams.
This is Newton's third law: if body A exerts a force on body B, then B exerts a force on A that is equal in magnitude and opposite in direction.
These are often called action-reaction pairs. The forces occur together; "reaction" does not mean a later response. Identify the pair by naming both bodies and then reversing their roles.
One contact interaction, two different bodies. Only the paired contact forces are shown.
The table pushes the book upwards
Selected body: book
The book pushes the table downwards
Selected body: table
Balanced forces and interaction pairs answer different questions
| Two forces that balance on the book | A contact interaction pair |
|---|---|
| Table on book, upwards. Earth on book, downwards. | Table on book, upwards. Book on table, downwards. |
| Both act on the same body. Their resultant on that body is zero. | They act on different bodies. They belong to the same book-table interaction. |
The table's support and the book's weight can be equal and opposite, but that alone does not make them an interaction pair. One comes from contact with the table; the other comes from gravitational attraction by Earth.
The partner of Earth's gravitational force on the book is the book's gravitational force on Earth. These also have equal magnitudes and opposite directions. Earth's very large mass means its resulting acceleration from this interaction is extremely small.
Worked explanation
A hand pushing a wall
A hand pushes horizontally on a wall with a force of 12 N. The wall exerts a 12 N force on the hand in the opposite direction.
On a diagram of the hand, include the wall's force on the hand. The hand's force on the wall acts on the other body, so it does not belong on that diagram. The pair does not cancel within the hand's force diagram.
Equal forces do not require equal accelerations of the two bodies. Their masses and the other forces acting on each body matter.
Name the interaction, not just the arrow directions. "Force on the book by the table" pairs with "force on the table by the book". Swapping the bodies keeps you from confusing this pair with the book's weight.
Check your understanding
A table pushes upwards on a book resting on it. Which force is the other member of this interaction pair?
Show the answer
The book pushing downwards on the table.
This is the other force in the same book-table contact interaction. The forces have equal magnitudes and opposite directions, and act on different bodies.
03
Resultant force and acceleration
Make a guess. You are not marked.
A box is pushed along a rough floor and moves at a constant velocity of 2 m/s. What is the resultant force on the box?
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Zero, because the push and friction are balanced.
Constant velocity means zero acceleration, and F = ma gives zero resultant force. The push is exactly balanced by friction, so the forces cancel.
The resultant force determines the acceleration. Calculate it from all the forces on the body before using F = ma.
Acceleration is the change in velocity per unit time. Use a free-body diagram to find the resultant and choose a positive direction.
Fresultant = maFor a body of constant mass: force in N, mass in kg and acceleration in m/s2. Acceleration is in the direction of the resultant force.
This is Newton's second law. A mark scheme may write "by Newton's second law, F = ma".
A resultant force of 1 N gives a 1 kg mass an acceleration of 1 m/s2. The equation concerns acceleration, not the velocity at that instant. An object may be moving quickly while its resultant force and acceleration are both zero.
Unbalanced forces on a moving trolley
Selected body: 2.0 kg trolley
The trolley is moving right along a horizontal track.
Horizontal resultant: 7 - 3 = 4 N right. The vertical forces balance.
Worked example
Subtract the opposing force first
The trolley shown is moving forwards. Find its acceleration. Take forwards as positive.
- Find the horizontal resultant: Fresultant = 7 - 3 = +4 N. The forces oppose each other, so subtract their magnitudes.
- Rearrange: a = Fresultant / m.
- Substitute: a = 4 / 2.0 = 2.0 m/s2 forwards.
The balanced vertical forces give no vertical acceleration. Using 7 / 2.0 would treat the pull as the entire resultant and ignore the track's resistance.
Try it step by step
Work it out in steps: Resultant force in N, then Acceleration a = F / m in m/s2.
Show the answer
Problem 1.
Resultant force in N: F = 20 - 8.0 = 12 N forwards
Acceleration a = F / m in m/s2: a = 12 / 4.0 = 3.0 m/s2 (2 sf)
Problem 2.
Resultant force in N: F = 50 - 20 = 30 N forwards
Acceleration a = F / m in m/s2: a = 30 / 12 = 2.5 m/s2 (2 sf)
Problem 3.
Resultant force in N: F = 9.0 - 3.0 = 6.0 N forwards
Acceleration a = F / m in m/s2: a = 6.0 / 1.5 = 4.0 m/s2 (2 sf)
A forward-moving object can accelerate backwards
In a different interval, suppose the trolley is still moving forwards but its string pull is 3 N and track resistance is 7 N. Keeping forwards positive gives:
a = -4 / 2.0 = -2.0 m/s2The acceleration is backwards while the velocity is forwards, so the trolley slows during this interval.
The negative sign specifies direction. It does not mean the trolley is already moving backwards. If the trolley reaches a stop, reassess the forces before predicting what happens next; the same resistance direction cannot simply be assumed throughout a reversal.
How mass affects acceleration
With the same resultant force, a larger mass has a smaller acceleration. A 4 N resultant gives a 2 kg trolley an acceleration of 2 m/s2, but a 4 kg trolley an acceleration of 1 m/s2. Doubling mass halves acceleration when the resultant is unchanged.
Investigating the relationship
The following idealised data describe a trolley of fixed mass 0.50 kg. The forces listed are resultant forces, after accounting for resistance.
| Resultant force / N | Acceleration / (m/s2) |
|---|---|
| 0.20 | 0.40 |
| 0.40 | 0.80 |
| 0.60 | 1.20 |
Doubling the resultant from 0.20 to 0.40 N doubles acceleration from 0.40 to 0.80 m/s2. The ratio Fresultant / a is 0.50 kg in every row, consistent with the fixed mass. An acceleration-against-resultant-force graph would be a straight line through the origin for this model.
For a real investigation, keep the trolley's total mass unchanged and use a level track. Change the horizontal pull, estimate acceleration from motion readings, and account for resistance when finding the resultant. A motion sensor or timed velocity readings can provide the changes needed for a = (v - u) / t.
Repeat readings to judge variation. A slightly sloping track adds a component of weight along the track, while changing resistance can make a measured pull different from the assumed resultant. Address those causes instead of calling every disagreement "human error". Real data need not fall exactly on the model values.
Watch out for this
A student says: "A ball thrown upwards keeps rising because the force of the throw is still acting on it." What is wrong with this?
Show the answer
The throwing force stops the moment the ball leaves the hand. The ball keeps rising because it already has an upward velocity, and its weight, now the resultant force, slows it down.
Check your understanding
A 4.0 kg trolley has a horizontal pull of 14 N to the right and a resistance of 6 N to the left. Its vertical forces balance. What is its acceleration?
Show the answer
2.0 m/s2 to the right.
The horizontal resultant is 14 - 6 = 8 N to the right. Using resultant force = mass x acceleration gives a = 8 / 4.0 = 2.0 m/s2 to the right.
04
Friction and motion
Make a guess. You are not marked.
A heavy box rests on a rough floor. You push it gently sideways and it does not move. What is the friction force on it?
Show the answer
Equal and opposite to your push, so the resultant is zero.
The box stays at rest, so the resultant force is zero. Friction matches your push exactly, up to the largest friction the surfaces can give.
Friction acts between surfaces in contact. It opposes their relative sliding, or their tendency to slide.
Always identify the body whose force you are describing. The direction of friction must follow the particular contact, just like any other force on a free-body diagram.
Friction can slow a sliding object
A box slides to the right across a stationary floor. Its lower surface slides right relative to the floor, so friction from the floor on the box acts left. If this is the only horizontal force, the resultant is leftwards and the box slows.
Friction on a box sliding right
Selected body: box
The box slides right over a stationary horizontal floor.
Friction opposes the relative sliding at this contact. Arrow lengths are schematic.
If you pull the box steadily across the same floor, the pull can balance friction. The box then moves at constant velocity even though both horizontal forces are present.
Friction can prevent sliding
A gentle horizontal push does not always make a box move. Static friction can balance the push while the surfaces remain at rest relative to each other. For example, if a 2 N push is balanced by 2 N of friction, the horizontal resultant is zero.
Static friction adjusts to the situation up to a limit. Increasing the push may increase the balancing friction while the box remains still; it does not mean friction always has one fixed magnitude. If the available friction cannot balance the push, the box starts sliding.
Friction can help you move forwards
As you push off to start walking forwards, your planted foot pushes backwards on the ground. The foot would tend to slip backwards without enough grip. Friction from the ground on the foot acts forwards, helping accelerate you forwards.
Friction helps a person start walking
Selected body: planted foot
When pushing off, the foot tends to slip backwards (left) over the ground.
Ground-on-foot friction acts forwards. Only this contact force is shown.
This is why "friction always opposes the object's motion" is too broad. In the sliding-box case it acts backwards; in this walking case it acts forwards. In both, the direction follows the relative sliding or tendency to slide at the contact.
Grip, braking and resistance
Friction provides grip between a tyre and the road. Friction between brake pads and a rotating wheel or brake disc helps slow the rotation. It can also cause unwanted wear and resistance in moving machine parts. Reducing friction can make some movements easier, while reducing grip where it is needed can make slipping more likely.
Air resistance is a resistive force from the surrounding air. Its direction opposes motion relative to the air. An object falling downwards through still air experiences upward air resistance.
Optional explanation: a box stays still under a 3 N horizontal push
Assume the floor's friction is the only other horizontal force. What is the friction on the box, and why?
The box remains at rest, so horizontal acceleration and resultant force are zero. Friction is therefore 3 N opposite to the push. There is no horizontal motion, but there is still a force of friction.
Revision summary
Choose the body, identify its interactions and find the resultant before predicting its motion.
- Weight and gravitational field
- W = mg. Weight W in N; mass m in kg; gravitational field strength g in N/kg. Near Earth, g is approximately 10 N/kg. Use the value given.
- Resultant force and acceleration
- Fresultant = ma, so a = Fresultant / m. Acceleration is in the resultant's direction. Use all the relevant forces, not just the applied pull.
From forces to motion
- Zero resultant: zero acceleration. The body stays at rest or continues at constant velocity.
- Non-zero resultant: velocity changes. The body can start, speed up, slow down or change direction.
- A fast-moving object need not have a large resultant force. A slowing object still has acceleration.
- Draw only forces on the chosen body. Identify the interaction behind each arrow; motion is not an extra force.
Newton's three laws
- First law: with no resultant force, a body stays at rest or keeps moving at constant velocity.
- Second law: a resultant force causes an acceleration in its direction, Fresultant = ma.
- Third law: if A exerts a force on B, B exerts an equal and opposite force on A.
Balanced forces or an interaction pair?
| Balanced forces | Action-reaction pair |
|---|---|
| Act on the same body and sum to zero. | Act on different bodies in the same interaction. |
| Example: table on book upwards and Earth on book downwards. | Example: table on book upwards and book on table downwards. |
Mass, weight and force types
Mass is a measure of matter and is measured in kg; weight is gravitational force and is measured in N. An unchanged object's mass stays the same when its gravitational field changes. For the same resultant force, a larger mass has a smaller acceleration.
Contact examples: normal force, tension, friction and air resistance. Non-contact examples: gravitational, electrostatic and magnetic forces. Normal force is perpendicular to the contact surface; tension pulls along a string.
Friction direction
Identify the contact and the relative sliding or tendency to slide. Friction can slow a sliding box, prevent a stationary box from sliding, or push a walking person forwards during push-off.
Check a practical explanation
State which mass or force you measured and its unit. For a force-motion comparison, keep total mass controlled, account for resistance and check that a slope has not introduced an extra force along the track. Name a limitation and explain its effect.
Back to forces on one bodyPractise exam questions
Write your answer first, then tick the mark points it makes.
Exam-style question
A car of mass 1200 kg moves along a straight, level road. The driving force on it is 3000 N and the total resistive force is 600 N. Calculate the acceleration of the car. [3]
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Exam-style question
A book rests on a table. The Earth pulls down on the book with a gravitational force of 4.0 N. Newton's third law says this force forms a pair with another force. State the size and direction of that other force, and the body it acts on. [3]
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Exam-style question
A box is pulled along a rough floor at a constant speed by a horizontal rope. Name the four forces acting on the box and state their directions. State the resultant force on the box and explain your answer. [3]
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Exam-style question
Explain why a car travelling at the same speed needs a longer distance to stop on a wet road than on a dry road. [3]
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Quick recall
A trolley travels along a straight level track at a constant 2 m/s. What must be true about the forces on it?