Dynamics

01

Forces, mass and weight

A force is a push or pull arising from an interaction between objects. To understand a force, identify what it acts on and what exerts it.

A string pulls on a trolley. A table pushes on a book. Earth attracts both objects. Each description names two interacting bodies, even when only one of them is moving.

Contact and non-contact forces

A contact force involves objects or materials touching. Its direction depends on the interaction.

Normal contact force

A surface pushes on an object in a direction perpendicular to the surface. A table supports a book in this way.

Tension

A stretched string or rope pulls on the object attached to it, along the string and away from the object.

Friction

Surfaces in contact exert a force that opposes their relative sliding, or their tendency to slide.

Air resistance

Air exerts a force on an object moving through it. Air is matter, so this is a contact interaction even though we cannot see the air.

Non-contact forces act without the objects touching. Gravitational attraction acts between masses; electrostatic forces act between electric charges; magnetic forces act between magnets or between a magnet and a suitable magnetic material. A magnet can attract a steel paper clip across a gap.

Forces are vectors, so give both magnitude and direction. For example, 6 N upwards describes a force of magnitude 6 newtons acting upwards. The quantity force may be written F or f; its unit is the newton, N.

Mass describes the object; weight is a force

Mass

A measure of the amount of matter in a body. Mass may be written m or M. It is a scalar, measured in kilograms (kg).

Weight

The gravitational force acting on a body. Weight is a vector, measured in newtons (N). Near Earth it acts downwards, towards Earth's centre.

A gravitational field is a region in which a mass experiences a force due to gravitational attraction. The gravitational field strength, g, is the gravitational force per unit mass at that point.

Near Earth's surface, g is approximately 10 N/kg: each kilogram experiences about 10 N of gravitational force. This corresponds to the approximately 10 m/s² downward acceleration of free fall. Field strength is not 10 N/kg everywhere; use the value supplied in a question.

Measuring mass and weight

Use an electronic balance for a mass reading and a spring balance or newton-meter for a force reading. Choose a suitable range and check the zero. When measuring weight with a spring balance, hang the object freely and wait until it is stationary. The upward tension then has the same magnitude as the downward weight.

For example, an object gives a balance reading of 0.150 kg and a steady spring-balance reading of 1.5 N. These are different quantities. Their ratio gives g = 1.5 / 0.150 = 10 N/kg. Record the quantity and unit for each reading; do not call the force reading a mass.

For a small mass, the unit mg means milligram: 1 mg = 0.001 g = 0.000001 kg. In the equation W = mg, the letters instead mean the product m x g, mass multiplied by gravitational field strength. A unit after a mass reading and two quantities in an equation have different roles.

Convert grams to kilograms before using W = mg with g in N/kg. For example, 150 g = 0.150 kg. The prefix conversion changes the number and unit, not the amount of matter.

02

Forces on one body

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.

1. Choose the body: for example, the book, rather than the book and table together.
2. Identify its interactions: Earth attracts the book, and the table pushes on it.
3. Draw only forces on that body: the book's force on the table belongs on a different diagram.
4. Check the directions and labels: weight is downwards; the normal contact force is perpendicular to the table.

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.

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.

03

Pairs of forces

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.

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.

Balanced forces and interaction pairs answer different questions

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.

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.

04

Resultant force and acceleration

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.

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.

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:

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.

Mass and inertia

Inertia is resistance to a change in motion. Mass is the property of a body associated with this resistance. A larger mass has greater inertia.

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.

When a car brakes, a passenger tends to continue with the existing forward velocity. The seat belt exerts a force that slows the passenger with the car. There is no new forward "inertia force" to add to the passenger's force diagram; inertia describes the body's resistance to changing its motion.

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.

Doubling the resultant from 0.20 to 0.40 N doubles acceleration from 0.40 to 0.80 m/s². The ratio F_resultant / 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.

05

Friction and motion

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.

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.

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; see falling with air resistance for its changing effect on motion.

06

Three-force equilibrium

For a stationary point mass, the forces have zero resultant. Three force vectors drawn head to tail therefore form a closed triangle.

Use graphical vector addition: preserve each arrow's direction, choose a scale, then measure lengths. If needed, review W = mg to find the known weight first.

A point-mass model represents a small body as a single point. Here we use it to find the tensions in two supporting strings. Each string pulls away from the mass along its own direction.

Draw the forces before drawing the triangle

A 0.50 kg mass hangs stationary from two strings. The left string is 30° above the horizontal to the left; the right string is 45° above the horizontal to the right. Take g = 10 N/kg.

The weight is W = mg = 0.50 x 10 = 5.0 N downwards. There are three forces on the mass. Their directions are known, but the two tensions' magnitudes are not.

The free-body diagram shows the forces acting at the mass. The separate triangle moves those arrows head to tail to add them; it does not show the physical positions of the strings. Returning to A means that the resultant vector is zero.

Use a paper construction for ruler measurements. A screen diagram changes size with the display and need not preserve physical centimetres. Use the stated scale and angles, draw carefully, then measure. Small differences in graphical answers are expected.

Do not add the two tension magnitudes and set that sum equal to the weight. Their directions differ, so they must be added as vectors. Drawing C to B instead of B to C would also reverse the left tension and break the intended head-to-tail sequence.

Try a different arrangement

This optional paper task uses a 6.0 N downward weight, supported by two strings that each make 60° with the horizontal on their respective sides. Draw the free-body diagram, choose a scale and construct the closed force triangle. There is no answer to submit.

07

Falling with air resistance

A falling object can speed up while its acceleration gets smaller. Increasing air resistance reduces the downward resultant force.

Weight is mg, and acceleration comes from resultant force / mass. Take downwards as positive in the examples on this page.

Without significant air resistance

For a body of constant mass in a uniform gravitational field, weight is constant. If air resistance is negligible, weight is the only force, so:

The mass cancels, which explains why objects of different masses have the same free-fall acceleration in this model. Gravity has not disappeared: it is the force causing the acceleration. With no air resistance, this model does not produce a terminal speed.

As a body falls through still air

Consider the same body released from rest, with its shape and orientation unchanged. At release its speed relative to the air is zero, so air resistance is zero. As its downward speed increases, upward air resistance grows, while its weight remains the same.

Terminal velocity

When air resistance equals weight, the resultant and acceleration are zero. The object continues downwards at a constant, non-zero velocity called its terminal velocity, while these conditions remain unchanged.

Read the two features separately: increasing graph height means increasing downward velocity; decreasing gradient means decreasing acceleration. A horizontal line at a positive velocity represents steady downward motion, not rest.

The supplied force values let us calculate acceleration at the three stages. They do not tell us the times of those stages or the numerical terminal velocity. A real falling object may reach the ground before it approaches terminal motion.

What if the air resistance changes?

If an already downward-moving object suddenly experiences air resistance greater than its weight, the resultant is upwards. Its acceleration is then upwards, so it slows while still travelling downwards. Zero acceleration occurs only when the forces balance.

Terminal velocity does not mean zero weight or zero velocity. Both weight and air resistance still act. Their balance gives zero acceleration and a constant downward velocity.