Resultant forces — AQA GCSE Physics
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Resultant forces explained
When several forces act on one object, they can be replaced by a single force that has exactly the same effect on that object as all of them acting together.
Read the full explanation
Along a straight line, forces pointing the same way add and forces pointing opposite ways subtract. A brick lying on the bottom of a pool has three forces on it: its weight downwards, upthrust from the water upwards and the normal contact force from the floor upwards. It stays still because the weight equals the upthrust plus the normal contact force, so the single equivalent force is zero. A hailstone at terminal velocity is the same story with two forces: air resistance has grown until it equals the weight, so the single force replacing them is zero and the velocity stops changing.
This single force is called the resultant force.
The single force that has the same effect as all the forces acting on an object is called the resultant force. Its size and its direction together decide what the object does. If the resultant force is zero the object either stays at rest or carries on at constant velocity; that is the case for a hailstone at terminal velocity, where air resistance has grown until it equals the weight. If the resultant force is not zero the object accelerates in the direction of that resultant, which is why a hailstone accelerates at the start of its fall, when its weight is greater than the air resistance. Because the resultant is a vector, an answer is only complete when it gives both a size in newtons and a direction.
Students should be able to calculate the resultant of two forces that act in a straight line.
For two forces on the same straight line, the resultant is found by simple addition and subtraction once you have fixed a positive direction. Start by writing down which way you are calling positive. Forces pointing that way count as positive, forces the other way as negative, and the resultant is the total. A car with a 4200 N driving force and 3000 N of drag has a resultant of 1200 N forwards. Two equal opposing forces give a resultant of zero, which is what happens once a car reaches maximum speed, when the driving force and the resistive forces are equal. Always finish with both the magnitude in newtons and the direction, because force is a vector.
(HT only) Students should be able to: describe examples of the forces acting on an isolated object or system use free body diagrams to describe qualitatively examples where several forces lead to a resultant force on an object, including balanced forces when the resultant force is zero.
Higher tier only. A free body diagram isolates one object and shows only the forces acting on that object, each drawn as a labelled arrow starting on the object and pointing the way the force acts, with the length showing its relative size. Forces the object exerts on other things do not appear. A book on a table gets weight down and normal contact force up; a car at constant speed gets driving force forwards, drag backwards, weight down and normal contact force up. When opposing arrows are equal in length, the forces are balanced and the resultant force is zero, so the object stays at rest or keeps a constant velocity.
(HT only) A single force can be resolved into two components acting at right angles to each other.
Higher tier only. A single force acting at an angle can be replaced by two components at right angles to each other, usually one horizontal and one vertical. Draw the force as the diagonal of a rectangle whose sides run along the two chosen directions, and the two sides are the components. Each component is smaller than the original force. Resolving explains real situations: a sled pulled by a rope at an angle is only driven forward by the horizontal component of the pull, while the vertical component effectively lifts the sled and so reduces the normal contact force from the surface, which in turn reduces the friction. On a scale drawing, the components are measured from the rectangle using the stated scale.
The two component forces together have the same effect as the single force.
The two components are not an approximation: together they have exactly the same effect on the object as the single force they came from. That is why you may replace a force by its components, work with them separately, and still be describing the same physics. It also works in reverse. Given two perpendicular forces, draw them as the sides of a rectangle from a common starting point, complete the rectangle, and the diagonal from that point is the single force equivalent to both. Measure the diagonal and convert it with the stated scale to get the magnitude, then measure the angle from a stated reference direction to get the direction. The components can never simply be added as numbers.
(HT only) Students should be able to use vector diagrams to illustrate resolution of forces, equilibrium situations and determine the resultant of two forces, to include both magnitude and direction (scale drawings only).
Higher tier only, and scale drawings only: no trigonometry is required. To find the resultant of two forces, state a scale such as 1 cm represents 20 N, draw the first force to length and direction, then draw the second starting from the tip of the first, keeping its own direction. The resultant runs from the tail of the first arrow to the tip of the second; measure it, convert with the scale, and measure its angle from a stated direction. For equilibrium, three forces acting on an object with a zero resultant form a closed triangle when drawn tip to tail, so if the drawing does not close, the forces are not balanced. Resolution is the same construction used in reverse.
Your focus
- Replace a set of forces acting along one line with a single equivalent force.
- Identify every force acting on an object resting on the bottom of a pool and show how they combine.
- Explain why the combined effect is zero for an object moving at terminal velocity.
Show all 21 objectives
- State what a zero resultant force implies about an object's motion, covering both possible cases.
- Explain why a falling hailstone first accelerates and then reaches terminal velocity, in terms of the resultant force.
- Give a resultant force as a complete vector answer, with size and direction.
- Calculate the resultant of two forces acting along the same line, giving magnitude and direction.
- Deduce an unknown force from the fact that an object is moving at constant speed.
- Set out a positive direction convention and apply it consistently through a calculation.
- Draw a labelled free body diagram for a named object in a described situation.
- Deduce the direction of the resultant force from the relative lengths of the arrows on a given free body diagram.
- Explain what a zero resultant force means for an object's motion, linking balanced forces to constant velocity or rest.
- Resolve a single force into two perpendicular components on a scale diagram.
- Explain why only part of an angled pull moves an object forwards.
- Describe how the vertical component of an angled pull changes the normal contact force and the friction.
- Construct the single force equivalent to two perpendicular components using a scale drawing.
- Explain why two perpendicular components cannot simply be added as numbers.
- Give the direction of a constructed force as an angle from a stated reference direction.
- Determine the resultant of two forces by a tip to tail scale drawing, giving magnitude and direction.
- Use a closed triangle of forces to test whether an object is in equilibrium.
- Choose a scale for a vector diagram that keeps the measurement error small, and state it on the drawing.
Resultant forces exam tips
Marking Points
- one mark for the single force having exactly the same effect on the object as all the original forces together
- one mark for adding forces that act in the same direction and subtracting those that act in opposite directions
- one mark for a worked case, such as the weight of a brick equalling the upthrust plus the normal contact force
- one mark for the single force being zero when the original forces are balanced, as for a hailstone at terminal velocity
- one mark for naming the single equivalent force the resultant force
- one mark for a resultant force of zero meaning the object stays at rest or moves at constant velocity
- one mark for a non-zero resultant force meaning the object accelerates in the direction of the resultant
- one mark for applying this, such as a hailstone accelerating because there is a resultant force, its weight being greater than the air resistance
- one mark for the correct subtraction of the opposing force, such as 4200 - 3000
- one mark for adding forces acting the same way and subtracting the one acting the opposite way
- one mark for the magnitude of the resultant in newtons
- one mark for the direction of the resultant, named in words or shown by the sign
- one mark for a free body diagram showing the chosen object alone with every force acting on it
- one mark for each force drawn as a labelled arrow from the object, with the length showing its relative size
- one mark for describing a balanced case, where the arrows cancel and the resultant force is zero
- one mark for linking a zero resultant force to the object remaining at rest or moving at constant velocity
- one mark for splitting the single force into two components at right angles to each other
- one mark for identifying which component does the useful job, such as only the horizontal component pulling the sled forward
- one mark for the effect of the other component, such as the vertical component effectively lifting the sled
- one mark for the consequence of that, a reduction in the force of the surface on the object
- one mark for a scale drawing in which the two components form the sides of a rectangle with the original force as its diagonal
- one mark for stating that the two perpendicular components together have the same effect as the original single force
- one mark for constructing the original force as the diagonal of the rectangle drawn on the two components
- one mark for measuring that diagonal and converting it to a force using the stated scale
- one mark for giving the direction as an angle measured from a stated reference line
- one mark for a stated scale, such as 1 cm represents 20 N
- one mark for the two forces drawn tip to tail, each to the correct length and direction
- one mark for the resultant drawn from the tail of the first force to the tip of the second
- one mark for measuring the resultant and converting it to newtons using the stated scale
- one mark for the direction of the resultant given as an angle from a stated reference direction
Examiner Tips
- 💡List every force acting before you combine them; the missing force is where most marks go.
- 💡State the direction you are calling positive, then let the signs do the work.
- 💡For a floating or submerged object, always ask whether upthrust belongs in the list.
- 💡For 'why does it accelerate', the one-mark answer is that there is a resultant force acting; adding which force is larger makes it safe.
- 💡Use 'constant velocity' rather than 'not moving' when the resultant is zero, since both cases are allowed.
- 💡Quote a direction with every resultant, even when the arithmetic feels obvious.
- 💡Write the positive direction at the top of your working; it costs one line and prevents sign errors.
- 💡If the object is at constant speed, you already know the resultant is zero, which can give you the missing force.
- 💡State the direction in words, such as 'forwards' or 'to the left', rather than relying on a minus sign alone.
- 💡Name the object your diagram is about before you draw; a diagram of two objects at once cannot be a free body diagram.
- 💡Make the arrow lengths tell the story: equal for balanced, clearly unequal for accelerating.
- 💡Choose your two directions to match the question, usually along the motion and at right angles to it.
- 💡Sketch the rectangle even when you only need one component; it keeps the geometry honest.
- 💡Say what each component physically does, because the marks are for the consequences, not the drawing alone.
- 💡Say which direction your angle is measured from, for example 'above the horizontal', or the direction mark cannot be awarded.
- 💡Use a set square or the corner of a ruler to keep the rectangle's angles at 90 degrees.
- 💡Check that your diagonal is longer than either side; if it is not, the drawing is wrong.
- 💡Choose a scale that uses most of the graph paper, since a bigger drawing gives a smaller percentage error.
- 💡For an equilibrium question, check whether your triangle closes; a gap means you have made a drawing error or the forces are unbalanced.
- 💡Leave the construction lines on the page, because the method marks come from the drawing itself.
Common Mistakes
- adding the magnitudes of forces that point in opposite directions
- leaving out one of the forces acting, most often upthrust on a submerged object
- assuming balanced forces must mean the object is stationary, when constant velocity also qualifies
- treating the single equivalent force as an extra force acting in addition to the originals
- claiming a resultant of zero only if all the individual forces are zero
- giving the name 'resultant force' when the question asks for its size and direction
- saying a resultant force of zero means no forces act on the object
- claiming an object at terminal velocity has stopped moving
- assuming the resultant force always points in the direction the object is travelling
- treating the resultant force as a separate physical force pushing the object in addition to the real ones
- always subtracting the two forces regardless of which way they point
- giving a size with no direction
- adding a 400 N driving force and 400 N of drag to get 800 N instead of zero
- forgetting that the resultant is zero when the object moves at a constant speed
- changing the positive direction between two parts of the same question
- drawing forces that the object exerts on other objects instead of forces acting on it
- drawing all arrows the same length when the object is accelerating
- including invented forces such as 'the force of motion' or momentum
- leaving arrows unlabelled, so the examiner cannot identify the forces
- omitting one real force, most often the normal contact force or drag
- halving the force to get each component
- drawing two components that are not at right angles to each other
- making a component longer than the original force
- forgetting that the vertical component changes the normal contact force and therefore the friction
- resolving along directions that do not match the ones the question is interested in
- adding the two component magnitudes arithmetically to get the original force
- drawing the single force as one side of the rectangle rather than as the diagonal
- omitting the angle, so the answer is not a complete vector
- measuring the diagonal but forgetting to apply the scale
- measuring the angle from an unstated or inconsistent reference direction
- using Pythagoras or trigonometry when the question specifies a scale drawing
- drawing the two forces from the same point and joining the tips as if that were the resultant
- forgetting to write the scale on the diagram
- quoting the answer in centimetres instead of newtons
- giving only the magnitude when the question asks for magnitude and direction