R: Forces and Newton’s laws — AQA A-Level Mathematics
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R: Forces and Newton’s laws explained
A force is a push or pull that can change an object's motion or shape, measured in newtons and represented as a vector with magnitude and direction.
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Newton's first law states that a body remains at rest or moves with constant velocity unless acted on by a resultant force. This means zero resultant force gives zero acceleration, not necessarily zero velocity. For example, a book on a table has weight downwards and normal contact upwards; these cancel, so the book stays still. If the surface is frictionless and the book is sliding, it continues at constant speed. To use the law, identify all forces, resolve if needed, and check whether the resultant is zero. If it is, acceleration is zero; if not, motion changes. This underpins later work on equilibrium and dynamics.
Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2D vectors); extend to situations where forces need to be resolved (restricted to 2 dimensions).
Newton's second law states that the resultant force on an object is equal to its mass times acceleration: F = ma. For motion in a straight line, you resolve forces along the direction of motion and perpendicular to it. In two perpendicular directions, apply F = ma independently in each direction. For example, a block on a rough horizontal surface: vertically, normal reaction balances weight; horizontally, applied force minus friction equals mass times acceleration. If forces are given as 2D vectors, add them to find the resultant, then use F = ma. When forces act at angles, resolve into components using trigonometry. Always define a positive direction, draw a free-body diagram, and check units. This law connects force, mass and acceleration and is fundamental to dynamics.
Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g , and its value in SI units to varying degrees of accuracy. (The inverse square law for gravitation is not required and g may be assumed to be constant, but students should be aware that g is not a universal constant but depends on location).
Weight is the force of gravity on a mass: W = mg. Near Earth's surface, g ≈ 9.8 m s⁻², but it varies with location (e.g., 9.81 in UK, 9.78 at equator, 9.83 at poles). For motion in a straight line under gravity alone, use constant acceleration equations (suvat) with a = ±g, taking up as positive or negative consistently. Example: a ball thrown vertically upwards at 20 m s⁻¹: time to highest point t = u/g ≈ 2.04 s, maximum height s = u²/(2g) ≈ 20.4 m. Weight acts downwards; mass is scalar and constant. In SI, g is measured in m s⁻²; use g = 9.8 or 9.81 as required. Remember g is not universal; it depends on planet and altitude.
Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2D vectors); application to problems involving smooth pulleys and connected particles; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces.
Newton’s third law: if A exerts a force on B, B exerts an equal and opposite force on A; these act on different bodies. For a particle in equilibrium, the resultant force is zero: resolve forces in two perpendicular directions and set sum of components to zero. For motion in a straight line, use F = ma along the direction of motion. Resolve forces into components using trigonometry (e.g., F cos θ, F sin θ). For smooth pulleys and connected particles, treat the system as a whole or separately, noting tension is the same throughout a light inextensible string over a smooth pulley. Example: two masses connected over a pulley; write equations for each mass and solve for acceleration and tension. Coplanar forces can be represented as 2D vectors; equilibrium requires vector sum zero.
Understand and use addition of forces; resultant forces; dynamics for motion in a plane.
This topic develops the vector nature of forces and their role in two-dimensional motion. You learn to combine multiple forces into a single resultant, often by resolving into perpendicular components or using vector addition, then apply Newton’s second law in vector form: F_net = ma. For motion in a plane, set up separate equations for two perpendicular directions, typically horizontal and vertical or parallel and perpendicular to a slope. A common method is to draw a force diagram, choose axes, resolve all forces, sum components, and solve for acceleration or unknown forces. Interpret equilibrium as zero resultant force, leading to constant velocity (which may be zero) rather than necessarily rest. For example, a 10 N force at 30° to the horizontal has components 10 cos 30° ≈ 8.66 N horizontally and 10 sin 30° = 5 N vertically.
Understand and use the F ≤ μR model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics.
This topic introduces the empirical model of friction, where the frictional force F satisfies F ≤ μR, with μ the coefficient of friction and R the normal reaction. You learn to distinguish static and dynamic situations: when a body is in limiting equilibrium, friction takes its maximum value F = μR; when it moves, friction is usually modelled as F = μR opposing motion; when it is stationary but not at the point of sliding, F < μR. For a body on a rough horizontal surface, R is often equal to weight, but on an incline R = mg cos θ. You apply Newton’s second law along the surface, including friction, to find acceleration or required forces, and use equilibrium conditions for statics problems.
Your focus
- Define force and describe its vector nature.
- State Newton's first law and explain its meaning in terms of resultant force and acceleration.
- Apply Newton's first law to determine whether an object is in equilibrium or accelerating.
Show all 18 objectives
- State and apply Newton's second law in the form F = ma.
- Resolve forces into two perpendicular components and use them to find resultant force or acceleration.
- Solve problems involving motion in a straight line with forces given as 2D vectors or at angles.
- Calculate weight using W = mg and state the SI unit of weight.
- Solve problems involving vertical motion under gravity using constant acceleration equations.
- Explain that g varies with location and use an appropriate value to a given degree of accuracy.
- State and apply Newton’s third law to identify force pairs acting on different bodies.
- Solve equilibrium problems by resolving coplanar forces in two perpendicular directions.
- Analyse motion of connected particles over a smooth pulley using F = ma and tension.
- Resolve forces into perpendicular components and combine them to find a resultant force.
- Apply Newton’s second law in vector form to solve for acceleration or unknown forces in two-dimensional motion.
- Analyse equilibrium situations and distinguish between zero resultant force and zero velocity.
- Apply the friction inequality F ≤ μR to determine whether a body remains in equilibrium or begins to move.
- Calculate the normal reaction and frictional force for bodies on rough horizontal and inclined surfaces.
- Solve dynamics and statics problems involving friction using Newton’s laws and equilibrium conditions.
R: Forces and Newton’s laws exam tips
Marking Points
- Defines force as a vector quantity with magnitude and direction, measured in newtons.
- States Newton's first law: a body remains at rest or moves with constant velocity unless acted on by a resultant force.
- Explains that zero resultant force implies zero acceleration, not zero velocity.
- Applies the law to identify equilibrium situations, such as a stationary object or one moving at constant speed.
- Resolves forces in simple cases to show the resultant is zero or non-zero.
- Distinguishes between balanced forces and the absence of forces.
- States Newton's second law as F = ma, where F is the resultant force.
- Resolves forces into two perpendicular directions, typically parallel and perpendicular to motion.
- Applies F = ma independently in each direction, recognising that acceleration perpendicular to motion may be zero.
- Adds forces given as 2D vectors to find the resultant force before using F = ma.
- Uses trigonometry to resolve forces at angles into components.
- Solves problems involving motion in a straight line with forces in two perpendicular directions.
- Weight is calculated as W = mg, where m is mass in kg and g is gravitational acceleration in m s⁻²; weight is a force in newtons.
- For vertical motion under gravity, use constant acceleration equations with a = g downwards (or -g if upwards is positive).
- At maximum height, vertical velocity is zero; time to highest point is u/g (if upwards positive).
- g varies with location: approximately 9.81 m s⁻² in the UK, but lower at the equator and higher at the poles; it is not a universal constant.
- When solving problems, state the value of g used and its direction; keep sign conventions consistent throughout.
- State Newton’s third law correctly: forces come in pairs, equal in magnitude, opposite in direction, acting on different bodies.
- For equilibrium, resolve forces in two perpendicular directions and set the sum of components in each direction to zero.
- For motion, apply F = ma along the direction of motion; if forces are at angles, resolve first.
- For connected particles over a smooth pulley, the tension is the same on both sides of the pulley and the acceleration magnitude is the same for both particles (if string is inextensible).
- Resolve forces using F cos θ and F sin θ, where θ is the angle to the chosen axis; ensure correct signs.
- Resolve a force of magnitude F at angle θ to a given direction into F cos θ and F sin θ components, maintaining consistent sign conventions. For example, a 10 N force at 30° to the horizontal gives 10 cos 30° ≈ 8.66 N horizontally and 10 sin 30° = 5 N vertically.
- Sum components in two perpendicular directions to obtain the resultant force vector, or to set up equations of motion using F_net = ma in each direction. For instance, if horizontal forces are 8 N right and 3 N left, the resultant horizontal component is 5 N right.
- Apply Newton’s second law in vector form, recognising that acceleration is in the direction of the resultant force and that perpendicular components are independent. For example, on a smooth slope, resolve weight parallel and perpendicular to the slope; the perpendicular component balances the normal reaction, while the parallel component causes acceleration.
- Interpret equilibrium as a zero resultant force, leading to constant velocity (which may be zero) rather than necessarily rest. For example, a skydiver at terminal velocity has zero resultant force but is moving at constant speed.
- Use kinematic equations in two dimensions when acceleration is constant, treating horizontal and vertical motions separately but linking them through time. For example, projectile motion: horizontal velocity is constant while vertical motion has acceleration g.
- Solve problems involving forces at angles, including tension in strings, normal reaction on inclined planes, and motion on a smooth surface. For example, find the acceleration of a block on a smooth slope by resolving weight parallel to the slope and using F_net = ma.
- State and apply the friction inequality F ≤ μR, recognising that equality holds at limiting equilibrium or when sliding.
- Calculate the normal reaction R correctly for horizontal and inclined rough surfaces, including cases with additional vertical forces.
- Determine the direction of friction as opposing impending or actual motion along the surface.
- Set up and solve equations of motion along a rough surface using F_net = ma, incorporating friction as μR when moving or at limiting equilibrium.
- Solve statics problems on rough surfaces by resolving forces and using equilibrium conditions, often with F = μR at the point of sliding.
- Interpret the coefficient of friction as a dimensionless constant relating maximum friction to normal reaction, and use it to compare roughness.
Examiner Tips
- 💡Always draw a free-body diagram showing all forces acting on the object before applying Newton's first law.
- 💡State clearly that the resultant force is zero when velocity is constant, and link this to zero acceleration.
- 💡Use the wording 'constant velocity' rather than 'constant speed' when direction is fixed, as velocity includes direction.
- 💡Draw a clear free-body diagram with all forces labelled and a defined positive direction.
- 💡Write down the equation F = ma for each direction separately, showing the resultant force as a sum of components.
- 💡Check that all forces are in newtons and mass in kilograms before calculating acceleration.
- 💡Always write down the suvat equation you are using and substitute values with correct signs; this gains method marks even if arithmetic slips.
- 💡If a question gives g to a specific accuracy (e.g., 9.8 or 9.81), use that value; do not default to 9.81 if told otherwise.
- 💡For multi-stage vertical motion (e.g., up then down), split the motion into stages and treat each with consistent sign conventions.
- 💡Draw a clear diagram showing all forces acting on each particle; label angles and directions.
- 💡For connected particles, write separate equations for each particle and solve simultaneously; check the direction of acceleration for each.
- 💡When resolving, choose axes that simplify the problem (e.g., along the direction of motion and perpendicular to it).
- 💡Draw a clear force diagram with labelled angles and axes; this often earns method marks even if arithmetic slips later.
- 💡State the direction you are taking as positive and stick to it consistently throughout the calculation.
- 💡When motion is on a slope, resolve parallel and perpendicular to the slope to simplify equations.
- 💡Check whether the question asks for magnitude, direction, or both; give the direction as an angle or bearing where appropriate.
- 💡Draw a free-body diagram showing all forces, including friction and normal reaction, and label the direction of motion.
- 💡For inclined plane problems, resolve parallel and perpendicular to the plane to simplify equations.
- 💡If a body is on the point of moving, state F = μR explicitly before solving.
- 💡Check whether the surface is rough or smooth; if smooth, friction is zero.
Common Mistakes
- Thinking that a moving object must have a resultant force acting on it. Correction: constant velocity means zero resultant force.
- Confusing mass and force, using kilograms for force. Correction: force is measured in newtons; mass in kilograms.
- Assuming that if an object is at rest, no forces act on it. Correction: forces may act but cancel to give zero resultant.
- Using the applied force instead of the resultant force in F = ma. Correction: always calculate the vector sum of all forces first.
- Forgetting to resolve forces at angles into components. Correction: draw a diagram and use sin/cos to find horizontal and vertical components.
- Mixing up mass and weight in the equation. Correction: use mass in kilograms for m, and weight (mg) as a force if needed.
- Confusing mass and weight: mass is in kg and constant; weight is in N and depends on g. Correction: always use W = mg for weight and state units clearly.
- Using g = 9.8 m s⁻² but treating it as positive when the object is moving upwards. Correction: choose a positive direction and apply signs consistently; if upwards is positive, acceleration is -g.
- Assuming g is the same everywhere on Earth. Correction: recognise that g varies slightly with latitude and altitude; use the value given in the question or standard 9.81 m s⁻² unless told otherwise.
- Applying Newton’s third law to forces on the same body. Correction: third-law pairs act on different bodies; for example, weight of a book on a table and the normal force from the table on the book are not a third-law pair (they act on the same body).
- Forgetting to resolve forces when they are not along the axes. Correction: always resolve into two perpendicular directions before summing.
- Assuming tension is different on each side of a smooth pulley. Correction: for a light inextensible string over a smooth pulley, tension is the same throughout.
- Adding force magnitudes directly without considering direction. Correction: treat forces as vectors; resolve into components or use vector triangle methods. For example, 3 N and 4 N perpendicular forces give a resultant of 5 N, not 7 N.
- Assuming that a zero resultant force means the object is stationary. Correction: zero resultant force means zero acceleration, so the object moves with constant velocity, which could be non-zero. For example, a car cruising at steady speed has zero resultant force but is moving.
- Mixing up sine and cosine when resolving. Correction: always identify the angle with a diagram; the component adjacent to the angle uses cosine, the opposite uses sine. For example, for a force at 30° to the horizontal, the horizontal component is F cos 30° and the vertical component is F sin 30°.
- Forgetting to include all forces, such as weight or normal reaction, when forming equations. Correction: draw a complete free-body diagram before resolving. For example, on a slope, include weight, normal reaction, and any friction or applied force.
- Assuming friction always equals μR. Correction: use F = μR only when the body is at limiting equilibrium or sliding; otherwise F ≤ μR and must be found from equilibrium or motion equations.
- Using R = mg on an inclined plane. Correction: resolve perpendicular to the plane to get R = mg cos θ, or include other perpendicular forces.
- Taking friction in the same direction as motion. Correction: friction always opposes relative motion or impending motion.
- Forgetting that μ is dimensionless and misusing units. Correction: treat μ as a number with no units; check that F and R are both forces.