Mechanics
Forces & Newton's Laws
Year 12 · Year 13
- ✓Calculate the resultant force acting on an object and determine its effect on motion.
- ✓Apply Newton's Second Law (F=ma) to solve problems involving constant acceleration in one or two dimensions.
- ✓Analyse and solve problems involving connected particles, such as those linked by strings or in contact.
- ✓Model and solve problems involving friction on both horizontal and inclined planes.
Key concepts
The single force that has the same effect as all the individual forces acting on an object. It is the vector sum of all forces. If the resultant force is zero, the object is in equilibrium (at rest or moving with constant velocity).
An object remains at rest or continues to move with constant velocity unless acted upon by a resultant external force. This is also known as the law of inertia.
The resultant force acting on an object is directly proportional to its mass and its acceleration, and acts in the same direction as the acceleration.
For every action, there is an equal and opposite reaction. When object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A. These forces act on different objects.
The force of gravity acting on an object. It always acts vertically downwards towards the centre of the Earth.
The force exerted by a surface on an object in contact with it, acting perpendicularly away from the surface. It prevents the object from passing through the surface.
The force transmitted through a string, rope, cable, or similar one-dimensional continuous object when pulled tight by forces acting from opposite ends. It acts along the string, away from the object.
A force that opposes the relative motion or tendency of motion between two surfaces in contact. It acts parallel to the surfaces. For limiting friction (when an object is on the point of moving or is moving), F_f = μR.
A dimensionless scalar quantity that describes the ratio of the force of friction between two bodies and the force pressing them together. It depends on the nature of the surfaces in contact.
A system of two or more objects that are linked together, often by a string or in direct contact. They typically share the same acceleration (if moving together) or have related accelerations. Forces like tension or contact forces act between them.
A flat surface tilted at an angle to the horizontal. When an object is on an inclined plane, its weight needs to be resolved into components parallel and perpendicular to the plane to apply Newton's laws effectively.
Key facts to remember
- 1Newton's First Law: An object remains in its state of motion unless a resultant force acts on it.
- 2Newton's Second Law: F = ma, where F is the resultant force, m is mass, and a is acceleration. This is a vector equation.
- 3Newton's Third Law: For every action, there is an equal and opposite reaction.
- 4Weight (W) = mg, where g is the acceleration due to gravity (approximately 9.8 m/s²).
- 5Friction (F_f) opposes motion or the tendency of motion and its maximum value is F_f = μR, where μ is the coefficient of friction and R is the normal reaction.
- 6When resolving forces on an inclined plane, resolve weight into components parallel (mg sinθ) and perpendicular (mg cosθ) to the plane.
- 7For connected particles, apply F=ma to each particle separately, remembering that tension in a light inextensible string is uniform and acceleration is shared.
- 8Always draw a clear force diagram and choose a consistent positive direction for each object.
Worked examples
Example 1
A particle of mass 5 kg is acted upon by two forces, F₁ = (3i + 7j) N and F₂ = (2i - 3j) N. Find the acceleration of the particle.
Answer
The acceleration of the particle is (i + 0.8j) m/s².
Forces and acceleration are vector quantities.
Example 2
A particle P of mass 3 kg and a particle Q of mass 2 kg are connected by a light inextensible string. The string passes over a smooth fixed pulley. The system is released from rest. Find the acceleration of the particles and the tension in the string.
Answer
The acceleration of the particles is 1.96 m/s² and the tension in the string is 23.5 N (3 s.f.).
"Light inextensible string" means tension is uniform throughout and particles have the same magnitude of acceleration. "Smooth pulley" means no friction at the pulley.
Example 3
A block of mass 4 kg rests on a rough plane inclined at an angle of 30° to the horizontal. The coefficient of friction between the block and the plane is 0.2. A force P N acts on the block, parallel to the plane and directed up the plane. Find the minimum value of P required to prevent the block from sliding down the plane.
Answer
The minimum value of P required is 12.8 N (3 s.f.).
Always consider the direction of impending motion to determine the direction of friction.
Common mistakes
- ✗Confusing mass (scalar, in kg) with weight (vector force, in N).
- ✗Incorrectly resolving forces, especially the weight component on an inclined plane (e.g., using sinθ for the perpendicular component).
- ✗Forgetting to include the normal reaction force or friction force when appropriate.
- ✗Making sign errors when setting up F=ma equations, particularly when forces act in opposite directions.
- ✗Assuming tension in a string is always equal to the weight of a hanging mass in a pulley system when the system is accelerating.
Exam tips
- ★Always start by drawing a clear, labelled diagram showing all forces acting on each object in the system.
- ★Choose a consistent positive direction for acceleration and forces for each object, especially in connected particle problems.
- ★When dealing with inclined planes, resolve forces parallel and perpendicular to the plane.
- ★Clearly state any assumptions made (e.g., "light inextensible string", "smooth pulley", "particle").
- ★Use g = 9.8 m/s² unless specified otherwise, and give answers to an appropriate number of significant figures (usually 3 s.f.).
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.