In simple terms
A friendly intro before the formal notes — no formulas yet.
The Balancing Act of Forces
Equilibrium is all about balance. For an object to stay still or move at a steady speed, all the pushes and pulls (forces) on it must cancel each other out perfectly.
Imagine a tug-of-war. If both teams pull with exactly the same strength, the rope doesn't move – that's equilibrium. If one team pulls even slightly harder, the balance is broken and the rope accelerates. Mechanics problems are just a more advanced version of this tug-of-war, often with forces pulling in many different directions.
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A force is a vector, having both magnitude and direction. We can split any force into two perpendicular 'component' forces, which is essential for analysing forces that aren't purely horizontal or vertical.
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For an object to be in equilibrium (balanced), the net force must be zero. This means the sum of all force components in any direction must be zero; for example, total force up equals total force down.
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Friction is a force that opposes sliding, but it has a limit. When an object is on the verge of slipping, it's in 'limiting equilibrium', and the friction force is at its maximum value, given by .
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To solve any forces problem, always start by drawing a free-body diagram. This means drawing the object by itself and adding arrows to represent every single external force acting on it, such as weight, tension, and reaction forces.
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Force as vector
Force as vector — resolve into perpendicular components.
Key formulas
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Tap a symbol — great for exam definitions
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Full topic notes
Formal explanation with the rigour you need for the exam.
Representing Forces: Free-Body Diagrams
The first and most critical step in any mechanics problem is to create a clear picture of the forces involved. We do this using a free-body diagram. The process involves isolating the object of interest and drawing arrows to represent every single external force acting on that object. It is crucial not to include forces that the object exerts on its surroundings.
Draw the object as a simple shape or particle.
Draw the weight (W = mg) acting vertically downwards from the centre of mass.
If the object is on a surface, draw the Normal Reaction (R) acting perpendicular to the surface.
If a string or rope is attached, draw a Tension (T) force pulling away from the object along the string.
If there is friction, draw the frictional force (F) acting parallel to the surface, opposing the direction of motion or potential motion.
Include any other applied forces (pushes or pulls) given in the problem.
The Conditions for Equilibrium
An object is in equilibrium if it is not accelerating. This means it is either stationary or moving at a constant velocity. For this to happen, the net force acting on the object must be zero. Since force is a vector, this means the vector sum of all forces must be the zero vector.
In a 2D system, we can make this condition much more practical by 'resolving' forces. This involves choosing two perpendicular directions (like horizontal and vertical) and stating that the sum of the force components in each of these directions must be zero. This gives us two independent equations to work with.
Friction and Limiting Equilibrium
When a surface is 'rough', it can exert a frictional force. This force opposes motion and acts parallel to the surface. The maximum frictional force a surface can provide is proportional to the normal reaction force. The constant of proportionality is the coefficient of friction, .
The inequality shows that the frictional force, , can take any value from zero up to its maximum, . The system will generate just enough friction to maintain equilibrium, if possible. If an object is on the verge of sliding, we say it is in 'limiting equilibrium', and in this specific case, the friction is at its maximum value: .
In questions asking for a 'range of values' for a force to maintain equilibrium on a rough slope, you must consider two separate limiting cases: 1) The object is about to slide UP the slope (so friction acts DOWN). 2) The object is about to slide DOWN the slope (so friction acts UP). This will give you the maximum and minimum values for the required force.
Worked examples
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A particle is held in equilibrium by three forces, , and . Given that N and N, find the force and calculate its magnitude and the angle it makes with the positive x-axis.
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For the particle to be in equilibrium, the vector sum of the forces must be zero.
N.\
A block of mass 20 kg rests on a rough plane inclined at to the horizontal. The coefficient of friction between the block and the plane is 0.3. A force of magnitude N acts on the block, parallel to a line of greatest slope. Find the range of possible values of for the block to remain in equilibrium. (Use )
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First, draw a free-body diagram and resolve the weight ( N) into components parallel and perpendicular to the plane.
Weight component perpendicular to plane: .
Weight component parallel to plane: .\
How it all connects
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Glossary
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Quick check
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Revision flashcards
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What is a 'force' in mechanics?
A push or a pull on an object that can cause it to accelerate (change its velocity). It is a vector quantity, possessing both magnitude (in Newtons, N) and direction.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
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Draw the object as a simple shape or particle.
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Draw the weight (W = mg) acting vertically downwards from the centre of mass.
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If the object is on a surface, draw the Normal Reaction (R) acting perpendicular to the surface.
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If a string or rope is attached, draw a Tension (T) force pulling away from the object along the string.
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If there is friction, draw the frictional force (F) acting parallel to the surface, opposing the direction of motion or potential motion.
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Include any other applied forces (pushes or pulls) given in the problem.
Practice — then mark it
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Test your understanding with exam-style questions on Forces and Equilibrium.
Test your understanding with exam-style questions on Forces and Equilibrium.
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Checkpoint
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