Forces and equilibrium in A Level Math Paper 4 (Mechanics): common mistakes and examiner advice

This guide covers forces in equilibrium, with and without friction: resolving forces, tensions in strings, inclined planes, limiting friction and the conditions for a body to stay at rest or begin to slip. The reports show that most students know what to do. The marks go missing in the resolving and in the direction of friction. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Wrong components. The most repeated comment. Students mixed up sine and cosine, used the wrong angle, or had a sign error in one of the two resolving equations. Reports also mention students who used an incorrect trigonometric relationship when solving for the angle.
• A missing force. The weight, the weight component on a slope, or one of the given forces (such as a 15 N force) was omitted from an equation. In a car and trailer question, students forgot the weight component of the trailer.
• Not saying that the resultant is zero. Some students resolved forces but did not state that the resultant in each direction was zero, so did not produce two equations. Others did not realise that the resultant force in the y-direction was zero.
• Wrong assumptions about tensions. Common errors were to assume that the tensions in two strings were equal, to assume that the other two angles in a triangle were both 45°, or to include the tension in a coupling in the equation for the whole system.
• Lami's theorem. One report says students tried to use it, but it applies only to a system of three forces.
• Resultant versus the force needed for equilibrium. Students gave the magnitude and direction of the resultant force rather than the additional force needed for equilibrium.
• Direction of a resultant. Giving only an angle is insufficient. The direction needs words, such as "above the positive x-axis", or a diagram.
• Negative tension. A common error was to obtain a negative tension from the vertical equation and then make it positive before substituting into the horizontal equation.
• Normal reaction. The normal reaction is not simply the component of the weight when other forces (such as a force P) also have a component perpendicular to the plane. Students omitted the term in P, used the wrong component, or had a sign error.
• Friction. The frictional force was in the wrong direction (when the body is about to slip down the plane, friction acts up the plane, and for the least value of P it also acts up), or was assumed to be unchanged when a force changed direction. Others applied F = μR with the wrong value of R, or wrote R = μF.
• Inequalities. Students introduced the inequality only on the final line and so did not offer a complete solution.
• Accuracy. Premature rounding led to tensions that were not correct to 3 significant figures.
• Poor layout. Reports describe working that was unclear, and tables of components that were not combined correctly.

What examiners advise

• Draw a clear force diagram with every force and its direction, including friction and the normal reaction.
• Keep intermediate calculations to 4 or 5 decimal places, or at least 4 significant figures.
• When a triangle is a 3, 4, 5 triangle or the sine or cosine of an angle is given, there is no need to evaluate the angle.
• Use exact trigonometric values until the final stage and only then calculate a numerical value.
• Solve simultaneous equations on a calculator only if you show enough working. One report says no marks can be awarded if it is done incorrectly and no working is shown.
• State the direction of a force clearly and uniquely.

Habits that help

• Resolve parallel and perpendicular to a slope for slope questions, and horizontally and vertically otherwise.
• Write "resultant = 0" at the start of each equation to remind yourself what you are doing.
• Check whether friction opposes the motion or the tendency to move.

Questions students ask

Why do I get the wrong components when resolving forces?

Examiners call it the most repeated comment. Students mix up sine and cosine, use the wrong angle or make a sign error in one resolving equation. Resolve parallel and perpendicular to a slope for slope questions, and horizontally and vertically otherwise.

What does it mean to say the resultant is zero?

State that the resultant is zero in each direction, so that you form two equations. Some students resolved forces without saying so.

Is the normal reaction just the component of the weight?

Not when other forces, such as a force P, also have a component perpendicular to the plane. Students omitted the term in P or used the wrong component.

Which way does friction act?

Opposite to the motion or to the motion that would occur. When a body is about to slip down the plane, friction acts up the plane, and for the least value of P it also acts up. Check the direction on your force diagram.

How do I give the direction of a resultant force?

With words or a diagram, such as "above the positive x-axis", not just an angle. Also check what is asked: the resultant, or the additional force needed for equilibrium.

Can I use Lami's theorem?

Only for a system of three forces. One report notes students tried to use it in other cases.

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