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Advanced Higher Mathematics of Mechanics formula sheet: the SQA formulae list

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Formula sheet

The card below opens today's Advanced Higher Mechanics paper with this list in the drawer. The contents follow in our own layout. For the centre of mass of a triangle, the list says it is two thirds of the way along the median from the vertex. Check against the list in the 2025 question paper before you rely on this page.

Formula sheet

Advanced Higher Mechanics

Open formula sheet

Gravitation

F=\frac{GMm}{r^2}Newton's inverse square law of gravitation

Simple harmonic motion

v^2=\omega^2(a^2-x^2)speed in terms of displacement
x=a\sin(\omega t+\alpha)displacement

Centre of mass

\frac{4r}{3\pi}uniform semicircle of radius r: distance of the centre of mass from the diameter, along the axis of symmetry
\overline{x}=\frac{1}{A}\int_a^bxy\ dxuniform lamina bounded by y = f(x), the x-axis, x = a and x = b: horizontal coordinate
\overline{y}=\frac{1}{A}\int_a^b\frac{1}{2}y^2\ dxuniform lamina: vertical coordinate

Standard derivatives

\frac{d}{dx}(\tan x)=\sec^2xderivative of tan x
\frac{d}{dx}(\cot x)=-\csc^2xderivative of cot x
\frac{d}{dx}(\sec x)=\sec x\tan xderivative of sec x
\frac{d}{dx}(\csc x)=-\csc x\cot xderivative of cosec x
\frac{d}{dx}(\ln x)=\frac{1}{x}derivative of ln x
\frac{d}{dx}(e^x)=e^xderivative of e^x

Standard integrals

\int\sec^2(ax)\ dx=\frac{1}{a}\tan(ax)+cintegral of sec² ax
\int\frac{1}{x}\ dx=\ln|x|+cintegral of 1/x
\int e^{ax}\ dx=\frac{1}{a}e^{ax}+cintegral of e^(ax)

In the gravitation law, G is the gravitational constant, M and m the two masses and r the distance between them. In the simple harmonic motion formulae, a is the amplitude, ω the angular frequency and α the phase.

What is not on the Advanced Higher Mechanics formula sheet

  • Kinematics. The constant acceleration equations and the projectile motion results.
  • Newton's laws and friction. F = ma, the friction law F = μR, and the equations for motion on a slope and in a pulley system.
  • Momentum and impulse. Conservation of momentum, collisions and impulse.
  • Work, energy and power. None of the relationships for them are printed.
  • Circular motion. The acceleration v²/r and the angular formulae are not printed.
  • The value of g. The 2025 paper tells you to take g as 9.8 m s⁻² where appropriate.

This is a short list for a course where most of the work is deciding which relationship applies. Practise the projectile and force questions in the Laws of Motion course, then press Mark.

Find the flight time or the landing distance of a ball fired sideways off a cliff.

Press Mark to have your answer marked in the Laws of Motion course.

How the paper uses the list

Mathematics of Mechanics is one question paper worth 100 marks, with a time of 3 hours, and a calculator is allowed. The course specification says the question paper has 100 marks out of a total of 100. In the 2027 timetable it is on Thursday 27 May from 13:00 to 16:00 (Exam Timetable 2027, version 02). Our Advanced Higher Mechanics guide covers the course in full.

Using the formulae list under exam pressure

Simple harmonic motion

A particle moves with simple harmonic motion with amplitude 0.5 m and angular frequency 4 rad/s. Find its greatest speed, and its speed when the displacement is 0.3 m. The greatest speed is at x = 0, so

v=\omega\sqrt{a^2-x^2}=4\times0.5=2\ \text{m s}^{-1}
v=4\sqrt{0.5^2-0.3^2}=4\sqrt{0.16}=4\times0.4=1.6\ \text{m s}^{-1}

Centre of mass

A uniform semicircular lamina has radius 6 cm. How far is its centre of mass from the diameter, along the axis of symmetry?

\frac{4r}{3\pi}=\frac{4\times6}{3\pi}=\frac{8}{\pi}=2.55\ \text{cm}

Gravitation

A satellite moves from an orbit of radius r to an orbit of radius 2r. The list gives F proportional to 1/r², so the force from the planet becomes one quarter of what it was. You never need G to answer a ratio question like this one.

A standard integral

Find the integral of sec²(3x) from 0 to π/12. The list gives ∫sec²(ax) dx = (1/a) tan(ax) + c, so

\int_0^{\pi/12}\sec^2(3x)\ dx=\left[\frac{1}{3}\tan3x\right]_0^{\pi/12}=\frac{1}{3}\tan\frac{\pi}{4}-0=\frac{1}{3}

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the sheet open beside you, mark it, and the worked answers show which line each question wanted. We do not publish official or past paper answers.

Use the sheet with today's paper

Uni-Sine publishes a free daily paper for Advanced Higher Mechanics. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full SQA paper and not set by SQA. Many schools set a longer prelim in winter: see our prelims guide. Open the sheet, set a timer, and write the line you use at the start of each answer.

Common mistakes

  • Using v² = ω²(a² − x²) with a wrong a. The amplitude a is the greatest displacement, not the displacement at the start.
  • Taking the centre of mass from the wrong edge. The semicircle result is measured from the diameter, along the axis of symmetry.
  • Assuming a constant acceleration. SHM has an acceleration that changes with displacement, so the SUVAT equations do not apply.
  • Forgetting radians. Use radians for ωt + α and for sin and cos in SHM.
  • Looking for g on the list. It is not there. The paper tells you what value to use.

Common questions

Do you get a formula sheet in Advanced Higher Mechanics?
Yes. SQA prints a formulae list at the front of the Mathematics of Mechanics question paper. It is short.
What is on the Advanced Higher Mechanics formulae list?
Newton's inverse square law of gravitation, two simple harmonic motion formulae, centre of mass results for a triangle, a semicircle and a lamina, and a few standard derivatives and integrals.
Are the SUVAT equations given in Advanced Higher Mechanics?
No. The constant acceleration equations, momentum, energy and friction are not on the list.
Is a calculator allowed in Advanced Higher Mechanics?
Yes. The question paper says you may use a calculator.
How long is the Advanced Higher Mechanics exam?
The paper is 3 hours and is worth 100 marks. In 2027 it is on Thursday 27 May from 13:00.

Revision cards

Say the answer first, then flip the card to check it.

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