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From ZuluNotes - Free Leaving Cert Notes

Introduction

Applied Mathematics - more commonly known at third level as Mathematical Physics - is concerned with the mathematical side of Physics. Whereas the subject of Physics focuses primarily on theory, the Applied Mathematics course is aimed at providing for students who choose to study it, a better understanding of the computational, analytical and numerical side of Physics problems.

The course consists of 12 topics in total, including all the Honours Level topics. The examination consists of one paper and is 2.5 hours long. The exam paper contains 10 questions at Honours Level and 9 at Ordinary Level, with students answering 6 questions in both cases, spending about 25 minutes on each question. The exam is generally at the end of the Leaving Certificate, and in 2007 it will be the very final exam along with Agricultural Science on the 22nd of June finishing at 4.30pm.

Applied Mathematics is considered to be one of the most difficult of all the subjects offered in the Leaving Certificate. However, approximately 1450 students sit the examination each year, with a staggering 92% of all students who choose to sit it, doing so at the Honours Level^[1].

Since there is a high amount of courses at third level that study Applied Mathematics (or Mathematical Physics) extensively, there is a large over-lap in terms of the content on the Leaving Certificate and a typical first year introductory course at third level. Therefore students hoping to study engineering in particular, find it very beneficial to study Applied Mathematics not only as a "pre-introductory" course but also to familiarise themselves with the difficult mathematics that it involves.

Course Structure

The following are the topics that should be studied by students for the Leaving Certificate course, as layed out by the Department Of Education (Honours Level topics are italicised) :

• Motion of a particle which includes:
  • Displacement and velocity as vectors.
  • Application of the Vector Addition Law.
  • Description of vectors in terms of unit perpendicular vectors.
  • Elementary treatment of Relative Motion.

• Newton's Laws which includes:
  • Mass, Momentum and Acceleration
  • Force as vectors.
  • Units and dimensions.

• Motion in a straight line under uniform acceleration which includes:
  • (Includes, but is not exclusive to) Motion under gravity and motion on smooth and rough inclined planes.
  • Work, Potential energy, Kinetic Energy and Power.
  • Application of energy conservation.
  • Motion of connected particles.

• Equilibrium of particles under concurrent forces which includes:
  • Situations where frictional forces must be taken into account.

• Centre of gravity of simple bodies and systems of particles which includes:
  • Moments/Couples in the system.
  • Equilibrium of a rigid body.
  • Equilibrium of rigid bodies.

• Liquid Pressure which includes:
  • Thrust on a horizontal surface.
  • Archimedes Principle.

• Projectiles which includes:
  • Projectiles on the inclined plane.

• Circular Motion which includes:
  • Angular Velocity
  • Circular Orbits.
  • The conical pendulum.

• Conservation Of Momentum which includes:
  • Elastic/Inelastic direct collisions.
  • Oblique Collisions between two smooth elastic spheres in two dimensions.

• Simple Harmonic Motion of a particle in a straight line which includes:
  • Applications to a simple pendulum.

• Motion of a rigid body about a fixed axis which includes:
  • Calculations of moments of inertia for a rod or rectangular/circular lamina, or thereby formed compound bodies. (Note that this does not include spheres)
  • Application of the Parallel Axis Theorem/Perpendicular Axis Theorem.
  • The concept of Radius Of Gyration.
  • Application of the Conservation Of Energy to a rotating body.
  • Application of the Principle Of Angular Momentum.
  • Simple applications of the compound pendulum.

• Ordinary Differential Equations (ODE's) and their applications which includes:
  • First order and second order ODE's with separable variables, or equations that can be reduced to first order or second order ODE's.
  • Applied equations to Power, Time, Forces or Distance.

Honours/Ordinary Level Differentiation

At Ordinary level all topics are studied except for:

• Simple Harmonic Motion.
• Projectiles on the inclined plane.
• Oblique Collisions.
• Rigid Body Rotation.
• Connected Particles with relative accelerations.
• Differential Equations.
• Circular Motion on/around a vertical circle.

Exam Paper Structure

• Paper 1 (Ordinary Level): (Students answer 6 out of 9 questions)

1. Linear Motion
2. Relative Motion
3. Projectiles
4. Connected Particles
5. Collisions
6. Centre Of Gravity
7. Statics
8. Circular Motion
9. Hydrostatics

• Paper 1 (Honours Level): (Students answer 6 out of 10 questions)

1. Linear Motion
2. Relative Motion
3. Projectiles
4. Connected Particles
5. Collisions
6. Circular & Simple Harmonic Motion
7. Statics
8. Rigid Body Motion
9. Hydrostatics
10. Differential Equations

Solutions

Higher Level

2007 Paper

Notes