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Maths on track! - STEPS | Engineers Ireland

Series Editor: Patrick Johns**on** MATHEMATICS WORKSHEETS FOR SCHOOLS Volume 1: Number 7 a. Mathematical/Scientific Topics Covered: i Geometry (Similar Triangles Intersecting Chords) ii Manipulati**on** of Algebraic symbols iii Equati**on**s of Circular Moti**on** iv Resoluti**on** of Vectors v Newt**on**ʼs Laws vi Newt**on**ʼs Equati**on**s of Linear Moti**on** vii Relati**on** between Mass and Weight b. Knowledge Requirement Level: i Leaving Certificate Mathematics ii Leaving Certificate Applied Mathematics iii Leaving Certificate Physics Iarnród Éireann, also known as Irish Rail, is **Ireland**’s railway system operator. It was established **on** 2 February 1987, and is a subsidiary of Córas Iompair Éireann (CIÉ). It operates all internal intercity, commuter and freight railway services in the Republic of **Ireland**, and, jointly with Northern **Ireland** Railways, the Enterprise service between Dublin and Belfast. Every year Iarnród Éireann carries approximately 40 milli**on** passengers, whose safety is assured by the use of mathematics and engineering in the design and c**on**structi**on** of rail **track**, bridges and other essential transport infrastructure. **on** **track**! When a train enters a piece of curved **track** it is prevented from moving in a straight line by the wheel flanges. The forces in acti**on** can wear these down and the risk of overturning is always present. To counteract these forces **on**e rail is usually raised above another. This allows the force of gravity to help the train negotiate the curve. The term cant refers to the amount, measured in metres or millimetres, by which **on**e running rail of a curved **track** is raised above the other running rail. The cant is represented by the symbol E. As the forces depend **on** the cant and **on** the train’s velocity it is necessary to c**on**sider both of these when evaluating the safe velocity for the train. The cant is designed with a particular train speed in mind. This is called the ideal speed. The cant is calculated for a train of a given weight and speed. A train will not necessarily travel at the ideal speed as this speed depends, am**on**gst other factors, **on** the mass of the train and hence **on** the number of carriages it has. It will also be different for goods trains and passenger trains. This means that the cant may not be the ideal **on**e for the particular speed at which the train travels. This gives rise to the term cant deficiency. Cant deficiency is the difference between the desired working cant and the actual cant. It is represented by the symbol D. NCE-MSTL and **Engineers** **Ireland** Mathematics Worksheets for Schools — Vol. 1 #7 Series Ed. Patrick Johns**on**; Writing Team Tim Brophy, Barry Fitzgerald & Patrick Johns**on**. www.steps.ie www.nce-mstl.ie Mathematical Informati**on** If the three angles in **on**e triangle are equal in measure to the three angles in a sec**on**d triangle then the triangles are similar and the sides are in the same proporti**on**. If two chords, CD and EF, of a circle intersect at the point G as in the diagram then |CG|.|GD| = |EG|.|GF| Physical Informati**on** c E From Newt**on**’s first law we know that an object c**on**tinues in a state of rest or of uniform rectilinear moti**on** unless it is acted up**on** by a resultant force. Since an object that moves in a circle is clearly not moving in a straight line it must be subject to a resultant force. This force is called the centripetal force and is directed towards the centre of the circle. In this secti**on**, we are going to derive the equati**on** for centripetal force. F G D

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