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# Maths on track! - STEPS | Engineers Ireland

Maths on track! - STEPS | Engineers Ireland

## Maths on track! - STEPS | Engineers

Series Editor: Patrick Johnson MATHEMATICS WORKSHEETS FOR SCHOOLS Volume 1: Number 7 a. Mathematical/Scientific Topics Covered: i Geometry (Similar Triangles Intersecting Chords) ii Manipulation of Algebraic symbols iii Equations of Circular Motion iv Resolution of Vectors v Newtonʼs Laws vi Newtonʼs Equations of Linear Motion vii Relation 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 million passengers, whose safety is assured by the use of mathematics and engineering in the design and construction of rail track, bridges and other essential transport infrastructure. ong>Mathsong> 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 action can wear these down and the risk of overturning is always present. To counteract these forces one 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 one 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 consider 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, amongst 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 one 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 Johnson; Writing Team Tim Brophy, Barry Fitzgerald & Patrick Johnson. www.steps.ie www.nce-mstl.ie Mathematical Information If the three angles in one triangle are equal in measure to the three angles in a second triangle then the triangles are similar and the sides are in the same proportion. If two chords, CD and EF, of a circle intersect at the point G as in the diagram then |CG|.|GD| = |EG|.|GF| Physical Information c E From Newton’s first law we know that an object continues in a state of rest or of uniform rectilinear motion unless it is acted upon 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 section, we are going to derive the equation for centripetal force. F G D

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