Leaving Cert Mathematics Functions: In engineering, a crank-and-slid

Question

In engineering, a crank-and-slider mechanism can be used to change circular motion into motion back and forth in a straight line.

In the diagrams below, the crank [OD] rotates about the fixed point [O]. The point [C] slides back and forth in a horizontal line. [CD] is the rod that connects [C] to the crank. The diagrams below show three of the possible positions for [C] and [D].

**(a)** The diagram below shows a particular position of the mechanism with $ \angle LCD = 15^\circ $. Find $ LCO $, correct to the nearest degree.

**(b)** As [D] moves in a circle around [O], the angle $ \alpha $ in the diagram below increases. The distance $ |CX| $ can be considered to be a function of $ \alpha $ and written as $ f(\alpha) $.

(i) Write down the period and range of $ f $.

**Period =**
**Range =**

(ii) Complete the table below for $ f(\alpha) $.
Give your answers correct to 2 decimal places where appropriate.  
(Note: Diagram 1 at the start of this question represents $ \alpha = 0^\circ $).  
 $ \alpha $   $ 0^\circ $   $ 90^\circ $   $ 180^\circ $   $ 270^\circ $   $ 360^\circ $
 $ f(\alpha) $   $ (cm) $
 $ 30 $   $ 18.28 $   $ 10 $   $ 18.28 $   $ 30 $

(iii) Use your values from the table to draw a rough sketch of $ f $ in the domain $ 0^\circ \leq \alpha \leq 360^\circ $.

Draw your graph on this grid.

This question continues on the next page.

(iv) Referring to Diagrams 1, 2, and 3 near the start of this question, for which of the three positions of the mechanism will a 1 degree change in $ \alpha $ cause the greatest change in the position of [C]? Explain your answer.

**(c)** The diagram below shows another crank-and-slider mechanism with different dimensions. In the diagram, $ |AB| = 36 $ cm, $ |AX| = 31 $ cm, and $ \angle BAO = 10^\circ $. (Note: $ \angle OBA \neq 90^\circ $)
Find $ r $, the length of the crank. Give your answer in cm, correct to the nearest cm.

In engineering, a crank-and-slider mechanism can be used to change circular motion into motion back and forth in a straight line - Leaving Cert Mathematics - Question 9 - 2018

Worked Solution & Example Answer:In engineering, a crank-and-slider mechanism can be used to change circular motion into motion back and forth in a straight line - Leaving Cert Mathematics - Question 9 - 2018

Find \( LCO \), correct to the nearest degree.

Write down the period and range of \( f \).

Complete the table below for \( f(\alpha) \).

Draw a rough sketch of \( f \) in the domain \( 0^\circ \leq \alpha \leq 360^\circ \).

for which of the three positions of the mechanism will a 1 degree change in \( \alpha \) cause the greatest change in the position of [C]? Explain your answer.

Find \( r \), correct to the nearest cm.

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