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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

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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 [... show full transcript

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

Step 1

Find |∠COD|, correct to the nearest degree.

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Answer

To find |∠COD|, we can use the sine rule. We know the following lengths:

  • [OD] = 10 cm
  • [DC] = 30 cm
    Using the angle |∠DCO| = 15°, we apply the sine rule:

10sin⁡(15°)=30sin⁡(x)\frac{10}{\sin(15°)} = \frac{30}{\sin(x)}

Rearranging gives:

sin⁡(x)=30⋅sin⁡(15°)10\sin(x) = \frac{30 \cdot \sin(15°)}{10}
Calculating this, we find:
( \sin(x) = 3 \cdot \sin(15°) = 0.77645 )
Thus,
( x = \sin^{-1}(0.77645) \approx 51° ) Therefore, |∠COD| = 51°.

Step 2

Write down the period and range of f.

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Answer

The function f(α) describes the distance |CX| as D moves around O, which is periodic due to its circular motion. The period of the function is:

Period=2π rad\text{Period} = 2\pi \, \text{rad}

In degrees, this corresponds to 360°. The range of f(α) is the minimum and maximum values of |CX| as D completes its circular path. Given the movement described, the range is:

Range=[10,30] cm\text{Range} = [10, 30] \text{ cm}

Step 3

Complete the table below for f(α).

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Answer

Using the known lengths in the mechanism,

α (°)090180270360
f(α) (cm)3018.281018.2830

Step 4

Draw your graph on this grid.

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Answer

The graph of f(α) will oscillate between the range of 10 cm to 30 cm, showing periodic behavior as D moves around O. The x-axis represents α, while the y-axis represents |CX|. Start at (0, 30), peak at (90, 18.28), minimum at (180, 10), peak again at (270, 18.28), and return to (360, 30).

Step 5

For which of the three positions of the mechanism will a 1 degree change in α cause the greatest change in the position of C?

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Answer

Referring to the steepness of the graph of f(α) at the specified angles,

  • Diagram 2 is likely to give the greatest change as |∠DCO| is smaller and thus a small change in α will yield a larger distance change in |CX| due to the geometry of the connecting rod. Specifically, when the angle is closer to 90°, the change in position due to a small change in angle is maximized.

Step 6

Find r, the length of the crank.

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Answer

To find the length of the crank (r), we can use the cosine rule as follows:

r2=362+(31+r)2−2(36)(31)(cos⁡(10°))r^2 = 36^2 + (31 + r)^2 - 2(36)(31)(\cos(10°))

This simplifies to:

r2=1296+961+62r+r2−(2232⋅0.9848)r^2 = 1296 + 961 + 62r + r^2 - (2232 \cdot 0.9848)

After rearranging and solving: 0=6r−91⇒r≈70 = 6r - 91 \Rightarrow r \approx 7 Thus, the length of the crank is approximately 7 cm.

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