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The diagram is a representation of a robotic arm that can move in a vertical plane - Leaving Cert Mathematics - Question 8 - 2012
Question 8
The diagram is a representation of a robotic arm that can move in a vertical plane. The point P is fixed, and so are the lengths of the two segments of the arm. The ... show full transcript
Worked Solution & Example Answer:The diagram is a representation of a robotic arm that can move in a vertical plane - Leaving Cert Mathematics - Question 8 - 2012
Step 1
Determine the angles α and β from given distances
Answer
To find the angles α and β, we can use the cosine law in triangle PQR.
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Calculate the distance |PR|:
|PR|^2 = |PQ|^2 + |QR|^2 - 2 |PQ| |QR| imes ext{cos}(eta)
Given |PQ| = 20 cm, |QR| = 12 cm, and |PR| as the hypotenuse:
Substitute the known values:
|PR|^2 = 20^2 + 12^2 - 2 (20)(12) imes ext{cos}(eta)
This will allow us to compute β when |PR| is known.
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Set the equations based on the coordinates of point R (24 cm right and 7 cm higher than P):
Using the formula for vertical and horizontal components:
Calculate γ using inverse tan:
→ approximately 16.26°.
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Then find angle α using the sine rule or further cosine law calculations, leading to:
α ≈ 44°, β ≈ 100°.
Step 2
Error analysis for positions of α and β
Answer
To determine which angle affects the position of R more:
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Analyze the arcs resulting from a 1° error:
- A 1° error in α causes R to move along an arc of radius 25 cm.
- A 1° error in β causes R to move along an arc of radius 12 cm.
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Since the distance moved due to error is proportional to the radius:
- Therefore, a 1° error in α creates a greater distance change for R compared to β.
- Thus, it can be concluded that the error in α has a greater impact on R's position.
Step 3
Conditions affecting sensitivity to errors in α and β
Answer
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R is more sensitive to errors in α when |PR| > 12.
- This occurs when β × cos(5/6) > 12.
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R is more sensitive to errors in β when |PR| < 12.
- In this case, the adjustment impacts the smaller radius more significantly.
- The borderline case occurs when ∆PQR becomes isosceles with |QR| = |RP|, leading to equal sensitivity.