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The end A of a uniform rod AB, of length 2a and weight W, is freely hinged to a vertical wall - CIE - A-Level Further Maths - Question 11 - 2016 - Paper 1
Question 11
The end A of a uniform rod AB, of length 2a and weight W, is freely hinged to a vertical wall. The end B of the rod is attached to a light elastic string of natural ... show full transcript
Worked Solution & Example Answer:The end A of a uniform rod AB, of length 2a and weight W, is freely hinged to a vertical wall - CIE - A-Level Further Maths - Question 11 - 2016 - Paper 1
Step 1
Find the angle between AB and CA produced by θ
Answer
Denote the angle between AB and CA produced by θ and the midpoint of BC by E. Using the geometry of the setup, we identify that:
- Angle EAB (which is θ) is required as part of our calculations, for which we shall denote it as θ.
- In triangle ABC, we can express the relationships in terms of the lengths and trigonometric ratios.
Step 2
Find tension T in terms of W
Answer
We start by substituting the lengths:
- The length AB = 2a and AE = 2a sin(θ).
- The weight W acts vertically downward at point A.
Taking moments about A, we have: ext{T} imes rac{3}{2}a imes ext{sin } heta = W imes 2a
Solving this for tension T gives: T = rac{W imes 4}{3 imes ext{sin } heta}
Step 3
Find the magnitude of the reaction force at A
Answer
At point A, we can find the vertical force Y and horizontal force X:
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The vertical force Y can be expressed as:
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The horizontal force X can be found as the component of the tension acting horizontally:
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To find the magnitude of the reaction force at A: . Substitute the expressions for Y and X to get the final answer.