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Either 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
Either 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 ... show full transcript
Worked Solution & Example Answer:Either 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 $ ext{cos} heta$
Answer
To find , we can use the triangle ABC in the given diagram. Here, you notice that AC is the vertical length (2a) and AB is the diagonal created by the rod.
Using the cosine definition:
In triangle ABC:
- The length AC = 2a (vertical)
- The length AB is found using the Pythagorean theorem:
Solving gives us BC = \sqrt{( rac{1}{2}a)^2 - (2a)^2} leading to simplification that allows for calculating showing that:
Step 2
Find the tension in the string
Answer
Using Hooke’s Law, we can express the tension T in the string. The extension in the string can be derived from the lengths:
- Natural length = rac{3}{2} a
- Current length = the angled length from A to C (which can be calculated using lengths AC and BC).
The extension, therefore, is:
Using Hooke’s Law:
where k is the spring constant. Substituting known values gives the tension in the string in terms of W.
Step 3
Find the magnitude of the reaction force at the hinge
Answer
The forces at hinge A can be analyzed. Vertically, we can assume equilibrium:
- Sum of vertical forces = 0:
Horizontally:
- Sum of horizontal forces = 0:
Combining these two equations will provide the magnitude for the reaction force at the hinge.