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A bench consists of a plank which is resting in a horizontal position on two thin vertical legs - Edexcel - A-Level Maths Mechanics - Question 4 - 2009 - Paper 1
Question 4
A bench consists of a plank which is resting in a horizontal position on two thin vertical legs. The plank is modelled as a uniform rod $PS$ of length 2.4 m and mass... show full transcript
Worked Solution & Example Answer:A bench consists of a plank which is resting in a horizontal position on two thin vertical legs - Edexcel - A-Level Maths Mechanics - Question 4 - 2009 - Paper 1
Step 1
Part (a): the magnitude of the normal reaction between the plank and the leg at Q and the magnitude of the normal reaction between the plank and the leg at R
Answer
To find the magnitudes of the normal reactions at points and , we begin by applying the equilibrium conditions for forces and moments on the plank.
Let:
- = normal reaction at
- = normal reaction at
Step 1: Sum of forces The total downward force due to weights is given by: (where weight of the plank plus weights of Arthur and Beatrice equals 120N)
Step 2: Moments about one end of the plank
Taking moments about point :
This simplifies to (), which can be solved to yield:
Step 3: Solving for C: Substituting back into the first equation gives:
ightarrow C = 90 ext{N}$$ The magnitudes of the normal reactions are: - $C = 90 ext{N}$ - $D = 30 ext{N}$Step 2
Part (b): find the distance QX
Answer
After Beatrice remains at , Arthur will now sit at point , and we know that the normal reaction at is now twice that of .
Let (the normal reaction at ).
Step 1: New upward force equation
Thus:
This gives:
and therefore:
.
Step 2: New moments about point Q:
Solving this will give:
On simplifying, we find: x = rac{16}{15} ext{ m} ext{ (or approximately 1.07 m)}
Therefore, the distance is approximately 1.07 m.