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A uniform beam AB has mass 12 kg and length 3 m - Edexcel - A-Level Maths: Mechanics - Question 6 - 2005 - Paper 1
Question 6
A uniform beam AB has mass 12 kg and length 3 m. The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at end A, the... show full transcript
Worked Solution & Example Answer:A uniform beam AB has mass 12 kg and length 3 m - Edexcel - A-Level Maths: Mechanics - Question 6 - 2005 - Paper 1
Step 1
Find the reaction on the beam at C.
Answer
To find the reaction at point C, we can use the principle of moments. Consider the beam balanced about point A.
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Calculate the weight of the beam:
Weight of beam = mass × gravity = 12 kg × 9.81 m/s² ≈ 117.72 N.
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Set up the moment equation about point A:
Moment due to the beam's weight about A = Weight of beam × distance from A to center of beam = 117.72 N × 1.5 m.
Moment due to reaction R at point C = R × 2 m.
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At equilibrium, moments about A must balance, so:
117.72 N × 1.5 m = R × 2 m.
R = ( \frac{117.72 \times 1.5}{2} )
Therefore:
R ≈ 88.29 N.
Step 2
Find the distance AD.
Answer
To find the distance AD when the woman is standing on the beam at point D, we can set up the balance of moments again, knowing that the reactions at A and C are equal (S).
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Set the equations for moments again about point A:
The total moment at A is now influenced by the weight of the woman and the beam.
S × 2 = 48 kg × 9.81 m/s² + 12 kg × 9.81 m/s².
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Calculate S:
S = ( \frac{48 kg × 9.81 + 12 kg × 9.81}{2} )
S = ( \frac{48 × 9.81 + 12 × 9.81}{2} ) = 30 N.
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Using the newly found value of S, set up another equation to find distance AD:
30 N × 2 = 12 kg × 9.81 × 1.5 + 48 kg × 9.81 × x,
where x is the distance from A to D. Solving for x gives:
x = 0.88 m or 0.875 m.