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A uniform rod AB has length 3 m and weight 120 N - Edexcel - A-Level Maths Mechanics - Question 6 - 2003 - Paper 1
Question 6
A uniform rod AB has length 3 m and weight 120 N. The rod rests in equilibrium in a horizontal position, smoothly supported at points C and D, where AC = 0.5 m and A... show full transcript
Worked Solution & Example Answer:A uniform rod AB has length 3 m and weight 120 N - Edexcel - A-Level Maths Mechanics - Question 6 - 2003 - Paper 1
Step 1
Show that $W = \frac{60}{1 - x}$
Answer
To solve for W, we consider moments about point A:
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The weight of the rod acts at its center, which is 1.5 m from point A. Therefore, the moment due to the weight of the rod is (120 \times 1.5).
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The reaction at point C, which we'll denote as R, must balance the moments:
[ 3R + 120 \times 1.5 = R \times 2 + 2R \times 1 ] (where 2R captures the moment arm for the reaction at D).
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This leads to:
[ 3R - R + 120 \times 1.5 = 0 ]
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Rearranging gives:
[ 3R + 180 - 3R = W ]
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Substituting W into the equation yields:
[ W(1 - x) = 60 ]
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Finally, we arrive at:
[ W = \frac{60}{1 - x} ]
Step 2
Hence deduce the range of possible values of x
Answer
From the equation (W = \frac{60}{1 - x}), we need to ensure that W is positive:
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For (W > 0), we require (1 - x > 0), which simplifies to (x < 1).
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Additionally, since W represents a weight, it must also be positive, implying (x > 0).
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Therefore, combining these inequalities, we deduce the range for x to be:
[ 0 < x < 1 ]