A beam AB has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points C and D, where AC = 1 m and DB = 1 m. Two children, Sophie and Tom, each of weight 500 N, stand on the beam with Sophie standing twice as far from the end B as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at D is three times the magnitude of the reaction at C. By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end B.

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A beam AB has length 6 m and weight 200 N - Edexcel - A-Level Maths Mechanics - Question 4 - 2010 - Paper 1

Question 4

A beam AB has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points C and D, where AC = 1 m and DB = 1 m. Two children, ... show full transcript

Worked Solution & Example Answer:A beam AB has length 6 m and weight 200 N - Edexcel - A-Level Maths Mechanics - Question 4 - 2010 - Paper 1

Step 1

Calculate Moments about Point B

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Answer

We will take moments about point B. Let the distance from end B to Tom be denoted by x m. The moment due to Tom is given by:

$500x$

The moment due to Sophie, who is standing twice as far from B, will be:

$500(2x) = 1000x$

The moment due to the weight of the beam (200 N), which acts at its center (3 m from B) is:

$200 imes 3 = 600$

In equilibrium, the sum of moments about point B must equal zero, thus:

$1000x + 500x + 600 = R imes 5$

Step 2

Establish the Vertical Force Equilibrium

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Answer

For vertical force equilibrium, the total up forces must equal the total down forces. Let R be the reaction at C and S be the reaction at D:

$R + S = 500 + 500 + 200 = 1200$

We also know that the reaction at D (S) is three times that at C (R), thus:

$S = 3R$ .

Substituting into the vertical force equilibrium equation gives:

ightarrow 4R = 1200 ightarrow R = 300$$ From this, we can find S: $$S = 3 imes 300 = 900$$

Step 3

Solve for x

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Answer

Substituting R into the moment equation we derived previously:

$1000x + 500x + 600 = 300 imes 5$

This simplifies to:

$1500x + 600 = 1500$

Thus,

$1500x = 1500 - 600$

$1500x = 900$

Dividing both sides by 1500 gives:

$x = rac{900}{1500} = 0.6 m$

Finally, to find how far Tom is from point B, we can conclude:

Tom is standing 0.6 m from the end B.

A beam AB has length 6 m and weight 200 N - Edexcel - A-Level Maths Mechanics - Question 4 - 2010 - Paper 1

Worked Solution & Example Answer:A beam AB has length 6 m and weight 200 N - Edexcel - A-Level Maths Mechanics - Question 4 - 2010 - Paper 1

Calculate Moments about Point B

Establish the Vertical Force Equilibrium

Solve for x

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