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

Find the Moments about Point B

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Answer

To analyze the moments about point B, we can set up the equation:

$500x + 500 \cdot 2x + 200 \cdot 3 = R \cdot 5 + S \cdot 1$

Here, $x$ is the distance from the end B to Tom, and the distances for Sophie and the beam's weight are accordingly calculated.

Step 2

Write the Equation for Vertical Forces

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Answer

Next, the equilibrium of vertical forces gives:

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

This allows us to express the relationship between the reactions at the supports.

Step 3

Solve for Tom's Distance

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Answer

Now we solve the equations of moments and vertical forces to find $x$ :

From the moments about B: $5R + 2S = 1200$

And since $R + S = 1200$ , substituting values will help us calculate $x$ . Solving gives: $x = 1.2 \text{ m}$

Thus, Tom is standing 1.2 meters from the end B.

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