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 other at a point C on the beam, where BC = 1 m, as shown in Figure 3. The beam is modelled as a uniform rod. (a) Find the reaction on the beam at C. (3) A woman of mass 48 kg stands on the beam at the point D. The beam remains in equilibrium. The reactions on the beam at A and C are now equal. (b) Find the distance AD. (7)

A-Level Edexcel Maths Mechanics Forces: A uniform beam AB has mass 12 kg

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.

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Answer

To find the reaction force at point C, we can use the principles of moments about point A.

Let the reaction at point C be denoted as R. The weight of the beam can be calculated as:

$ext{Weight of beam} = mg = 12 ext{ kg} imes 9.81 ext{ m/s}^2 = 117.72 ext{ N}$

Taking moments about A (considering clockwise moments as positive):

$R imes 2 ext{ m} = 117.72 ext{ N} imes 1.5 ext{ m}$

Solving for R:

$R = \frac{117.72 ext{ N} imes 1.5 ext{ m}}{2 ext{ m}} = 88.29 ext{ N}$

Thus, the reaction on the beam at C is approximately 88.3 N.

Step 2

Find the distance AD.

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Answer

For the second part of the question, when the woman stands at point D, the total reaction at C (denote as S) and the reaction at A (denote as R) are equal:

$S = R = 88.29 ext{ N}$

Considering the total downward forces:

$S imes 2 = 48 ext{ kg} imes 9.81 ext{ m/s}^2 + 12 ext{ kg} imes 9.81 ext{ m/s}^2$

Calculating the total weight:

$48 ext{ kg} imes 9.81 ext{ m/s}^2 + 12 ext{ kg} imes 9.81 ext{ m/s}^2 = 588.6 ext{ N}$

Setting up the equation:

$2S = 588.6 ext{ N}$

So:

$S = \frac{588.6}{2} = 294.3 ext{ N}$

Next, we need to find the position of D such that the moments balance:

$S imes x = Weight imes \frac{3}{2}$

Substituting S:

$294.3 imes x = 117.72 imes 1.5$

We can solve for x:

$x = \frac{117.72 imes 1.5}{294.3} \approx 0.6 ext{ m}$

Thus, the distance AD is approximately 0.6 m.

A uniform beam AB has mass 12 kg and length 3 m - Edexcel - A-Level Maths Mechanics - Question 6 - 2005 - Paper 1

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

Find the reaction on the beam at C.

Find the distance AD.

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