13. A uniform rod, AB, has length 7 metres and mass 4 kilograms. The rod rests on a single fixed pivot point, C, where AC = 2 metres. A particle of weight W newtons is fixed at A, as shown in the diagram. The system is in equilibrium with the rod resting horizontally. (a) Find W, giving your answer in terms of ... (b) Explain that the weight of the rod acts through the midpoint or the centre of mass at the midpoint (of the rod) OE.

A-Level AQA Maths Pure Vectors in 2 Dimensions: 13. A uniform rod, AB, has lengt

13. A uniform rod, AB, has length 7 metres and mass 4 kilograms - AQA - A-Level Maths Pure - Question 13 - 2020 - Paper 2

Question 13

13. A uniform rod, AB, has length 7 metres and mass 4 kilograms. The rod rests on a single fixed pivot point, C, where AC = 2 metres. A particle of weight W newton... show full transcript

Worked Solution & Example Answer:13. A uniform rod, AB, has length 7 metres and mass 4 kilograms - AQA - A-Level Maths Pure - Question 13 - 2020 - Paper 2

Step 1

Find W, giving your answer in terms of ...

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Answer

To find the weight W, we will use the principle of moments. The moment about point C due to the weight W is given by:

$ext{Moment} = W imes AC = W imes 2 \text{ m}$

The centre of mass of the rod, which is a uniform rod, will be at the midpoint, which is 3.5 m from point C. The weight of the rod can be calculated as:

$ext{Weight of rod} = ext{mass} \times g = 4 \text{ kg} \times 9.81 \text{ m/s}^2 = 39.24 \text{ N}$

The moment about point C due to the weight of the rod acting downward is:

$ext{Moment} = ext{Weight of rod} \times \text{distance to C} = 39.24 \times 3.5$

Setting the moments equal for equilibrium:

$W \times 2 = 39.24 \times 3.5$

Solving for W gives us:

$W = \frac{39.24 \times 3.5}{2} = 68.37 \text{ N}$

Step 2

Explain that the weight of the rod acts through the midpoint or the centre of mass at the midpoint (of the rod) OE.

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Answer

The weight of a uniform rod acts through its centre of mass, which is located at the midpoint of the rod when the rod is uniform. For a rod of length 7 m, the centre of mass is found at:

$p = \frac{7}{2} = 3.5 \text{ m from A}$

Thus, considering the entire rod is uniform, we can conclude that the weight of the rod acts vertically downward through this midpoint (the midsection of the rod).

13. A uniform rod, AB, has length 7 metres and mass 4 kilograms - AQA - A-Level Maths Pure - Question 13 - 2020 - Paper 2

Worked Solution & Example Answer:13. A uniform rod, AB, has length 7 metres and mass 4 kilograms - AQA - A-Level Maths Pure - Question 13 - 2020 - Paper 2

Find W, giving your answer in terms of ...

Explain that the weight of the rod acts through the midpoint or the centre of mass at the midpoint (of the rod) OE.

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