A metal rod, of mass m kilograms and length 20 cm, lies at rest on a horizontal shelf. The end of the rod, B, extends 6 cm beyond the edge of the shelf, A, as shown in the diagram below. 20 cm Shelf A B 6 cm 14 (a) The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of AB. Show that m = 0.21 [3 marks] 14 (b) The object of mass 0.28 grams is removed. A number, n, of identical objects, each of mass 0.048 kg, are hung from the rod at all a distance of 1 cm from B. Find the maximum value of n such that the rod remains horizontal. [4 marks] 14 (c) State one assumption you have made about the rod. [1 mark]

A-Level AQA Maths Pure Vectors in 2 Dimensions: A metal rod, of mass m kilograms

A metal rod, of mass m kilograms and length 20 cm, lies at rest on a horizontal shelf - AQA - A-Level Maths Pure - Question 14 - 2019 - Paper 2

Question 14

A metal rod, of mass m kilograms and length 20 cm, lies at rest on a horizontal shelf. The end of the rod, B, extends 6 cm beyond the edge of the shelf, A, as shown... show full transcript

Worked Solution & Example Answer:A metal rod, of mass m kilograms and length 20 cm, lies at rest on a horizontal shelf - AQA - A-Level Maths Pure - Question 14 - 2019 - Paper 2

Step 1

The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of AB.

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Answer

To show that the mass of the rod, m, equals 0.21 kg, we can use the principle of moments. We take moments about point A.

The moment caused by the weight of the hanging mass, which is 0.28 kg, acting 0.03 m from A (the midpoint of AB), is:

$0.28 ext{ kg} imes 0.03 ext{ m} = 0.0084 ext{ kg m}$

Let the distance from A to B be 0.06 m, thus the moment due to the weight of the rod is:

$mg imes 0.06 ext{ m}$

Setting these moments equal gives:

$mg imes 0.06 = 0.0084$ $m = rac{0.0084}{0.06} = 0.14 ext{ kg}$

Solving for m gives:

$m = 0.21$ .

Step 2

Find the maximum value of n such that the rod remains horizontal.

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Answer

When the object of mass 0.28 has been removed, we must find a new balance point with identical objects of mass 0.048 kg at a distance of 0.01 m from B.

Taking moments about point A:

The total moment produced by n objects is:

$n imes 0.048 ext{ kg} imes 0.01 ext{ m}$

And the moment due to the weight of the rod is:

$0.06 ext{ m} imes 0.21 ext{ kg} = 0.0126 ext{ kg m}$

Setting the moments equal:

$0.0126 = n imes 0.048 imes 0.01$

Solving for n:

$n = rac{0.0126}{0.00048} = 26.25$ .

Since n must be a whole number, the maximum value of n is 3.

Step 3

State one assumption you have made about the rod.

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Answer

The assumption made about the rod is that it is rigid.

A metal rod, of mass m kilograms and length 20 cm, lies at rest on a horizontal shelf - AQA - A-Level Maths Pure - Question 14 - 2019 - Paper 2

Worked Solution & Example Answer:A metal rod, of mass m kilograms and length 20 cm, lies at rest on a horizontal shelf - AQA - A-Level Maths Pure - Question 14 - 2019 - Paper 2

The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of AB.

Find the maximum value of n such that the rod remains horizontal.

State one assumption you have made about the rod.

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