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. The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of AB. Show that $m = 0.21$. The object of mass 0.28 kilograms is removed. A number, $n$, of identical objects, each of mass 0.048 kg, are hung from the rod at all of a distance of 1 cm from B. Find the maximum value of $n$ such that the rod remains horizontal. State one assumption you have made about the rod.

A-Level AQA Maths Mechanics Moments: A metal rod, of mass $m$ kilogra

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 - AQA - A-Level Maths Mechanics - 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 show... 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: The end of the rod, B, extends 6 cm beyond the edge of the shelf, A, as shown in the diagram below - AQA - A-Level Maths Mechanics - Question 14 - 2019 - Paper 2

Step 1

Show that the rod is in equilibrium

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Answer

To determine the mass of the rod, we will take moments about point A.

Let the distance from A to the midpoint of AB be 10 cm (0.1 m). Therefore, the distance from A to B is 0.06 m.
The moment caused by the weight of the hanging mass (0.28 kg) is:

$0.28 imes 9.81 imes 0.03$ (since 0.03 m is the distance from the midpoint to A)
The moment caused by the weight of the rod (mass $m$ ) is:

$0.06 imes (0.21 imes 9.81)$
Setting the moments equal to maintain equilibrium gives:

$0.28 imes 9.81 imes 0.03 = 0.06 imes (0.21 imes 9.81)$
Simplifying:

$0.0084 imes 9.81 = 0.00126 imes 9.81$
Therefore, the mass of the rod, $m$ , is found to be:

$m = 0.21 ext{ kg}.$

Step 2

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

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Answer

Next, we analyze the situation where an object of mass 0.28 kg is removed, and we need to find the maximum number of objects $n$ of mass 0.048 kg that can be added.

Taking moments about point A again, we have:

$0.21 imes 9.81 imes 0.06 = 0.048 imes n imes 0.01$
This simplifies to:

$0.00126 = 0.00048n$
Solving for $n$ :

$n = rac{0.00126}{0.00048} = 2.625$
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

One assumption made about the rod is that the rod is uniform, meaning its weight is distributed evenly along its length.

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 - AQA - A-Level Maths Mechanics - 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: The end of the rod, B, extends 6 cm beyond the edge of the shelf, A, as shown in the diagram below - AQA - A-Level Maths Mechanics - Question 14 - 2019 - Paper 2

Show that the rod is in equilibrium

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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