A uniform rod AB has length 1.5 m and mass 8 kg. A particle of mass m kg is attached to the rod at B. The rod is supported at the point C, where AC = 0.9 m, and the system is in equilibrium with AB horizontal, as shown in Figure 2. (a) Show that m = 2. (b) Find the distance AD.

A-Level Edexcel Maths Mechanics Moments: A uniform rod AB has length 1.5

A uniform rod AB has length 1.5 m and mass 8 kg - Edexcel - A-Level Maths Mechanics - Question 3 - 2007 - Paper 1

Question 3

A uniform rod AB has length 1.5 m and mass 8 kg. A particle of mass m kg is attached to the rod at B. The rod is supported at the point C, where AC = 0.9 m, and the ... show full transcript

Worked Solution & Example Answer:A uniform rod AB has length 1.5 m and mass 8 kg - Edexcel - A-Level Maths Mechanics - Question 3 - 2007 - Paper 1

Step 1

Show that m = 2.

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Answer

To prove that $m = 2$ , we start by analyzing the torque about point C. The rod's weight acts at its center of mass, which is at 0.75 m from A. Thus, the torque due to the rod is:

$ext{Torque}_{ ext{rod}} = 8g imes (0.9 - 0.75) = 8g imes 0.15$

For the particle at B, the torque is given by:

$ext{Torque}_{ ext{particle}} = mg imes (1.5 - 0.9) = mg imes 0.6$

Setting the counterclockwise torque equal to the clockwise torque for equilibrium, we have:

$8g imes 0.15 = mg imes 0.6$

Simplifying, we find:

$1.2 = 0.6m$

Solving for m gives:

$m = rac{1.2}{0.6} = 2.$

Step 2

Find the distance AD.

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Answer

To find the distance AD, we need to set up the moments around point D.

Let the distance between point A and D be $x$ . Then, the distance from D to B is $(1.5 - x)$ .

The moment about point D will be:

$M(D) : 5g imes x = 8g imes (0.75 - x) + 2g imes (1.5 - x)$

Substituting the known values:

$5g imes x = 8g imes (0.75 - x) + 2g imes (1.5 - x)$

After canceling out $g$ , we get:

$5x = 8(0.75 - x) + 2(1.5 - x)$

Expanding and rearranging gives:

$5x = 6 + 0.5x$

This simplifies to:

$5x - 0.5x = 6$

Thus,

$4.5x = 6 \\ x = rac{6}{4.5} = rac{4}{3} = 0.6 ext{ m}.$

Therefore, the distance AD is:

$AD = x = 0.6 ext{ m}.$

A uniform rod AB has length 1.5 m and mass 8 kg - Edexcel - A-Level Maths Mechanics - Question 3 - 2007 - Paper 1

Worked Solution & Example Answer:A uniform rod AB has length 1.5 m and mass 8 kg - Edexcel - A-Level Maths Mechanics - Question 3 - 2007 - Paper 1

Show that m = 2.

Find the distance AD.

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