A uniform plank AB has mass 40 kg and length 4 m. It is supported in a horizontal position by two smooth pivots, one at the end A, the other at the point C of the plank where AC = 3 m, as shown in Fig. 1. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to R newtons. By modelling the plank as a rod and the man as a particle, find a) the value of R, b) the distance of the man from A.

A-Level Edexcel Maths Mechanics Moments: A uniform plank AB has mass 40 k

A uniform plank AB has mass 40 kg and length 4 m - Edexcel - A-Level Maths Mechanics - Question 1 - 2003 - Paper 1

Question 1

A uniform plank AB has mass 40 kg and length 4 m. It is supported in a horizontal position by two smooth pivots, one at the end A, the other at the point C of the pl... show full transcript

Worked Solution & Example Answer:A uniform plank AB has mass 40 kg and length 4 m - Edexcel - A-Level Maths Mechanics - Question 1 - 2003 - Paper 1

Step 1

a) the value of R

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Answer

To find the value of R, we can set up the equilibrium condition for vertical forces. The total weight on the plank consists of the weight of the plank and the weight of the man:

Distribution of weights: The weight of the plank is given by its mass multiplied by the gravitational acceleration (g):

$F_{plank} = 40 ext{ kg} imes g$

The weight of the man:

$F_{man} = 80 ext{ kg} imes g$
Summing the forces in the vertical direction:

$R_{A} + R_{C} = F_{plank} + F_{man}$

Where R_A and R_C are the reactions at the pivots A and C, respectively.
Since both reactions are equal: $2R = 80g + 40g$

Substituting the values:

$2R = 120g$

Solving for R:

$R = 60g$

Assuming g = 9.8 m/s², we can calculate R:

$R = 60 imes 9.8 ext{ N} = 588 ext{ N}$

Step 2

b) the distance of the man from A

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Answer

To find the distance of the man from A, we can use the principle of moments (torque). The sum of moments about point A must equal zero for equilibrium:

The moment due to the weight of the man around A is:

$M(man) = 80g imes x$

where x is the distance from A to the man.
The moment due to the weight of the plank is:

$M(plank) = 40g imes 2$

(as the center of mass of the plank is at 2 m from A).
Setting up the equation for moments:

$80g imes x = 40g imes 2$

Simplifying:

ightarrow x = 1.5 ext{ m}$$

Therefore, the distance of the man from A is 1.5 m.

A uniform plank AB has mass 40 kg and length 4 m - Edexcel - A-Level Maths Mechanics - Question 1 - 2003 - Paper 1

Worked Solution & Example Answer:A uniform plank AB has mass 40 kg and length 4 m - Edexcel - A-Level Maths Mechanics - Question 1 - 2003 - Paper 1

a) the value of R

b) the distance of the man from A

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