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 Forces: 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

the value of R

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Answer

To find the value of R, we first consider the forces acting on the plank. The total downward force acting on the system is the combined weight of the plank and the man:

$ext{Total Downward Force} = 80g + 40g = 120g$

Using the equilibrium conditions, we can set the sum of forces in the vertical direction to zero:

$R_A + R_B = 120g$

Taking moments about point A (considering clockwise moments positive), we have:

Moment due to the man (80 kg at 2 m from A): $80g imes 2$
Moment due to the plank's weight (40 kg at its center of gravity, which is 2 m from A): $40g imes 2$

Setting the total moments equal to the moments around A:

$R imes 4 = 80g imes 2 + 40g imes 2$ $R imes 4 = 160g + 80g = 240g$

Now, substituting for the total weight, we find:

$R = \frac{240g}{4} = 60g$

With $g$ typically approximated as $9.8 ext{ m/s}^2$ , the value of R is:

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

Step 2

the distance of the man from A

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Answer

To calculate the distance of the man from point A, we use the moment balance equation:

Setting the total clockwise moments around A equal to the total counterclockwise moments:

Total weight's clockwise moments about A:

$80g imes x + 40g imes 2 = 60g imes 3$

Now substituting $g$ into the equation simplifies to (using $60g$ as R):

$80g imes x + 80g = 180g$

This simplifies to:

$80gx = 180g - 80g$ $80gx = 100g$

Dividing both sides by $80g$ gives:

$x = \frac{100}{80} ext{ m} = 1.25 ext{ m}$

Thus, the distance of the man from A is:

$x = 1.25 ext{ m}$

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

the value of R

the distance of the man from A

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