A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m. A child of mass 40 kg stands on the plank at a distance of 2 m from P and a block of mass M kg is placed on the plank at the end R. The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at P is equal to the force exerted on the plank by the support at Q. By modelling the plank as a uniform rod, and the child and the block as particles, a) (i) find the magnitude of the force exerted on the plank by the support at P, (ii) find the value of M. b) State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.

A-Level Edexcel Maths Mechanics Forces: A plank PQR, of length 8 m and m

A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m - Edexcel - A-Level Maths Mechanics - Question 5 - 2011 - Paper 1

Question 5

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A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m. A child of mass 40 kg stands on th... show full transcript

Worked Solution & Example Answer:A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m - Edexcel - A-Level Maths Mechanics - Question 5 - 2011 - Paper 1

Step 1

find the magnitude of the force exerted on the plank by the support at P

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Answer

To find the force exerted at support P, we use the principle of moments about point Q. The total clockwise moments must equal the total counterclockwise moments.

Taking moments about Q:

Weight of the child (40 kg) at 2 m from P:

$ext{Moments} = 40 imes 9.81 imes 4$

Weight of the plank (20 kg) at its midpoint (4 m from P):

$ext{Moments} = 20 imes 9.81 imes 3$

Let X be the force at P:

$X imes 6 = (40 imes 9.81 imes 4) + (20 imes 9.81 imes 3)$

Solving for X gives:

$X = rac{(40 imes 9.81 imes 4) + (20 imes 9.81 imes 3)}{6}$

Calculating gives the total force at P.

Step 2

find the value of M

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Answer

Using the equilibrium condition, we know the forces at P and Q must be equal:

Let:

Force at Q = X
Weight of the block = M imes 9.81

Using the moment equation about P:

$X imes 2 = 20 imes 9.81 imes 4 + M imes 9.81 imes 6$

Substituting for X from above and solving for M results in:

$M = 4 ext{ kg}$ .

Step 3

State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.

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Answer

In the calculations, the child and the block are treated as point masses. This simplifies the analysis by assuming that all the weight acts at a single point (the center of mass). This is valid as the dimensions of the child and the block are negligible compared to the length of the plank.

A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m - Edexcel - A-Level Maths Mechanics - Question 5 - 2011 - Paper 1

Worked Solution & Example Answer:A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m - Edexcel - A-Level Maths Mechanics - Question 5 - 2011 - Paper 1

find the magnitude of the force exerted on the plank by the support at P

find the value of M

State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.

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