Statics, Moments of Forces

Q. Statics, Moments of Forces.

Question 
A plank of wood AB has length 5.4m. It lies on a horizontal platform, with 1.2m projecting over the edge. When a girl of mass 50 kg stands at point C on the plank, where BC = 0.3 m, the plank is on the point of tilting. By modelling the plank as a uniform rod and the girl as a particle:

a) find the mass of the plank.

The girl places a rock on the end of the plank at A.By modeling the rock also as a particle:

b) find the smallest mass of the rock which will enable the girl to stand on the plank at B without it tilting.

Answer 

a) 
Let m = mass of the plank in kg.

Taking moments about the edge of the platform,

Moment of the weight of part of plank from edge to the point A 
= moment of the weight of the girl + moment of the weight of the part of the plank from the edge to the point B.
=> mg * [(5.4 - 1.2) / (5.4)] * (1/2)*(5.4 - 1.2) = 50(1.2 - 0.3)g + mg*[(1.2) / (5.4)] * (1/2)*(1.2)
=> 1.633m = 45 + 0.133m
=> 30 kg

 b) 
Moment of the weight of rock + moment of the weight of part of the plank on A-side of edge 
= moment of the weight of girl at B + moment of the weight of part of the plank on B-side of the edge. 
If m' = mass of rock,
m'*(4.2)g + mg*(4.2)/(5.4)* (1/2)*(4.2) = 50g*(1.2) + mg*(1.2)/(5.4)*(1/2)*(1.2)
=> (4.2)m' + 1.633m = 60 + 0.133m
=> (4.2)m' = 60 - 1.5m = 60 - 1.5(30) 
=> m' = 3.57 kg.