9. 11 As shown in the figure, a beam is supported by two pillars separated by a

Question

9. 11

As shown in the figure, a beam is supported by two pillars separated by a distance l. The beam has a mass M, a length L, and a girl of mass m is walking from the left end toward the right. Find a symbolic expression for the normal force exerted on the beam by the pillar on the right, when the beam is on the verge of tipping. (Use the following as necessary: M, m, and g.) N2 Find a symbolic expression for the girl's position, when the beam is on the verge of tipping. (Use the following as necessary: M, m, L and l.) Find a symbolic expression for the minimum value of l that will allow the girl to reach the end of the beam without it tipping. (Use the following as necessary: M, m, and L.)

Explanation / Answer

while tipping the beam will rotate about the right pillar ... when the girl is on the right side of the right pillar at a distance (x-l)

the mass of the beam acts at the centre of the beam at a distance of (l-L/2) to the left of the right pillar

on the verge of tipping there will no no contact with the left pillar so the entire load is bore by the right pillar

the normal force exerted by the pillar = sum of weights of the beam and the girl

Normal force (N)=(M+m)g

taking moments about the right pillar we have

for the beam to tip

mg*(x-l)> Mg(l-L/2)

x=(M/m)*(l-L/2) + l

when the girl is at the extreme end she is at a distance of (L-l) from the right pillar

taking moments about right pillar

for the beam not to tip

Mg(l-L/2)>mg(L-l)

(M+m)l >mL+ML/2

l > (2m+M)L/2(M+m)

there fore minimum value of l = (2m+M)L/2(M+m)

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