Beam on platform

A homogeneous wooden beam of length 6.00 m and mass 72.0 kg lies on a horizontal platform high above the ground and overhangs by 1.80 m over the edge of the platform

a) Decide whether a person with a mass of 60.0 kg can stand on the overhanging end of the beam.
b) Determine the maximum mass of a person who can stand on the end of this beam so as not to tip over with the beam.
c) Determine to what distance from the end of the beam a person with a mass of 75.0 kg can stand so that the beam does not tip over.
d) Determine the maximum length by which the beam can overhang over the edge so that a person with a mass of 60.0 kg standing at its end does not tip over with the beam.

Final Answer:

b =  48 kg
c =  1.152 m
d =  1.6364 m

Step-by-step explanation:

$$\begin{gathered} h=72/6=12 \\ {r}_1=1.8 \\ {r}_2=6-r_1=6-1.8=\frac{21}{5}=4.2 \\ {\sum }_{\{}{r}_1{\}}^{\{}{r}_2\}{m}_i{r}_i=r_1\cdot b \\ \underset{\{}{\int }{r}_1{\}}^{\{}{r}_2\}hrdr=r_1\cdot b=M \\ M=h\cdot (r_2^2/2-r_1^2/2)=12\cdot (4.2^2/2-1.8^2/2)=\frac{432}{5}=86.4 \\ b=M/r_1=86.4/1.8=48\text{kg} \end{gathered}$$

$c=M/75.0=86.4/75.0=1.152\text{m}$

$$\begin{gathered} \underset{\{}{\int }d{\}}^{\{}6-d\}hrdr=60d \\ h((6-d{)}^2/2-d^2/2)=60\cdot d \\ h(6^2-12d+d^2-d^2)=120d \\ d=h\cdot 6^2/(120+12\cdot h)=12\cdot 6^2/(120+12\cdot 12)=1.6364\text{m} \end{gathered}$$

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