Question

A uniform plank of length 2.00 m and mass 30.0 kg is supported by three ropes. Find the tension in each rope when a 700-N person is 0.500 m from the left end.

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asked 2021-02-09
A uniform plank of length 2.00 m and mass 30.0 kg is supported by three ropes. Find the tension in each rope when a 700-N person is 0.500 m from the left end.

Answers (1)

2021-02-10

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Solution: \(\displaystyle\sum{T}={0}\)
condition of rotational equilibrium, we take the left end as a pivot point
\(\displaystyle-{700}{\left({0.500}\right)}-{\left({9.8}\cdot{30}\right)}{\left({1.00}\right)}+{T}_{{1}}{\sin{{\left({40}\right)}}}{\left({2.00}\right)}={0}\)
Solving for \(T_1\)
\(T_1=501\ N\)
Then we apply the translational equilibrium condition, In the horizontal
\(\displaystyle\sum{F}_{{x}}={0}\)
\(\displaystyle{T}_{{1}}{\cos{{\left({40}\right)}}}-{T}_{{3}}={0}\)
\(\displaystyle{T}_{{3}}={501}{\cos{{\left({40}\right)}}}={384}\ {N}\)
In the vertical axis:
\(\displaystyle\sum{F}_{{y}}={0}\)
\(\displaystyle{T}_{{2}}-{30}\cdot{9.8}-{700}+{T}_{{1}}{\sin{{\left({40}\right)}}}={0}\)
\(\displaystyle{T}_{{2}}-{30}\cdot{9.8}-{700}+{501}{\sin{{\left({40}\right)}}}={0}\)
solving for \(T_2:\)
\(\displaystyle{T}_{{2}}={672}\ {N}\)

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