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.

Other
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.

2021-02-10

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 $$T_2:$$
$$\displaystyle{T}_{{2}}={672}\ {N}$$