Question

problem 4: a cat balcony was taken over by a very possessive english mastiff dog. the mass of the dog is 100 kg. the dog walks to the end of the cat balcony, which is made of a plank that is 2 meters long and has a mass of 20 kg. given that the plank is pivot connected and has a cable supporting it with an angle of 35° (see diagram), find the tension on the cable. if the cable can withstand up to 1000 n of tension, will the dog stay or fall?

Explanation:

Step1: Calculate the torques

The torque due to the dog's weight: $\tau_{dog}=m_{dog}g\times L$, where $m_{dog} = 100\space kg$, $g = 9.8\space m/s^{2}$, $L = 2\space m$. So $\tau_{dog}=100\times9.8\times2=1960\space N\cdot m$.
The torque due to the plank's weight: $\tau_{plank}=m_{plank}g\times\frac{L}{2}$, where $m_{plank} = 20\space kg$. So $\tau_{plank}=20\times9.8\times1 = 196\space N\cdot m$.
The torque due to the tension in the cable: $\tau_{tension}=T\times L\times\sin\theta$, where $\theta = 35^{\circ}$.

Step2: Set up the torque - equilibrium equation

Since the plank is in rotational equilibrium, $\sum\tau=0$. So $\tau_{dog}+\tau_{plank}-\tau_{tension}=0$.
Substituting the values: $1960 + 196-T\times2\times\sin35^{\circ}=0$.
First, simplify the left - hand side: $2156=T\times2\times\sin35^{\circ}$.
We know that $\sin35^{\circ}\approx0.574$. Then $T=\frac{2156}{2\times0.574}$.

Step3: Calculate the tension

$T=\frac{2156}{1.148}\approx1878\space N$.

Step4: Compare the tension with the cable's maximum tension

The cable can withstand $T_{max} = 1000\space N$. Since $T\approx1878\space N>1000\space N$.

Answer:

The tension in the cable is approximately $1878\space N$. The dog will fall.