Romeo tries to reach Juliet by climbing with constant velocity up a rope which is knotted at point A. Any of the three segments of the rope can sustain a maximum force of 2 kN before it breaks. Determine if Romeo, who has a mass of 65 kg, can climb the rope, and if so, can he along with Juliet, who has a mass of 60 kg, climb down with constant velocity?
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
Let us first draw a free body diagram with just Romeo’s mass pulling down on the rope.
We can now write an equation of equilibrium for the y-axis forces first.
T_{AB}\text{sin}\,(60^0)\,-\,637.6\,=\,0
(Solve for T_{AB})
T_{AB}\,=\,736.2 N
We will now write an equation of equilibrium for the x-axis forces.
T_{AC}\,-\,736.2\text{cos}\,(60^0)\,=\,0
(Remember we found T_{AB}\,=\,736.2 N)
T_{AC}\,=\,368.1 N
✓We see that both of these values are less than 2000 N, therefore, Romeo can climb the rope.
We will now see if the rope can support both of their weight. Again, draw a free body diagram, this time with a total mass of 125 kg.
As before, we will write our equations of equilibrium, starting with the y-axis forces.
T_{AB}\text{sin}\,(60^0)\,-\,1226\,=\,0
(Solve for T_{AB})
T_{AB}\,=\,1416 N
Now, we will write an equation of equilibrium for x-axis forces.
T_{AC}\,-\,1416\text{cos}\,(60^0)\,=\,0
(Solve for T_{AC})
T_{AC}\,=\,708 N
✓Again, note that both tension values are less than 2000 N, including the total tension applied to the rope by the mass of both Romeo and Juliet. Therefore, Romeo and Juliet can both climb down the rope.