Question

with what minimum acceleration can a fireman slide down a rope while breaking strength of the rope is 2/3 of the weight of the fireman

Answer 1

HEY DEAR ...

Sum of the forces acting = ma. These types of questions can also be easily solved by drawing a free body diagram as shown below. If we add the inertia force (ma) to the FBD then we can solve it using dynamic equilibrium (all forces balance).

Edit: This method is D’Alembert’s principle.

Since he is accelerating downward, there is an inertia force F = ma upward (opposite the direction of acceleration). The only other two forces acting on the fireman are the rope tension upward and his weight = mg downward.


equilibrium: T - mg +ma = 0

setting rope tension = 2/3(mg) gives:

2/3(mg) - mg + ma = 0

or

a = (1/3)g downward

Note that if a=0, the rope tension is equal to his weight. And if he accelerates up the rope, the direction of the inertia force is downward and the rope tension will be higher than his weight (like the monkey-climbing-a-rope Quora question)

EDIT: If you prefer not to add the inertia force to the free body diagram, then simply use Newton’s second law. Acceleration is downward therefore negative:

ΣFy=mayΣFy=may

T−mg=m(−a)T−mg=m(−a)

(which is the same equation as above: T−mg+ma=0