The end of uniform rope of mass m and length L that is piled...

The end of uniform rope of mass $m$ and length $L$ that is piled on a platform is lifted vertically with a constant velocity $v$ by a variable force $F$. The value of $F$ as a lifted position $x$ of the rope is : (a) $\frac{m}{L}\left(g x+v^{2}\right)$ (b) $m\left(g x+v^{2}\right)$ (c) $\frac{m}{L^{2}}\left(g x^{2}+x v\right)$ (d) none of these

The end of uniform rope of mass $m$ and length $L$ that is piled on a platform is lifted vertically with a constant velocity $v$ by a variable force $F$. The value of $F$ as a lifted position $x$ of the rope is :
(a) $\frac{m}{L}\left(g x+v^{2}\right)$
(b) $m\left(g x+v^{2}\right)$
(c) $\frac{m}{L^{2}}\left(g x^{2}+x v\right)$
(d) none of these

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Key Concepts

Uniform Linear Mass Density

This concept refers to a constant mass per unit length, which in a uniform rope allows us to easily relate any segment’s length to its mass. It is essential for calculating the weight of the lifted portion and understanding how mass is distributed along the object.

Momentum Flux in Variable Mass Systems

When a rope is being lifted, new sections of the rope initially at rest are accelerated to the constant velocity. This change in momentum happens continuously and requires an additional force, calculated as the product of the rate at which mass is added and the velocity, which contributes to the overall force needed.

Newton’s Second Law in the Context of Changing Mass

Even though the lifted part of the rope moves at a constant velocity (implying zero acceleration for that section), the total system’s dynamics are influenced by the continuous addition of moving mass. Newton’s second law is applied to the whole system by accounting not only for the gravitational force acting on the lifted mass but also for the force required to accelerate additional mass from rest to the constant velocity.

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