Question 38: Tension in a Falling Rope
System Parameters
- Mass of rope: $m = 1.0 \text{ kg}$
- Length of rope: $l = 2.0 \text{ m}$
- Gravitational acceleration: $g = 10 \text{ m/s}^2$
- Linear mass density: $\lambda = \frac{m}{l} = \frac{1.0}{2.0} = 0.5 \text{ kg/m}$
Analysis of Motion
Let $y$ be the distance the free end has fallen below the fixed point. Since the free end falls under gravity, its velocity $v$ is given by free-fall kinematics:
$$v^2 = 2gy$$At any instant, the rope consists of a stationary part hanging from the support and a moving part. As the free end falls, the “fold” or bottom of the loop moves downwards, effectively transferring mass from the moving side to the static side. The tension $T$ at the support must balance two forces:
- Static Weight ($W_{static}$): The weight of the portion of the rope that is currently suspended.
- Dynamic Force ($F_{dyn}$): The force required to arrest the momentum of the falling segments of the rope as they straighten out.
1. Static Weight Component
The total length of the rope hanging is always $l/2$ (initial) plus the extension. More precisely, if the free end is at $y$, the length of the straightened portion plus the loop is effectively determined by the geometry. However, a simpler approach for the “folded” release is to note that at full extension ($y=l$), the entire weight $mg$ is supported.
Using the variable mass approach derived in similar impulse problems, the weight of the suspended portion at extension $y$ can be modeled as:
$$W_{static} = \lambda g \left(\frac{l+y}{2}\right)$$(Note: Initially at $y=0$, $W = \lambda g l / 2 = mg/2$. Finally at $y=l$, $W = mg$.)
2. Dynamic Force Component
The rate of change of momentum (thrust) due to the variable mass stopping is:
$$F_{dyn} = v \frac{dm}{dt} = \lambda \frac{v^2}{2}$$Substituting $v^2 = 2gy$:
$$F_{dyn} = \lambda \left(\frac{2gy}{2}\right) = \lambda g y$$Total Tension
Summing the components:
$$T = W_{static} + F_{dyn} = \lambda g \frac{(l+y)}{2} + \lambda g y $$ $$T_{max} = mg + mg$$ $$T_{max} = 2mg$$