Question

1. a heavy, uniform rope hangs from a ceiling and is not long enough to touch the floor. which of the following best compares the magnitude of the tension force in the middle of the rope, $f_{mid}$, to the magnitude of the tension at the top of the rope near the ceiling, $f_{top}$?

a. $f_{mid} = 2f_{top}$

b. $f_{mid} = f_{top}$

c. $f_{mid} = \frac{f_{top}}{2}$

d. $f_{mid} = \frac{f_{top}}{4}$

Explanation:

Step1: Define total rope mass

Let total mass of rope = $M$, so weight $W = Mg$

Step2: Analyze top tension

Top tension supports full rope weight:
$F_{top} = Mg$

Step3: Analyze middle tension

Middle tension supports half the rope (mass $\frac{M}{2}$):
$F_{mid} = \frac{M}{2}g$

Step4: Relate the two tensions

Substitute $F_{top}=Mg$ into $F_{mid}$:
$F_{mid} = \frac{F_{top}}{2}$

Answer:

c. $F_{mid} = \frac{F_{top}}{2}$