MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.01
W03D1_1 Group Problem: Tension in a Rope Hanging Between Two Trees Solution
Suppose a rope of mass m hangs between two trees. The ends of the rope are at the same
height and they make an angle ! with the trees.
a) What is the tension at the ends of the rope where it is connected to the trees?
b) What is the tension in the rope at a point midway between the trees?
Be sure you show any free body force diagrams or sketches that you plan to use.
Solution: We can solve this problem by a clever choice of system. Since we are
trying to fins the tension at the midpoint of the rope, we consider half of the rope. The
free body diagram is shown in Figure 1.
Figure 1
The rope is not accelerating so Newton’s Second Law becomes
î : Tmid ! Tend sin " = 0 .
(1)
m
g=0
2
(2)
ĵ : Tend cos! "
We can solve Eq. (2) for the tension at the end:
mg
Tend =
2cos!
(3)
We now substitute Eq. (3) into Eq. (1) and solve for the tension at the midpoint:
Tmid = Tend sin ! =
mg
mg
sin ! =
tan ! .
2cos!
2
(4)