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 θ with the trees.

(a) What is the tension at the ends of the rope where it is connected to the trees? Express your answer in terms of m, g, and θ (enter theta for θ).

(b) What is the tension in the rope at a point midway between the trees? Express your answer in terms of m, g, and θ (enter theta for θ).

The tension in a point of the rope is defined as follows. It is the magnitude of force that the part of the rope on one side of the point exerts on the part of the rope on the other side of the point. The direction of the force then depends on which sides you are considering here, but by Newton's third law, these forces are equal in magnitude.

The rope is at rest, so by Newton's second law:

F = m a

for zero acceleration, the total force F acting on the rope must be zero. Gravity exerts a force of m g in the downward direction, so the tensions at the points where the rope is connected to the trees must yield a force of m g in the vertical direction. Due to symmetry the two tensions are equal.

If the tension is T, then the force exerted by one of the trees in the vertical direction is T cos(theta), so you have that:

2 T cos(theta) = m g ---->

T = m g/(2 cos(theta)

Then tot find the tension at the midway point, consider the left half part of the rope from the leftmost tree to the midway point. Since this part is at rest the total force acting on it must be zero, If you consider the total vertical force, you will again find the above result for the tension. But let's now consider the total foce in the horizontal direction.

The tree exerts a force directed the the left of T sin(theta), while the part of the rope to the right of the midway point is exerting some force S on the part to the left of the midway point. Then S is, by definition, the tension in the midway point. Since the toal force must be zero, you have:

T sin(theta) - S = 0 --->

S = T sin(theta) = m g tan(theta)/2

To solve both parts of the problem, we need to consider the forces acting on the rope at different points.

(a) The tension at the ends of the rope where it is connected to the trees:
At each end of the rope, there are two forces acting: the tension T and the weight of the rope, mg.

Since the rope is at rest and the ends are at the same height, the vertical component of the tension must balance the weight of the rope:
T * cos(θ) = mg

Therefore, the tension at the ends of the rope is T = mg / cos(θ).

(b) The tension in the rope at a point midway between the trees:
At a point midway between the trees, the forces acting on the rope are the tension T and the weight of half the rope, (m/2)g.

Again, since the rope is at rest, the vertical component of the tension must balance the weight:
T * cos(θ/2) = (m/2)g

Therefore, the tension in the rope at a point midway between the trees is T = (m/2)g / cos(θ/2).

By substituting the given values of m, g, and θ into the above equations, you can calculate the tensions at the ends and midway of the rope.