Bethany, who weighs 560 N, lies in a hammock suspended by ropes tied to two trees. The left rope makes an angle of 45∘ with the ground; the right one makes an angle of 30∘.

A)

Find the tension in the left rope.

Express your answer with the appropriate units.

B)

Find the tension in the right rope.

Express your answer with the appropriate units.

Explanation:

Conditions for Equilibrium

There are three forces applied to Bethany: T1 to the right making an angle of 30° with the ground, T2 to the left making an angle of 45° with the ground, and the weight W down the y-axis. Since the system is assumed to be at rest, the forces must be in equilibrium.

Please refer to the image below.

In the y-axis:

The projections of the tensions in the vertical direction are

Thus, replacing into the first equation:

Now for the x-axis

The components of T1 and T2 in the horizontal direction are

Replacing

Solving for T1

Replacing in [1]

Computing the trigonometric values and simplifying

Solving

And therefore