A 54.0 kg person takes a nap in a (lightweight) backyard hammock. Both ropes supporting the hammock are at an angle of 16.9° above the horizontal. Find the tension in the ropes.

2Tsin16.9=mg=0

2Tsin16.9=mg
T=mg/2sin16.9
T=54(9.8)/2sin16.9
T=529.2/sin16.9
T=529.2/.581
T=912

2Tcos16.9=W,

but W=mg=54x9.8=529.2N,
T=W/2c0s16.9
T on each cord is to be 276.54N

both above wrong...stupid!!!!

M*g = 54 * 9.8 = 529.2 N. = Wt. of the person.

Eq1: T1*Cos16.9 = -T2*Cos(180-16.9).
0.957T1 = 0.957T2.
T1 = T2.

Eq2: T1*sin16.9 + T2*sin(180-16.9) = -529.2[270o].
0.291T1 + 0.291T2 = 529.2.
Replace T1 with T2:
0.291T2 + 0.291T2 = 529.2
0.581T2 = 529.2
T2 = 910 N. = T1.

To find the tension in the ropes supporting the hammock, we can use the concept of vector components. First, let's draw a free body diagram of the person lying in the hammock.

In the diagram, there are three forces acting on the person: the gravitational force pointing downward (mg), and the tensions in the ropes pointing upward (T1 and T2).

Since the hammock is in equilibrium (not accelerating), the vertical component of the tension in the ropes must balance the weight of the person. Therefore, we can write the following equation:

T1sin(θ) + T2sin(θ) - mg = 0

Here, θ represents the angle of the ropes from the horizontal, and m represents the mass of the person.

Given that the mass of the person is 54.0 kg and the angle of the ropes is 16.9°, we can substitute these values into the equation:

T1sin(16.9°) + T2sin(16.9°) - (54.0 kg)(9.8 m/s^2) = 0

Now, let's solve for T1 and T2. However, we also need one more equation to find the exact values of T1 and T2.

Since the hammock is in equilibrium, the horizontal component of the forces must also be balanced. Therefore, the horizontal component of T1 must equal the horizontal component of T2. This can be written as:

T1cos(θ) = T2cos(θ)

We now have two equations with two unknowns (T1 and T2). We can solve these equations simultaneously to find the tensions.

Using the trigonometric identities, we can simplify the equation:

T1sin(16.9°) + T2sin(16.9°) = (54.0 kg)(9.8 m/s^2)

T1cos(16.9°) = T2cos(16.9°)

Now, let's solve these equations simultaneously:

T1 = [(54.0 kg)(9.8 m/s^2) - T2sin(16.9°)] / sin(16.9°)

Substitute the value of T1 into the second equation:

[(54.0 kg)(9.8 m/s^2) - T2sin(16.9°)] / sin(16.9°) * cos(16.9°) = T2

Now, solve for T2:

[(54.0 kg)(9.8 m/s^2) - T2sin(16.9°)] * cos(16.9°) = T2 * sin(16.9°)

Simplify the equation:

(54.0 kg)(9.8 m/s^2) * cos(16.9°) - T2 * sin(16.9°) * cos(16.9°) = T2 * sin(16.9°)

Group the terms involving T2:

(54.0 kg)(9.8 m/s^2) * cos(16.9°) = T2 * sin(16.9°) * cos(16.9°) + T2 * sin(16.9°)

Now, solve for T2:

T2 = [(54.0 kg)(9.8 m/s^2) * cos(16.9°)] / [sin(16.9°) * cos(16.9°) + sin(16.9°)]

Finally, substitute the value of T2 back into the equation for T1 to find T1:

T1 = [(54.0 kg)(9.8 m/s^2) - T2sin(16.9°)] / sin(16.9°)

Calculate both T1 and T2 using the given values, and you will find the tension in the ropes supporting the hammock.