A 50.0-kg person takes a nap in a backyard hammock. Both ropes supporting the hammock are at an angle of 15.0degrees above the horizontal. Find the tention of the ropes.

2Tsin15=mg=0

2Tsin15=mg
T=mg/(2sin15)
T=(50)(9.8)/(2sin15)

T=490/(2sin15)

T=490/.517
T=948

M*g = 50 * 9.8 = 490 N.

Eq1: T1*Cos15 + T2*Cos165 = 0
T1*Cos15 = -T2*Cos165
T1 = T2

Eq2: T1*sin15 + T2*sin165 = -490[270].
0.259T1 + 0.259T2 = 0 + 490
Replace T1 with T2:
0.259T2 + 0.259T2 = 490
0.518T2 = 490
T2 = 947 N. = T1

Well, it seems like this 50.0-kg person really knows how to relax! Now, let's figure out the tension in those hammock ropes.

First, let's break down the forces acting on the hammock. We have the weight of the person pulling downward, which can be calculated as W = mg, where m is the mass (50.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).

Now, since we have an angle involved, we need to consider the vertical and horizontal components of the tension force. The vertical component will counterbalance the weight, so we can write T * cos(15°) = mg.

Next, we have the horizontal component, which has to cancel out since the hammock isn't moving horizontally. So, T * sin(15°) = 0.

Since sin(15°) = 0.2588 and cos(15°) = 0.9659, we can solve the equations:
T * 0.9659 = 50.0 kg * 9.8 m/s²,
T * 0.2588 = 0.

Solving the first equation, we get T ≈ 482.216 N. But since the second equation tells us that the horizontal tension is zero, the ropes must be providing a tension of zero in the horizontal direction.

So, the tension in the ropes is approximately 482.216 N in the vertical direction, but zero in the horizontal direction.

To find the tension in the ropes supporting the hammock, we can start by drawing a diagram. Let's denote the tension in the left rope as T1 and the tension in the right rope as T2.

Since the person is in equilibrium, the net force acting on them must be zero. The vertical component of the tension in the ropes is equal to the weight of the person, and the horizontal component of the tension in the ropes cancel each other out.

We can start by finding the vertical component of one of the tensions. Let's consider T1. To find the vertical component of T1, we can use the equation:

T1 * sin(15°) = Weight of the person

The weight of the person is equal to their mass times the acceleration due to gravity (9.8 m/s^2). So we can rewrite the equation as:

T1 * sin(15°) = 50.0 kg * 9.8 m/s^2

Next, we can solve for T1:

T1 = (50.0 kg * 9.8 m/s^2) / sin(15°)

Using a calculator, we find:

T1 ≈ 206.2 N

Since the hammock is in equilibrium, the tension in the left rope (T1) is equal to the tension in the right rope (T2). Therefore, the tension in both ropes is approximately 206.2 N.

Draw a free body diagram. Then perform a static force balance.