While fishing, you get bored and start to swing a sinker weight around in a circle below you on a 0.31- piece of fishing line. The weight makes a complete circle every 0.70 .

What is the angle that the fishing line makes with the vertical?
Express your answer using two significant figures.

To find the angle that the fishing line makes with the vertical, we can use the relationship between the angle and the length of the vertical component of the fishing line.

First, let's identify the relevant information from the problem:
- The weight makes a complete circle every 0.70 seconds.
- The length of the fishing line is 0.31 meters.

Since the weight makes a full circle every 0.70 seconds, we can calculate the angular speed (ω) using the formula:
ω = 2π / T , where T is the period (time) for one full rotation.

Plugging in the values:
ω = 2π / 0.70 ≈ 8.97 rad/s

Next, we need to find the length of the vertical component of the fishing line. This can be calculated using the formula for the circumference of a circle:
C = 2πr

Since the length of the fishing line is the circumference of the circle, we can write:
C = 0.31 meters = 2πr

Rearranging the equation to solve for r:
r = 0.31 / 2π ≈ 0.049 meters

Now, we can find the length of the vertical component (L) of the fishing line through basic trigonometry:
L = r * sin(θ), where θ is the angle between the fishing line and the vertical.

Rearranging the equation to solve for θ:
θ = arcsin(L/r)

Plugging in the values:
θ = arcsin(0.049 / 0.31) ≈ 0.159 radians

Finally, we can express the angle in degrees:
θ ≈ 0.159 * (180/π) ≈ 9.12 degrees

Therefore, the angle that the fishing line makes with the vertical is approximately 9.12 degrees.