While fishing, you get bored and start to swing a sinker weight around in a circle below you on a 0.24 m piece of fishing line. The weight makes a complete circle every 0.60 s. What is the angle that the fishing line makes with the vertical?
To find the angle that the fishing line makes with the vertical, we can use the concept of angular velocity.
Angular velocity (ω) is defined as the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s). In this case, the angular velocity can be calculated using the formula:
ω = 2π / T
Where:
ω = Angular velocity in radians per second (rad/s)
T = Time period for one complete circle in seconds (s)
π = Pi, a mathematical constant approximately equal to 3.14159
Given that the weight makes a complete circle every 0.60 s, we can substitute this value into the formula:
ω = 2π / 0.60
Calculating this, we get:
ω ≈ 10.47 rad/s
Now, to find the angle that the fishing line makes with the vertical, we need to consider the relationship between angular velocity and the angle of rotation.
The angle (θ) in radians is related to the angular velocity by the formula:
θ = ω * t
Where:
θ = Angle in radians (rad)
ω = Angular velocity in radians per second (rad/s)
t = Time elapsed in seconds (s)
In this case, we are interested in the angle after a certain time period. Since the weight is rotating continuously, we can consider that the time elapsed is equal to the time period for one complete circle.
Therefore, substituting the values into the equation, we get:
θ = ω * T
θ = 10.47 rad/s * 0.60 s
Calculating this, we find:
θ ≈ 6.28 radians
So, the angle that the fishing line makes with the vertical is approximately 6.28 radians.