While fishing, you get bored and start to swing a sinker weight around in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 sec. What is the angle that the fishing line makes with the vertical?
Centripetal acceleration, Ac (horizontal)
= (1/2)v²/r m/s²
Acceleration due to gravity, g (vertical)
= 9.8 m/s²
Angle with vertical, θ
= tan-1(Ac/g)
76.1°
To find the angle that the fishing line makes with the vertical, we can use the formula:
θ = (2π / T) * t
Where:
θ is the angle in radians
T is the time period for one complete circle (in this case, 0.50 sec)
t is the time at which we want to find the angle
Since the weight makes a complete circle every 0.50 sec, the time period (T) is 0.50 seconds.
Now, let's calculate the angle:
θ = (2π / 0.50) * t
Since we want to find the angle at any given time, we can substitute t with 0.50 sec to get the angle at one complete circle:
θ = (2π / 0.50) * 0.50
Simplifying the expression:
θ = (2π) * 1
θ = 2π radians
Therefore, the angle that the fishing line makes with the vertical is 2π radians.
To find the angle that the fishing line makes with the vertical, we can use the concept of angular velocity.
Angular velocity is defined as the change in angle per unit time. In this case, the sinker weight makes a complete circle every 0.50 sec. This means that it completes one revolution in 0.50 sec.
To find the angular velocity, we can divide the angle of revolution (360 degrees or 2π radians) by the time taken for one revolution (0.50 sec):
Angular velocity = 360° / 0.50 sec or 2π radians / 0.50 sec
Simplifying this, we get:
Angular velocity = 720°/sec or 4π radians/sec
Now, let's consider the triangle formed by the fishing line, the horizontal line, and the vertical line. The vertical line represents the direction of gravity.
The angle we are looking for is the angle between the fishing line and the vertical line. This is typically referred to as the angle of elevation.
Since we now have the angular velocity, we can relate it to the angle of elevation.
Angular velocity = Tangent of the angle of elevation
We can now find the angle of elevation by taking the arctangent (or inverse tangent) of the angular velocity:
Angle of elevation = arctan(Angular velocity)
Angle of elevation = arctan(4π radians/sec)
Calculating this using a calculator or math software, we find:
Angle of elevation ≈ 78.69 degrees
Therefore, the angle that the fishing line makes with the vertical is approximately 78.69 degrees.