A potter’s wheel is spinning with an angular speed of 4.5 rad/s. When the potter stops working, the wheel slows down with an angular acceleration of -0.5 rad/s 2. Calculate the angle, in radians, through which the wheel has turned by the time its angular speed reaches 1.4 rad/s.
V^2 = Vo^2 + 2a*d.
V = 1.4 rad/s, Vo = 4.5 rad/s, a = -0.5 rad/s^2, d = ?.
Note: d is angular distance in radians.
To calculate the angle through which the wheel has turned, we can use the equation:
𝜔^2 = 𝜔_0^2 + 2𝛼θ,
where 𝜔 is the final angular speed, 𝜔_0 is the initial angular speed, 𝛼 is the angular acceleration, and θ is the angle through which the wheel has turned.
Given:
𝜔_0 = 4.5 rad/s,
𝛼 = -0.5 rad/s^2,
𝜔 = 1.4 rad/s.
Rearranging the equation, we have:
𝜔^2 = 𝜔_0^2 + 2𝛼θ,
θ = (𝜔^2 - 𝜔_0^2) / (2𝛼).
Now substituting the values:
θ = (1.4^2 - 4.5^2) / (2 * -0.5).
Simplifying the equation further:
θ = (1.96 - 20.25) / -1.
θ = -18.29 / -1.
θ ≈ 18.29 radians.
Therefore, the wheel has turned approximately 18.29 radians by the time its angular speed reaches 1.4 rad/s.
To find the angle through which the wheel has turned when its angular speed reaches 1.4 rad/s, we need to use kinematic equations for rotational motion.
We have the following information:
Initial angular speed (ω₁) = 4.5 rad/s
Final angular speed (ω₂) = 1.4 rad/s
Angular acceleration (α) = -0.5 rad/s²
The kinematic equation for rotational motion is:
ω₂² = ω₁² + 2αθ
Rearrange the equation to solve for θ:
2αθ = ω₂² - ω₁²
θ = (ω₂² - ω₁²) / (2α)
Now, let's substitute the given values into the equation:
θ = (1.4² - 4.5²) / (2 * (-0.5))
Calculating the values:
θ = (1.96 - 20.25) / (-1)
θ = (-18.29) / (-1)
θ = 18.29 radians
Therefore, the wheel has turned by an angle of 18.29 radians when its angular speed reaches 1.4 rad/s.