The initial angular velocity was higher by 5.1*10 = 51 rad/s, so it was 51.74 rad/s. The average angular velocity was
(51.74 + 1.74)/2 = 26.74 rad/s.
Multiply that by 10 s for the angualr displacement in radians.
(51.74 + 1.74)/2 = 26.74 rad/s.
Multiply that by 10 s for the angualr displacement in radians.
We know that the initial angular velocity is 0 rad/s since the wheel starts from rest. The final angular velocity is +1.74 rad/s, and the angular acceleration is -5.1 rad/s². To find the angular displacement, we can use the following equation:
ω² = ω₀² + 2αθ
where ω is the final angular velocity, ω₀ is the initial angular velocity, α is the angular acceleration, and θ is the angular displacement.
In this case, ω₀ = 0 rad/s, ω = +1.74 rad/s, and α = -5.1 rad/s². Plugging these values into the equation, we can solve for θ:
1.74² = 0² + 2(-5.1)θ
θ = (1.74²) / (-10.2)
θ ≈ -0.294 rad
The angular displacement of the wheel is approximately -0.294 radians. Keep in mind that negative values indicate a clockwise rotation, while positive values indicate a counterclockwise rotation.
ωf = ωi + αt + 1/2 * α * t^2
Where:
- ωf is the final angular velocity
- ωi is the initial angular velocity
- α is the angular acceleration
- t is the time
Given:
- ωi = 0 rad/s (assuming the wheel starts from rest)
- ωf = +1.74 rad/s
- α = -5.1 rad/s^2
- t = 10.0 s
Substituting the given values into the equation:
1.74 = 0 + (-5.1) * 10.0 + 1/2 * (-5.1) * (10.0)^2
Simplifying:
1.74 = -51 + (-2.55) * 100
1.74 = -51 - 255
1.74 = -306
This equation is not possible, as it results in a contradiction. It appears there may be an error in the given values. Please double-check the data provided.
1. The equation that relates angular displacement (θ), initial angular velocity (ω0), angular acceleration (α), and time (t) is:
θ = ω0 * t + (1/2) * α * t^2
2. Given that the initial angular velocity is +1.74 rad/s, the angular acceleration is -5.1 rad/s^2, and the time is 10 seconds, we can substitute these values into the equation:
θ = (1.74 rad/s) * (10 s) + (1/2) * (-5.1 rad/s^2) * (10 s)^2
3. Evaluating this equation gives:
θ = 17.4 rad + (1/2) * (-5.1 rad/s^2) * 100 s^2
θ = 17.4 rad - 255 rad
θ = -237.6 rad
Therefore, the angular displacement of the wheel after 10 seconds is -237.6 radians.