A machine part rotates at an angular speed
of 0.38 rad/s; its speed is then increased to
3.46 rad/s using an angular acceleration of
0.61 rad/s2.
Find the angle through which the part ro-
tates before reaching this final speed.
Answer in units of rad
To find the angle through which the part rotates before reaching the final speed, we can use the kinematic equation that relates angular speed, angular acceleration, and the angle of rotation. The equation is:
ω^2 = ω0^2 + 2αθ
Where:
ω = final angular speed (3.46 rad/s)
ω0 = initial angular speed (0.38 rad/s)
α = angular acceleration (0.61 rad/s^2)
θ = angle of rotation
We can rearrange the equation to solve for θ:
θ = (ω^2 - ω0^2) / (2α)
Now we can substitute the given values into the equation:
θ = (3.46^2 - 0.38^2) / (2 * 0.61)
= (11.9716 - 0.1444) / 1.22
= 11.8272 / 1.22
≈ 9.674
Therefore, the angle through which the part rotates before reaching the final speed is approximately 9.674 radians.
To find the angle through which the part rotates before reaching the final speed, we can use the equation:
ω^2 = ω0^2 + 2αθ
where ω is the final angular speed, ω0 is the initial angular speed, α is the angular acceleration, and θ is the angle of rotation.
Given:
ω = 3.46 rad/s
ω0 = 0.38 rad/s
α = 0.61 rad/s^2
Substituting these values into the equation, we have:
(3.46)^2 = (0.38)^2 + 2(0.61)θ
Simplifying:
11.9716 = 0.1444 + 1.22θ
Rearranging the equation to solve for θ:
1.22θ = 11.9716 - 0.1444
1.22θ = 11.8272
θ = 9.6839
Therefore, the angle through which the part rotates before reaching the final speed is approximately 9.6839 radians.