A machine part rotates at an angular speed
of 0.77 rad/s; its speed is then increased to
1.76 rad/s using an angular acceleration of
0.29 rad/s
2
.
Find the angle through which the part rotates before reaching this final speed.
Answer in units of rad
wf^2 = wi^2 + 2 (alpha) * (angle of rotation (delta theta))
delta theta = (1.76^2 - 0.77^2) / 2*0.29
delta theta = 2.36
(check me on my math))
To find the angle through which the part rotates before reaching the final speed, we can use the kinematic equation of rotational motion:
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 part rotates.
Given:
ω_0 = 0.77 rad/s
ω = 1.76 rad/s
α = 0.29 rad/s^2
Let's plug in these values into the equation and solve for θ:
(1.76 rad/s)^2 = (0.77 rad/s)^2 + 2(0.29 rad/s^2)θ
3.0976 = 0.5929 + 0.58θ
Subtracting 0.5929 from both sides:
2.5047 = 0.58θ
Dividing both sides by 0.58:
θ = 2.5047 / 0.58
θ ≈ 4.32 rad
Therefore, the angle through which the part rotates before reaching the final speed is approximately 4.32 radians.