Question

At t = 0, a flywheel has an angular velocity of 4.3 rad/s, a constant angular acceleration of -0.26 rad/s², and a reference line at ?₀ = 0.

(a) Through what maximum angle ?_max will the reference line turn in the positive direction?

(b) At what times t will the line be at ? = 1/2 ?_max (consider both positive and negative values of t)

(c) At what times t will the line be at ? = -10.7 rad (consider both positive and negative values of t)

Answer 1

?(t) = ?? + ?? * t + 1/2 ? t^2
?? = 0
?? = 4.3 rad/s
? = - 2.6 rad/s^2

ergo,

?(t) = ?? * t + 1/2 ? * t^2

a. Max will be when
? = -? t
or
t.max = -?/?

[If you know calculus, you set
d?(t)/dt = 0
0 = ? + ? t
? = - ? t]

Sub in for t.max
?.max = ? * (-?/?) + 1/2 ? (- ?/?)^2
?.max = -1/2 ?^2/?
? max = 35.557.... rads

b. and c.

?(t) = ?? * t + 1/2 ? * t^2
Set ?(t) = 1/4 ?.max

1/4 ?.max = ?? * t + 1/2 ? * t^2
1/4 *(-1/2 ?^2/?) = ?? * t + 1/2 ? * t^2
0 = 1/8 ?^2/? + ?? * t + 1/2 ? * t^2

Divide out 1/2 ? (you don't have to, but it is cleaner
0 = 1/4(?/?)^2 + 2(?/?) t + t^2
0 = t^2 + 2(?/?) t + 1/4(?/?)^2

Solve quadric
t = - 2(?/?) /2