p11_015.pdf

15. The problem has (implicitly) speciﬁed the positive sense of rotation. The angular acceleration of magni-

tude 0 . 25 rad/s 2 in the negative direction is assumed to be constant over a large time interval,including

negative values (for t ).

(a) We specify θ max with the condition ω = 0 (this is when the wheel reverses from positive rotation

to rotation in the negative direction). We obtain θ max using Eq. 11-14:

θ max =

−

ω o

2 α =

−

4 . 7 2

=44rad .

−

0 . 25)

(b) We ﬁnd values for t 1 when the angular displacement (relative to its orientation at t =0)is θ 1 =

22 rad (or 22 . 09 rad if we wish to keep track of accurate values in all intermediate steps and only

round oﬀ on the ﬁnal answers). Using Eq. 11-13 and the quadratic formula,we have

θ 1 = ω o t 1 +

2 αt 1 =

⇒

t 1 = −

ω o ± ω o +2 θ 1 α

which yields the two roots 5 . 5sand32s.

−

θ 2 = ω o t 2 +

2 αt 2 =

⇒

t 2 = −

ω o ± ω o +2 θ 2 α

2 . 1sand40s.

(d) With radians and seconds understood,the graph of θ versus t is shown below (with the points found

in the previous parts indicated as small circles).

−

–20

10 . 5 rad. Using Eq. 11-13 and the quadratic formula,we have

which yields the two roots