Answer
a) $\alpha$ = -3.07012 rad/ $\sec^{2}$
b) t=12.44s
c) The car has travelled 95.03 meters before stopping.
Work Step by Step
a) To find the angular acceleration, the following formula can be used: $\omega_{f}^{2}$ = $\omega_{o}^{2}$ + 2$\alpha$$\theta$. This formula is best because we either know or can calculate all the unknown variables and no unnecessary variables, such as time, are present.
1. The given speeds 55km/h and 95km/h are the translational speeds of the wheels. However, by using the formula v=$\omega$r, they can be converted into angular speeds $\omega_{o}$ and $\omega_{f}$.
$\frac{55km}{h}$($\frac{1000m}{1km}$)($\frac{1h}{3600s}$) = $\frac{275m}{18s}$
$\frac{95km}{h}$($\frac{1000m}{1km}$)($\frac{1h}{3600s}$) = $\frac{475m}{18s}$
Translational speed (v) = $\omega$(radius)
($\frac{275m}{18s}$) $\div$ (0.4m) = $\frac{1375rad}{36s}$ = $\omega_{f}$
($\frac{475m}{18s}$) $\div$ (0.4m) = $\frac{2375rad}{36s}$ = $\omega_{o}$
2. $\theta$ is the total angle (in radians) through which the wheel has rotated as it slowed down. Since the tires make 75 revolutions and one revolution is 2$\pi$, the total angle $\theta$ = 75(2$\pi$) = 150$\pi$
3. Then we just plug all the variables in the formula $\omega_{f}^{2}$ = $\omega_{o}^{2}$ + 2$\alpha$$\theta$ and find the angular acceleration.
$\frac{1375rad}{36s}^{2}$ = $\frac{2375rad}{36s}^{2}$ + 2$\alpha$(150$\pi$)
$\alpha$ = -3.07012 rad/ $\sec^{2}$
b) The answer can be determined using the formula $\omega_{f}$ = $\omega_{o}$ + $\alpha$t
Stopping means that $\omega_{f}$ = 0rad/s and "how much more time" implies that $\omega_{o}$ = $\frac{1375rad}{36s}$.
Plug the variables in the formula and solve for t.
0rad/s = $\frac{1375rad}{36s}$ + (-3.07012 rad/$\sec^{2}$)t
t=12.44s
c) We again use the formula $\omega_{f}^{2}$ = $\omega_{o}^{2}$ + 2$\alpha$$\theta$, only this time to find $\theta$.
$\frac{0rad}{s}^{2}$ = $\frac{1375rad}{36s}^{2}$ + 2(-3.07012 rad/$\sec^{2}$)$\theta$
$\theta$ = 237.583 rad
The relationship between the distance traveled and the angle theta is d=$\theta$(radius)
d = (237.583)(0.4m) = 95.03 m
The car has travelled 95.03 meters before stopping
