Question

A wheel rotates with a constant angular acceleration of $2rad/s^2$. The total acceleration of the wheel becomes $13.6rad/s^2$after 1s of the beginning of motion. Determine the radius of the wheel.

Answer 1

Step 1: Given data:
Constant angular acceleration $\alpha =2rad/s^2$
Total acceleration of the wheel $a=13.6rad/s^2$
Time $t=1s$

Step 2: To find acceleration:

The acceleration of the wheel is given by,
$a=\sqrt{a_r^2+a_t^2}$,
The tangential acceleration of the wheel is
$a_t=r\alpha =2r$,
The radial acceleration $a_r={\omega }^{2}r$ ,
where, r is the radius, $\omega$ is the angular velocity $\alpha$is the angular acceleration. It is given by,
$\omega =\alpha t$
Therefore,
$a_r=(\alpha t{)}^{2}r$
$a_r=(2\times 1{)}^{2}r$
$a_r=4r$
Step 3: To find the radius:

Therefore,
$13.6=\sqrt{{\left(4r\right)}^2+{\left(2r\right)}^2}$
$13.6=\sqrt{20r^2}$
$13.6=4.47r$
$r=\frac{13.6}{4.47}$
$r=3.04m$
The radius of the wheel. is 3.04m