Question

A wheel is rotating about a fixed axis with constant angular acceleration 3 rad/s². At different moments, its angular speed is $-2rad/s,0,and+2rad/s$. For a point on the rim of the wheel, consider at these moments the magnitude of the tangential component of acceleration and the magnitude of the radial component of acceleration. Rank the following five items from largest to smallest: (a) $\left|a_t\right|$ when$\omega =-2rad/s$, (b) $\left|a_r\right|$ when $\omega =-2rad/s$, (c)$\left|a_r\right|$ when$\omega =0$, (d) $\left|a_t\right|$ when $\omega =2rad/s$ and (e)$\left|a_r\right|$ when data-custom-editor="chemistry" $\omega =2rad/s$.If two items are equal, show them as equal in your ranking. If a quantity is equal to zero, show that fact in your ranking.

Short answer

The ranking of tangential and radial acceleration is $b=e>a=d>c$.

Step-by-step solution

Identification of given data

The constant angular acceleration of the wheel is $\alpha =3rad/s^2$.

the angular speeds of the wheel are,

$$\begin{gathered} {\omega }_2=2rad/s \\ {\omega }_2=0 \\ {\omega }_2=-2rad/s \end{gathered}$$

Acceleration for  circular motion

For the particle moving on a circular path, two types of acceleration acts on the particle- tangential acceleration and radial acceleration. The tangential acceleration acts along the tangent, at a particular point on the circumference of the circle. the radial acceleration acts along the radius of the circular path and is responsible for change in direction of the particle only.

Determination of tangential and radial acceleration for all subparts.

The tangential acceleration of the wheel is given as:

$a_t=r·\alpha$

For the values given, the above equation becomes-

$$\begin{gathered} a_t=r\left(3rad/s^2\right) \\ =3rm/s^2 \end{gathered}$$

The above value of tangential acceleration remains constant for angular speeds -2 rad/s and 2 rad/s in part (a) and part (d).

The radial acceleration of the wheel is given as:

$a_r=r{\omega }^2$

Substitute $\omega =-2rad/s$ for radial acceleration in part (b) in the above equation.

$$\begin{gathered} a_r=r{\left(-2rad/s\right)}^2 \\ {a}_r=4rm/s^2 \end{gathered}$$

Substitute $\omega =0$ for radial acceleration in part (c) in the above equation.

$$\begin{gathered} a_r=r{\left(0rad/s\right)}^2 \\ {a}_r=0m/s^2 \end{gathered}$$

Substitute $\omega =2rad/s$ for radial acceleration in part (e) in the above equation.

$$\begin{gathered} a_r=r{\left(2rad/s\right)}^2 \\ {a}_r=4rm/s^2 \end{gathered}$$

Ranking of tangential and radial acceleration from largest to lowest for all subparts.

The radial acceleration in part (b) and part (e) are same and largest. The tangential acceleration of wheel in part (a) and part (d) are same and second in the ranking. The radial acceleration in the part (c) is zero so it is lowest. The radial acceleration in part (c) is zero because for this position the angle between radial and tangential acceleration become ${90}^0$.

Therefore, the ranking of tangential and radial acceleration is $b=e>a=d>c$.