Question

A disk 8.00 cm in radius rotates at a constant rate of 1200 rev / min about its central axis. Determine (a) its angular speed in radians per second, (b) the tangential speed at a point 3.00 cm from its center, (c) the radial acceleration of a point on the rim, and (d) the total distance a point on the rim moves in 2.00 s.

Short answer

1. The angular speed in radians per second of a disk is ${\omega }_i=126 rad/s$
2. The tangential speed at a pointfrom its center is$v=3.78 m/s$.
3. The radial acceleration of a point on a rim is$a_r=1270 m/s^2$ directed towards the centre.
4. The total distance a point on the rim moves in 2.00s is $d=20.2 m$.

Step-by-step solution

The tangential speed formula

The tangential speed can be expressed in terms of angular speed, $\mathit{v}\mathbf{=}\mathit{\omega }\mathit{r}$$\omega$

is the angular speed, ris the radius.

Calculation of the angular speed of a disk

(a)

To find the angular speed in radians per second, use the conversion factor,

Convert the unit from$(rev/\min )to(rad/s):$

$$\begin{gathered} {\omega }_i=1200\left(\frac{rex}{mix}\right)\left(\frac{2\pi rad}{1rex}\right)\left(\frac{1\min }{60s}\right) \\ =126 rad/s \end{gathered}$$

Therefore, the angular speed is${\omega }_i=126 rad/s$

The tangential speed

(b)

The formula for tangential speed is,

$v=\omega r$

Since, the point is in 3.00 cm distance, r = 0.03 m

Substitute the values,

$$\begin{gathered} \omega =126 rad/s, \\ r=0.03 m \end{gathered}$$

Therefore,

$v=\left(126\right)(0.03)=3.78m/s$

Therefore, the tangential speed of a disk at a point 3.00 cm is $v=3.78 m/s$.

Determination of the radial acceleration on a rim

(c)

Since, the radial acceleration can be expressed in terms of angular speed; the formula is given by,

$a_r=\frac{v^2}{r}={\omega }^{2}r$

Substitute the values,$\omega =126 rad/s, r=0.08 m$

$a_r=(126{)}^2(0.08)=1270 m/s^2$

Therefore, the radial acceleration on a rim is $a_r=1270 m/s^2$directed towards the centre.

 Step 5: Calculation of the total distance moved by the rim in 2s:

(d)

The formula for the total distance is,

$d=\omega rt$

Substitute the given values,

$$\begin{gathered} \omega =126 rad/s, \\ r=0.08 m \\ t=2 s \end{gathered}$$

Therefore,

$d=(0.08)\left(126\right)\left(2\right)=20.2 m$

Thus, the total distance moved by the rim in 2s is d=20.2m.