Chapter 4.1, Problem 109E

(a)

To determine

Find the angular speed of a Blu- ray disc rotates in radian per minute.

Answer to Problem 109E

  $333.33\text{π}\text{rad/sec}$

Explanation of Solution

Given:

The driver motor of the Blu Ray player is able to rotate up to 10000 revolutions per minute depending on what track is being read.

Concept Used:

Linear velocity is the change of position of the body with respect to the time when it is travelling along the straight path whereas the angular velocity is the change of position of the body with respect to the time when it is travelling along the circular path.

1 revolution = $2\text{π}\text{radians}$

  $\text{ω}$ = angular speed (radians per time) = $\frac{1\text{rev}}{\text{Time}}=\frac{2\text{π}}{\text{T}}$

Calculation:1 revolution = $2\text{π}\text{radians}$

Use the formula for $\text{ω}$ = angular speed (radians per time) = $\frac{1\text{rev}}{\text{Time}}=\frac{2\text{π}}{\text{T}}$

  $\text{ω}$ = angular speed (radians per time) $\text{π}=3.14$

  $\text{ω}=\frac{\text{10000}\text{rev}}{\text{1}\text{min}}·\frac{2\text{π}\text{rad}}{\text{1}\text{rev}}·\frac{1\min }{\text{60}\text{sec}}=\frac{1000\text{π}\text{rad}}{\text{3}\sec }=333.33\text{π}\text{rad/sec}$

The angular speed of the Blu − Ray disc $\text{ω}=333.33\text{π}\text{rad/sec}$

Thus the angular speed of the Blu − Ray disc is $333.33\text{π}\text{rad/sec}$ .

(b)

To determine

Find the maximum linear speed (in meter per second) of a point on the outermost track as the disc rotates.

Answer to Problem 109E

  $\text{62}\text{.8}\text{inches / sec}$

Explanation of Solution

Given:

The Blu-Ray disc is approximately 12 cm in diameter.

Concept Used:

Linear velocity is the change of position of the body with respect to the time when it is travelling along the straight path whereas the angular velocity is the change of position of the body with respect to the time when it is travelling along the circular path.

1 revolution = $2\text{π}\text{radians}$

The formula for linear speed: $\text{ν}\text{=}\text{r}·\text{ω}$

Calculation:

From the given angular speed $\text{ω}==\frac{1000\text{π}\text{rad}}{\text{3}\sec }=333.33\text{π}\text{rad/sec}$

We can find the linear speed by using the formula $\text{ν}\text{=}\text{r}·\text{ω}$

Where $\text{Radius = }\frac{12}{2}=6\text{cm}\text{=}\text{0}\text{.06}\text{m}$

  $\text{π}=3.14$

  $\text{ν}\text{=}\text{r}·\text{ω}=(0.06\text{in})·\frac{1000\text{π}\text{rad}}{\text{3}\text{sec}}=20\text{π inches / sec =}\text{62}\text{.8}\text{inches / sec}$

Linear speed = $\text{62}\text{.8}\text{inches / sec}$

Thus, the maximum linear speed of a point on the outermost track as the disc rotatesis $\text{62}\text{.8}\text{inches / sec}$