Chapter 4.1, Problem 112E

(a)

To determine

Find the angular speed of the blade in radian per minute.

Answer to Problem 112E

  $9600\text{π}\text{rad}\text{/}\text{min}$

Explanation of Solution

Given:

Diameter of the circular blade is 7.25 inches and rotates at 4800 revolutions per minutes.

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) = $\frac{1\text{rev}}{\text{Time}}=\frac{2\text{π}}{\text{T}}$

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

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) $\text{π}=3.14$

  $\text{ω}=\frac{\text{Number}\text{of}\text{revolutions}}{\text{1}\text{min}}·\frac{2\text{π}\text{rad}}{\text{1}\text{rev}}=\frac{\text{4800}\text{rev}}{\text{1}\text{min}}·\frac{2\text{π}\text{rad}}{\text{1}\text{rev}}=9600\text{π}\text{rad}\text{/}\text{min}$

Thus the angular speed of the blade is $9600\text{π}\text{rad}\text{/}\text{min}$ .

(b)

To determine

Find the linear speed of the saw teeth in inches per minute.

Answer to Problem 112E

  $\text{109327}\text{.42 inches / min}$

Explanation of Solution

Given:

Diameter of the circular blade is 7.25 inches and rotates at 4800 revolutions per minutes.

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) = $\frac{1\text{rev}}{\text{Time}}=\frac{2\text{π}}{\text{T}}$

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

Calculation:

From the given angular speed $\text{ω}=9600\text{π}\text{rad}\text{/}\text{min}\text{=}\text{30144 rad}\text{/}\text{min}$

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

Where $\text{r = radius = }\frac{7.25}{2}\text{inches}=3.625\text{'}\text{'}$

  $\text{π}=3.14$

  $\text{ν}\text{=}\text{r}·\text{ω}=(3.625\text{in})·\frac{9600\text{π}\text{rad}}{\text{minutes}}=(3.625\text{in})·\frac{30144}{\text{minutes}}\approx \text{ 109327}\text{.42 inches / min }$

Linear speed = $\text{109327}\text{.42 inches / min}$

Thus, the linear speed of the saw teeth is $\text{109327}\text{.42 inches / min}$