Chapter 8, Problem 17P

(a)

To determine

The angular acceleration of the high speed jet.

Answer to Problem 17P

The angular acceleration of the high speed jet is $\text{0}{\text{.0698 rad/sec}}^2$

Explanation of Solution

Given:

  $\begin{array}{l}{\text{ω}}_{\text{initial}}\text{ = 0 rpm (assumed)}\\ \text{Δθ = 20 revolutions}\\ \text{time (t) = 1 minute}\end{array}$

Formula used:

The angular acceleration is determined by the following formula,

  $\begin{array}{l}{\text{Δθ = ω}}_{\text{i}}\text{t + }\frac{\text{1}}{\text{2}}{\text{αt}}^{\text{2}}\\ \text{α = }\frac{\text{2}\times \text{Δθ}}{{\text{t}}^{\text{2}}}\end{array}$

Calculation:

Substitute the given values,

  $\text{α = }\frac{\text{2}\times \text{Δθ}}{{\text{t}}^{\text{2}}}\text{ = }\frac{\text{(2}\times \text{20)}}{{\text{(1 min)}}^{\text{2}}}{\text{ = 40 revolutions/min}}^{\text{2}}$

  $\begin{array}{c}{\text{α = 40 revolutions/min}}^{\text{2}}\\ \text{ = 40}\times \text{2π}\times \frac{\text{1}}{{\text{(60)}}^{\text{2}}}{\text{ rad/sec}}^{\text{2}}\end{array}$

  $\text{α = 0}{\text{.0698 rad/sec}}^2$

Conclusion:

The angular acceleration of the high speed jet is $\text{0}{\text{.0698 rad/sec}}^2$

(b)

To determine

The final angular speed of the high speed jet in rpm.

Answer to Problem 17P

The final angular speed of the high speed jet is 40 rpm.

Explanation of Solution

Given:

  $\begin{array}{l}{\text{ω}}_{\text{initial}}\text{ = 0 rpm (assumed)}\\ \text{Δθ = 20 revolutions}\\ \text{time (t) = 1 minute}\end{array}$

Formula used:

The final angular speed of the high speed jet ${\text{ω}}_{\text{f}}$ is determined by the following formula,

  $\begin{array}{l}\text{Δθ = }\frac{\text{1}}{\text{2}}\left({\text{ω}}_{\text{i}}{\text{+ω}}_{\text{f}}\right)\times \text{t}\\ {\text{ω}}_{\text{f}}\text{ = }\frac{\text{2}\times \text{Δθ}}{\text{t}}-{\text{ω}}_{\text{i}}\end{array}$

Calculation:

Substitute the given values,

  $\begin{array}{c}{\text{ω}}_{\text{f}}\text{ = }\frac{\text{2}\times \text{20}}{\text{1}}\left(\frac{\text{revoltion}}{\text{minute}}\right)-\text{0}\\ \text{ = 40 revolution / minute}\end{array}$

Conclusion:

Thus, the final angular speed ${\text{ω}}_{\text{f}}$ of the high speed jet is $\text{40 rpm}$