Chapter 8, Problem 16P

(a)

To determine

The angular acceleration of an automobile engine.

Answer to Problem 16P

The angular acceleration of an automobile engine is $-\text{138}{\text{.23 rad/sec}}^2$

Explanation of Solution

Given:

  $\begin{array}{c}{\text{ω}}_{\text{initial}}\text{ = 4500 rpm }\\ \text{= 4500}\times \text{2π}\times \text{(1/60) rad / sec}\\ \text{ = 471}\text{.24 rad/sec}\\ {\text{ω}}_{\text{final}}\text{ = 1200 rpm}\\ \text{= 1200}\times \text{2π}\times \text{(1/60) rad / sec}\\ \text{= 125}\text{.66 rad/sec}\end{array}$

t = 2.5 seconds.

Formula used:

The angular acceleration is determined by the following formula,

  $\text{Angular Acceleration (α) = }\frac{{\text{ω}}_{\text{final}}{\text{ - ω}}_{\text{initial}}}{\text{time}}{\text{ radian/sec}}^{\text{2}}$

Calculation:

Substitute the given values,

  $\begin{array}{c}\text{α = }\frac{125.66-471.24}{2.5}\\ \text{=}-\text{138}{\text{.23 rad / s}}^2\end{array}$

Conclusion:

The angular acceleration of an automobile engine is $-\text{138}{\text{.23 rad/sec}}^2$

(b)

To determine

The number of revolutions in given time.

Answer to Problem 16P

The number of revolutions ${\text{Δθ}}_{}$ in given time is $119$

Explanation of Solution

Given:

  $\begin{array}{l}{\text{ω}}_{\text{initial}}\text{ = 4500 rpm = 471}\text{.24 rad/sec}\\ {\text{ω}}_{\text{final}}\text{ = 1200 rpm = 125}\text{.66 rad/sec}\\ \text{time (t) = 2}\text{.5 seconds}\end{array}$

Formula used:

The number of revolutions ${\text{Δθ}}_{}$ is determined by the following formula,

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

Calculation:

Substitute the given values,

  $\begin{array}{l}\text{Δθ = }\frac{\text{1}}{\text{2}}\left(\text{471}\text{.24 + 125}\text{.66}\right)\times \text{2}\text{.5}\\ \text{Δθ = 746}\text{.125 radians}\end{array}$

To get the revolutions, the above angular displacement is multiplied by $\frac{\text{1 revolution}}{\text{2π radians}}$

  $\begin{array}{c}{\text{θ}}_{\text{f}}\text{ = 746}\text{.125 radians}\times \frac{\text{1 revolution}}{\text{2π radians}}\\ \text{ = 119 revolutions}\end{array}$

Conclusion:

Thus, the number of revolutions ${\text{Δθ}}_{}$ in given time is $119$