Chapter 8, Problem 15P

To determine

The number of revolutions in given time.

Answer to Problem 15P

The number of revolutions ${\text{θ}}_{\text{f}}$ in given time is $27,500$

Explanation of Solution

Given:

  $\begin{array}{l}{\text{ω}}_{\text{initial}}\text{ = 0 rpm = 0 rad/sec}\\ {\text{ω}}_{\text{final}}\text{ = 15000 rpm = 15000}\times \text{2π}\times \text{(1/60) rad / sec = 1571 rad/sec}\\ \text{time (t) = 220 seconds}\\ {\text{θ}}_{\text{i}}\text{ = 0}\end{array}$

Formula used:

The angular acceleration is determined by the following formula,

  $\begin{array}{l}\text{Angular Acceleration (α) = }\frac{{\text{ω}}_{\text{final}}{\text{ - ω}}_{\text{initial}}}{\text{time}}{\text{ radian/sec}}^{\text{2}}\\ {\text{Final Angular Displacement θ}}_{\text{f}}{\text{ = θ}}_{\text{i}}\text{ +}\left({\text{ ω}}_{\text{i}}\times \text{t}\right)\text{ + }\left(\frac{\text{1}}{\text{2}}\times \text{α}\times {\text{t}}^{\text{2}}\right)\end{array}$

Calculation:

Substitute the given values,

  $\text{α = }\frac{\text{1571}-\text{0}}{220}\text{ = 7}{\text{.14 rad / s}}^2$

Final angular momentum is,

  ${\text{θ}}_{\text{f}}\text{ = 0 + }\left(\text{0}\times \text{220}\right)\text{ + }\left(\frac{\text{1}}{\text{2}}\times \text{7}\text{.14}\times {\text{220}}^{\text{2}}\right)\text{ = 172788 radians}$

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

  ${\text{θ}}_{\text{f}}\text{ = 172788 radians}\times \frac{\text{1 revolution}}{\text{2π radians}}\text{ = 27,500 revolutions}$

Conclusion:

Thus, the number of revolutions ${\text{θ}}_{\text{f}}$ in given time is $27,500$