Chapter 8, Problem 12P

(a)

To determine

The angular acceleration wheel.

Answer to Problem 12P

The angular acceleration of the 70 cm diameter wheel is $\text{ 3}{\text{.93 rad/sec}}^2$

Explanation of Solution

Given:

  $\begin{array}{l}{\text{ω}}_{\text{initial}}\text{ = 130 rpm = 130}\times \text{2π}\times \text{(1/60) rad / sec = 13}\text{.6 rad/sec}\\ {\text{ω}}_{\text{final}}\text{ = 280 rpm = 280}\times \text{2π}\times \text{(1/60) rad / sec = 29}\text{.3 rad/sec}\\ \text{time (t) = 4}\text{.0 seconds}\end{array}$

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,

  $\text{α = }\frac{\text{29}\text{.3 - 13}\text{.6}}{4.0}\text{ = 3}{\text{.93 rad / s}}^2$

Conclusion:

The angular acceleration of the 70 cm diameter wheel is $\text{ 3}{\text{.93 rad/sec}}^2$

(b)

To determine

The radial and tangential components of the linear acceleration.

Answer to Problem 12P

Radial components of the linear acceleration is ${\text{162 m/sec}}^2$

Tangential components of the linear acceleration is $\text{1}{\text{.4 m/sec}}^2$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Radius (r) = }\frac{\text{0}\text{.70}}{\text{2}}\text{ = 0}\text{.35 m}\\ \text{Angular Acceleration (α) = 3}{\text{.93 rad/s}}^2\\ \text{Time (t) = 2}\text{.0 seconds}\end{array}$

Formula used:

The instantaneous angular velocity is determined by the following formula,

  ${\text{Instantaneous Angular Velocity (ω}}_{\text{f}}{\text{) = ω}}_{\text{i}}\text{ + (α}\times \text{t)}$

Radial and tangential components of the linear acceleration is calculated by following formula,

  $\begin{array}{l}{\text{Radial Components (a}}_{\text{rad}}{\text{) = ω}}_{\text{f}}{}^{\text{2}}\times \text{r}\\ {\text{Tangential Components (a}}_{\text{tan}}\text{)= α}\times \text{r }\end{array}$

Calculation:

Substitute the given values,

  ${\text{ω}}_{\text{f}}\text{= 13}\text{.6 + (3}\text{.93}\times \text{2) = 21}\text{.5 rad/s}$

Radial and tangential components are,

  $\begin{array}{l}{\text{a}}_{\text{rad}}\text{= (21}{\text{.5)}}^{\text{2}}\times \text{0}{\text{.35 = 162 m/s}}^{\text{2}}\\ {\text{a}}_{\text{tan}}\text{= 3}\text{.93}\times \text{0}\text{.35 = 1}{\text{.4 m/s}}^{\text{2}}\end{array}$

Conclusion:

Radial components of the linear acceleration is ${\text{162 m/sec}}^2$

Tangential components of the linear acceleration is $\text{1}{\text{.4 m/sec}}^2$