Chapter 19, Problem 1FP

The 60-kg wheel has a radius of gyration about its center O of k₀ = 300 mm. If it is subjected to a couple moment of M = (3t²) N m, where t is n seconds, determine the angular velocity of the wheel when t = 4 s, starting from rest.

To determine

The angular velocity ${\omega }_2$ of the wheel when $t=4\text{s}$ starting from rest.

Answer to Problem 1FP

The angular velocity of the wheel when $t=4\text{s}$ starting from rest is $\text{11}\text{.9}\text{rad}/\text{s}$.

Explanation of Solution

Given:

The mass of the wheel, $m=60\text{kg}$.

The radius of gyration of wheel about its mass center, $k_O=\left(300\text{mm}\right)\text{or}\text{0}\text{.3}\text{m}$.

The couple moment on wheel, $M=\left(3t^2\right)\text{N}\cdot \text{m}$.

The given time of rotation for wheel, $t=4\text{s}$.

Conclusion:

Initially the wheel is at rest, thus take the initial angular velocity of wheel as $0$.

${\omega }_1=0$

Express the mass moment of inertia of wheel about $O$.

$I_O=m{k}_O^2$ (I)

Substitute $\text{60}\text{kg}$ for $m$ and $\text{0}\text{.3}\text{m}$ for $k_O$ in Equation (I).

$\begin{array}{c}{I}_O=60{\left(0.3\right)}^2\\ =5.4\text{kg}\cdot {\text{m}}^{\text{2}}\end{array}$

Apply principle of impulse and momentum in clockwise direction.

$\begin{array}{l}{I}_O{\omega }_1+{\displaystyle \sum {\displaystyle {\int }_{t_1}^{t_2}{M}_{O}dt}=}{I}_O{\omega }_2\\ I_O{\omega }_1+{\displaystyle {\int }_{t_1}^{t_2}Mdt}=I_O{\omega }_2\end{array}$ (II)

Substitute $5.4\text{kg}\cdot {\text{m}}^{\text{2}}$ for $I_O$, $0$ for ${\omega }_1$, $0$ for $t_1$, $\text{4}\text{s}$ for $t_2$, and $\left(3t^2\right)\text{N}\cdot \text{m}$ for $M$ in Equation (II).

$\begin{array}{l}5.4\left(0\right)+{\displaystyle {\int }_0^{4}3t^{2}dt}=5.4{\omega }_2\\ 3{\left[\frac{t^3}{3}\right]}_0^4=5.4{\omega }_2\\ 3\left(21.33\right)=5.4{\omega }_2\end{array}$

$\begin{array}{l}5.4{\omega }_2=64\\ {\omega }_2=11.85\text{rad}/\text{s}\\ \simeq 11.9\text{rad}/\text{s}\end{array}$

Hence, the angular velocity of the wheel when $t=4\text{s}$ starting from rest is $\text{11}\text{.9}\text{rad}/\text{s}$.