Chapter 15.7, Problem 15.242P

A disk of 180-mm radius rotates at the constant rate ${\omega }_2=12$ rad/s with respect to arm CD, which itself rotates at the constant rate ${\omega }_1=8$ rad/s about the Y axis. Determine at the instant shown the velocity and acceleration of point A on the rim of the disk.

To determine

The velocity of point $A$.

The acceleration of point $A$.

Answer to Problem 15.242P

The velocity of point $A$ is $-\left(5.04\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}$.

The acceleration of point $A$ is $-\left(9.6\text{m}/{\text{s}}^2\right)\text{i}-\left(25.92\text{m}/{\text{s}}^{\text{2}}\right)\text{j}+\left(57.6\text{m}/{\text{s}}^{\text{2}}\right)\text{k}$.

Explanation of Solution

Given information:

The angular velocity of arm $CD$ is $8\text{rad}/\text{s}$, the angular velocities of disc is $12\text{rad}/\text{s}$ and radius of disc is $180\text{mm}$. Consider point $D$ as origin point of coordinate.

Figure-(1) represents the mechanism of disc system.

Figure-(1)

Write the expression for position vector of point $A$ with respect to point $D$.

${\overrightarrow{r}}_{A/D}={\overrightarrow{r}}_A-{\overrightarrow{r}}_D$ ...... (I)

Here, position vector of point $A$ is ${\overrightarrow{r}}_A$ and position vector of point $D$ is ${\overrightarrow{r}}_D$

Write the expression for position vector of point $A$ with respect to point $C$.

${\overrightarrow{r}}_{A/C}={\overrightarrow{r}}_A-{\overrightarrow{r}}_C$ ...... (II)

Here, position vector of point $C$ is ${\overrightarrow{r}}_C$.

Write the expression for velocity of point $A$ with respect to $D$.

${\overrightarrow{v}}_{A^{\prime }}={\overrightarrow{\omega }}_1\times {\overrightarrow{r}}_{A/D}$ ...... (III)

Here, angular velocity of frame $CD$ is ${\omega }_1$.

Write the expression for velocity of point $A$ with respect to $C$.

${\overrightarrow{v}}_{A/C}={\overrightarrow{\omega }}_2\times {\overrightarrow{r}}_{A/C}$ ...... (IV)

Here, angular velocity of disc is ${\omega }_2$.

Write the expression for velocity of point $A$.

${\overrightarrow{v}}_A={\overrightarrow{v}}_{A^{\prime }}+{\overrightarrow{v}}_{A/C}$ ...... (V)

Write the expression for acceleration of point $A$ with respect to point $D.$

${\overrightarrow{a}}_{A^{\prime }}={\overrightarrow{\omega }}_1\times {\overrightarrow{v}}_{A^{\prime }}$ ...... (VI)

Write the expression for acceleration of point $A$ with respect to point $C$

${\overrightarrow{a}}_{A/C}={\overrightarrow{\omega }}_2\times {\overrightarrow{v}}_{A/C}$ ...... (VII)

Write the expression for coriolis component of acceleration.

${\overrightarrow{a}}_C=2{\overrightarrow{\omega }}_1\times {\overrightarrow{v}}_{A/C}$ ...... (VIII)

Write the expression for net acceleration of point $A$.

${\overrightarrow{a}}_A={\overrightarrow{a}}_{A^{\prime }}+{\overrightarrow{a}}_{A/C}+{\overrightarrow{a}}_C$ ...... (IX)

Calculation:

Considering point $D$ as origin, the coordinate of point $C$ is $\left(150\text{mm},0\text{mm},-360\text{mm}\right)$, coordinate of point $A$ is $\left(150\text{mm},180\text{mm},-360\text{mm}\right)$

Substitute $\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}$ for ${\overrightarrow{r}}_A$ and $0\text{i}+0\text{j}+\text{0k}$ for ${\overrightarrow{r}}_D$ in Equation (I).

$\begin{array}{c}{\overrightarrow{r}}_{A/D}=\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}-0\text{i}+0\text{j}+\text{0k}\\ \text{=}\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}\end{array}$

Substitute $\left(8\text{rad}/\text{s}\right)\text{j}$ for ${\omega }_1$ and $\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}$ for ${\overrightarrow{r}}_{A/D}$ in Equation (III).

$\begin{array}{c}{\overrightarrow{v}}_{A^{\prime }}=\left(8\text{rad}/\text{s}\right)\text{j}\times \left[\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}\right]\\ =\left(8\text{rad}/\text{s}\right)\text{j}\times \left[\begin{array}{l}\left(150\text{mm}\times \frac{1\text{m}}{1000\text{mm}}\right)\text{i}+\left(180\text{mm}\times \frac{1\text{m}}{1000\text{mm}}\right)\text{j}\\ -\left(360\text{mm}\times \frac{1\text{m}}{1000\text{mm}}\right)\text{k}\end{array}\right]\\ =\left(8\text{rad}/\text{s}\right)\text{j}\times \left[\left(0.150\text{m}\right)\text{i}+\left(0.18\text{m}\right)\text{j}-\left(0.36\text{m}\right)\text{k}\right]\\ =\left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 0& 8\text{rad}/\text{s}& 0\\ 0.15\text{m}& 0.18\text{m}& -0.36\text{m}\end{array}\right|\end{array}$

$\begin{array}{c}{\overrightarrow{v}}_{A^{\prime }}=\left(-8\text{rad}/\text{s}\times 0.36\text{m}-0\right)\text{i}-\left(0\right)\text{j}+\left(0-8\text{rad}/\text{s}\times 0.15\text{m}\right)\text{k}\\ =-\left(2.88\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}\end{array}$

Substitute $\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}$ for ${\overrightarrow{r}}_A$ and $\left(150\text{mm}\right)\text{i}-\left(360\text{mm}\right)\text{k}$ for ${\overrightarrow{r}}_C$ in Equation (II).

$\begin{array}{c}{\overrightarrow{r}}_{A/C}=\left[\begin{array}{l}\left(150\text{mm}\right)\text{i}+\left(180\text{mm}\right)\text{j}-\left(360\text{mm}\right)\text{k}-\\ \left(150\text{mm}\right)\text{i}-\left(360\text{mm}\right)\text{k}\end{array}\right]\\ =\left(180\text{mm}\times \frac{1\text{m}}{1000\text{mm}}\right)\text{j}\\ \text{=}\left(0.18\text{m}\right)\text{j}\end{array}$

Substitute $\left(12\text{rad}/\text{s}\right)\text{k}$ for ${\omega }_2$ and $\left(0.18\text{m}\right)\text{j}$ for ${\overrightarrow{r}}_{A/C}$ in Equation (IV).

$\begin{array}{c}{\overrightarrow{v}}_{A/C}=\left(12\text{rad}/\text{s}\right)\text{k}\times \left(0.18\text{m}\right)\text{j}\\ =\left(-2.16\text{m}/\text{s}\right)\text{i}\end{array}$

Substitute $-\left(2.88\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}$ for ${\overrightarrow{v}}_{A^{\prime }}$ and $-\left(2.16\text{m}/\text{s}\right)\text{i}$ for ${\overrightarrow{v}}_{A/C}$ in Equation (V).

$\begin{array}{c}{\overrightarrow{v}}_A=-\left(2.88\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}-\left(2.16\text{m}/\text{s}\right)\text{i}\\ =-\left(5.04\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}\end{array}$

Substitute $\left(8\text{rad}/\text{s}\right)\text{j}$ for ${\omega }_1$ and $-\left(2.88\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}$ for ${\overrightarrow{v}}_{A^{\prime }}$ in Equation (VI).

$\begin{array}{c}{\overrightarrow{a}}_{A^{\prime }}=\left(8\text{rad}/\text{s}\right)\text{j}\times \left[-\left(2.88\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}\right]\\ =\left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 0& 8\text{rad}/\text{s}& 0\\ -2.88\text{m}/\text{s}& 0& -1.2\text{m}/\text{s}\end{array}\right|\\ =\left(-8\text{rad}/\text{s}\times 1.2\text{m}/\text{s}-0\right)\text{i}-\left(0\right)\text{j}+\left(8\text{rad}/\text{s}\times 2.88\text{m}/\text{s}\right)\text{k}\\ \text{=}-\left(9.6\text{m}/{\text{s}}^2\right)\text{i}+\left(23.04\text{m}/{\text{s}}^2\right)\text{k}\end{array}$

Substitute $\left(12\text{rad}/\text{s}\right)\text{k}$ for ${\omega }_2$ and $\left(-2.16\text{m}/\text{s}\right)\text{i}$ for ${\overrightarrow{v}}_{A/C}$ in Equation (VII).

$\begin{array}{c}{\overrightarrow{a}}_{A/C}=\left(12\text{rad}/\text{s}\right)\text{k}\times \left(-2.16\text{m}/\text{s}\right)\text{i}\\ \text{=}-\left(25.92\text{m}/{\text{s}}^{\text{2}}\right)\text{j}\end{array}$

Substitute $\left(8\text{rad}/\text{s}\right)\text{j}$ for ${\omega }_1$ and $\left(-2.16\text{m}/\text{s}\right)\text{i}$ for ${\overrightarrow{v}}_{A/C}$ in Equation (VIII).

$\begin{array}{c}{\overrightarrow{a}}_C=2\left(8\text{rad}/\text{s}\right)\text{j}\times \left(\left(-2.16\text{m}/\text{s}\right)\text{i}\right)\\ =\left(34.56\text{m}/{\text{s}}^{\text{2}}\right)\text{k}\end{array}$

Substitute $-\left(9.6\text{m}/{\text{s}}^2\right)\text{i}+\left(23.04\text{m}/{\text{s}}^2\right)\text{k}$ for ${\overrightarrow{a}}_{A^{\prime }}$, $-\left(25.92\text{m}/{\text{s}}^{\text{2}}\right)\text{j}$ for ${\overrightarrow{a}}_{A/C}$ and $\left(34.56\text{m}/{\text{s}}^{\text{2}}\right)\text{k}$ for ${\overrightarrow{a}}_C$.

$\begin{array}{c}{\overrightarrow{a}}_A=-\left(9.6\text{m}/{\text{s}}^2\right)\text{i}+\left(23.04\text{m}/{\text{s}}^2\right)\text{k}-\left(25.92\text{m}/{\text{s}}^{\text{2}}\right)\text{j}+\left(34.56\text{m}/{\text{s}}^{\text{2}}\right)\text{k}\\ \text{=}-\left(9.6\text{m}/{\text{s}}^2\right)\text{i}-\left(25.92\text{m}/{\text{s}}^{\text{2}}\right)\text{j}+\left(57.6\text{m}/{\text{s}}^{\text{2}}\right)\text{k}\end{array}$

Conclusion:

The velocity of point $A$ is $-\left(5.04\text{m}/\text{s}\right)\text{i}-\left(1.2\text{m}/\text{s}\right)\text{k}$.

The acceleration of point $A$ is $-\left(9.6\text{m}/{\text{s}}^2\right)\text{i}-\left(25.92\text{m}/{\text{s}}^{\text{2}}\right)\text{j}+\left(57.6\text{m}/{\text{s}}^{\text{2}}\right)\text{k}$.