Chapter 6

Planar Kinematics of Rigid Bodies

1.
What is kinematics?

Rigid Bodies and Types of Motions

1.
What is a rigid body?






2.
If an object deforms, but the deformation is small, one can $\rule{1.2in}{.01in}$ its motion by modeling it as a rigid body.

3.
What is a body-fixed coordinate system?







Translation

4.
When is a rigid body said to be in translation?






5.
If a rigid body is in translation, the motion of the body is completely described by the motion of a $\rule{1.0in}{.01in}$ $\rule{1.0in}{.01in}$.


Rotation About a Fixed Axis

6.
In pure rotation problems it is often assumed that the rotation is about the $\rule{0.4in}{.01in}$ axis.

Two-Dimensional Motion

7.
Another name for two-dimensional motion is $\rule{1.0in}{.01in}$ motion. Describe when a rigid body is said to undergo two-dimensional motion.






8.
What is the plane of motion?







Rotation About a Fixed Axis

1.
Sketch Figure 6.7.












2.
The $\rule{1.0in}{.01in}$ $\rule{0.4in}{.01in}$ is used to describe the object's orientaiton.

3.
Write the definition of the angular velocity $\omega$.





4.
Write the definition of the angular acceleration $\alpha$.





5.
Sketch Figures 6.8(a) and (b) below (side-by-side).












6.
Write an expression for the velocity v.

7.
Write an expression for at and an, the tangential and normal components of the acceleration.






8.
Sketch Figure 6.9. (Don't try drawing the teeth. Simply draw two circles intersecting at one point.)


















9.
The velocities of the gears must be equal at P. Write the resulting equation.





General Motions: Velocities

1.
Sketch Figure 6.11(a).















2.
Complete the following equations.

3.
Sketch Figures 6.12(a) and (b) (side-by-side).












4.
Describe what is meant by rolling.









5.
Complete the following equation.

6.
State the above equation in words.












7.
Sketch Figures 6.12(c) and (d) (side-by-side).












8.
Complete the following equation.

The Angular Velocity Vector

9.
At any instant a rigid body rotates about an axis called the instantaneous axis of rotation. Describe the relationship between this axis and the angular velocity vector.






10.
Write an expression for $\mbox{$\bar{{\bf v}}$}_{A/B}$.





11.
Use the above equation to write a new expression for $\mbox{$\bar{{\bf v}}$}_A$.





12.
Use this expression to derive the cartesian coordinate description of $\mbox{$\bar{{\bf v}}$}_A$ in Figure 6.16.












Instantaneous Centers









13.
Describe what is meant by instantaneous center.


General Motions: Accelerations

1.
Sketch Figures 6.25(a) (side-by-side).















2.
Complete the following equation.

3.
Sketch Figures 6.26(a). (Label the contact point with the ground C.)













4.
Show that $\mbox{$\bar{{\bf a}}$}_C = R\omega^2\mbox{$\bf{\hat j}$}$.












5.
Write the equation defining the angular acceleration vector $\mbox{$\bar{\bf \alpha}$}$.



6.
Beginning with $\mbox{$\bar{{\bf v}}$}_{A/B} = \mbox{$\bar{\bf \omega}$}\times\mbox{$\bar{{\bf r}}$}_{A/B}$, derive the expression for $\mbox{$\bar{{\bf a}}$}_{A/B}$ in terms of $\mbox{$\bar{\bf \alpha}$}$ and $\mbox{$\bar{\bf \omega}$}$.












7.
Rewrite equations (6.8) and (6.9) here.









8.
Recognizing that $\mbox{$\bar{\bf \omega}$}= \omega\mbox{$\bf{\hat k}$}$ and $\mbox{$\bar{{\bf r}}$}_{A/B} = r_{A/B}\mbox{${\bf{\hat e}}_r$}$, derive equation (6.10) starting with equation (6.9).













Sliding Contacts

1.
Sketch Figure 6.31.


















2.
Notice that the (x,y,z) coordinate system is body-fixed. Hence, the unit vectors $\mbox{$\bf{\hat i}$}$, $\mbox{$\bf{\hat j}$}$, and $\mbox{$\bf{\hat k}$}$ are not constant, because they $\rule{1.0in}{.01in}$ with the body-fixed coordinate system. Hence, the derivatives $\frac{d\mbox{$\bf{\hat i}$}}{dt}$, $\frac{d\mbox{$\bf{\hat j}$}}{dt}$, and $\frac{d\mbox{$\bf{\hat k}$}}{dt}$     are     are not     zero. (Circle the correct answer).

3.
Differentiate

\begin{displaymath} \mbox{$\bar{{\bf r}}$}_A = \mbox{$\bar{{\bf r}}$}_B + x\mbox{$\bf{\hat i}$}+ y\mbox{$\bf{\hat j}$}+ z\mbox{$\bf{\hat k}$}.\end{displaymath}

to obtain an expression for $\mbox{$\bar{{\bf v}}$}_A$.





4.
Use the relations

\begin{displaymath} \frac{d\mbox{$\bf{\hat i}$}}{dt} = \omega\times\mbox{$\bf{\h... ...{d\mbox{$\bf{\hat k}$}}{dt} = \omega\times\mbox{$\bf{\hat k}$},\end{displaymath}

to derive equation (6.11).












5.
Rewrite equations (6.11) and (6.12).





where






6.
Derive equation (6.12) for $\mbox{$\bar{{\bf a}}$}_A$.












7.
Rewrite equations (6.13) and (6.14).





where





8.
Write the simplification of equation (6.13) for two-dimensional motion.








Rotating Coordinate Systems

1.
Rewrite equations (6.19) and (6.20).









2.
These equations were derived for a body-fixed coordinate system. They     do     do not     apply to any coordinate system rotating with angular velocity $\mbox{$\bar{\bf \omega}$}$ and angular acceleration $\mbox{$\bar{\bf \alpha}$}$.


Chapter Summary

Read this section.