9.1 · Distinguish translation and rotation about a fixed axis

University of Alberta EN PH 131: Engineering Mechanics: Dynamics

Introduction to Rigid-Body Dynamics

Recognizing how a rigid body and its points move

What you will learn

• Define the system, observer, reference frame, coordinates, and positive directions for planar motion.
• Distinguish translation from rotation about a fixed axis using a body's orientation and the motion of its points.
• Relate angular motion of a body to the tangential motion of points on it.
• Check motion descriptions and calculations for consistent directions and SI units.

1. Set up a clear description of motion

• Name the system, observer, reference frame, axes, and positive directions.
• Displacement is not path length; linear quantities describe points, while angular quantities describe turning.
• A point may be stationary even when other points on the body move.

2. Translation: the body does not turn

• The body's orientation stays constant.
• All points have equal displacement over the same time interval.
• At an instant, all points have the same velocity and acceleration.

3. Rotation about a fixed axis

• The axis stays fixed relative to the chosen frame.
• Points share angular motion, but linear speed depends on distance from the axis.
• Use radians for angular displacement in the arc-length relationship.

4. A reliable way to distinguish the motions

• Use orientation and more than one point to classify a body's motion.
• A curved path alone does not prove rotation of the body.
• A stationary point on the axis does not mean the whole body is stationary.

Worked example

An elevator moving upward

1. Set the frame and interval
   The elevator is the system, and the observer is fixed to the building. Choose the $y$-axis upward, so upward displacement is positive. Compare the corner and floor-centre points over the same initial-to-final interval.
2. Classify the motion
   The elevator's orientation stays unchanged, so this is translation. Every point on a translating body has the same displacement over the interval.
   $$\Delta\mathbf{r}_{\text{corner}}=\Delta\mathbf{r}_{\text{centre}}=3.2\,\mathbf{j}\,\mathrm{m}$$

Worked example

Comparing speeds on a rotating disk

1. Set the frame and knowns
   The disk is the system, observed from a frame fixed to its stationary support. Counterclockwise angular motion is positive. Both points share the disk's angular velocity, while their distances from the axis differ.
   $$\omega=8.0\,\mathrm{rad/s},\quad r_A=0.10\,\mathrm{m},\quad r_B=0.30\,\mathrm{m}$$
2. Apply the fixed-axis relationship
   For circular motion about a fixed axis, tangential-speed magnitude is radius times angular-speed magnitude. This relationship applies because the points follow circles centred on the stationary axis.
   $$v_t=r|\omega|$$
3. Calculate both speeds
   Substitute each radius. Point $B$ is three times as far from the axis as point $A$, so its tangential speed is three times as great.
   $$v_A=(0.10)(8.0)=0.80\,\mathrm{m/s},\quad v_B=(0.30)(8.0)=2.4\,\mathrm{m/s}$$

Worked example

A rod turning about a fixed pin

1. Set the system and signs
   The rod is the system, and the observer is fixed to the stationary support. Take counterclockwise as positive. The point remains $0.60\,\mathrm{m}$ from the pin; its path is a circle centred on the pin.
   $$\Delta\theta=30^\circ=\frac{\pi}{6}\,\mathrm{rad}$$
2. Find angular speed
   At constant angular speed, angular displacement divided by elapsed time gives angular velocity. The result is positive because the stated rotation is counterclockwise.
   $$\omega=\frac{\Delta\theta}{\Delta t}=\frac{\pi/6}{0.50}=1.05\,\mathrm{rad/s}$$
3. Find the point's speed
   The point's tangential-speed magnitude is its radius from the fixed pin multiplied by the angular-speed magnitude.
   $$v_t=r|\omega|=(0.60)(1.05)=0.63\,\mathrm{m/s}$$

Common mistakes and how to avoid them

Lesson summary

• Translation leaves the body's orientation unchanged; every point has the same displacement over the same interval.
• Rotation about a fixed axis turns the body around an axis stationary relative to the observer's frame.
• Points on a rotating body share angular motion, while their linear speeds depend on distance from the axis.
• For fixed-axis rotation, $s=r\theta$ and $v_t=r|\omega|$, with angles in radians.

Check your understanding

Key terms

Rigid body

An idealized body whose shape and size remain unchanged during motion.

Translation

Motion in which the body's orientation does not change.

Fixed axis

An axis that remains stationary relative to the selected observer's reference frame.

Angular velocity

The rate of change of angular position, with a sign indicating the chosen direction of rotation.

Tangential speed

The speed of a point moving along its circular path about the fixed axis.

Continue through EN PH 131

View the complete EN PH 131 University of Alberta EN PH 131: Engineering Mechanics: Dynamics curriculum and lessons

• 9.2 · Relate angular position, velocity, and acceleration
• 9.3 · Relate velocities of points on a planar rigid body
• 9.4 · Relate accelerations of points on a planar rigid body
• 9.5 · Apply introductory planar rigid-body equations of motion
• 1.2 · Relate position, path length, and displacement
• 1.3 · Differentiate position to obtain velocity and acceleration

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