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S5.2 Angular positions, velocities and accelerations

[A] Using vectors to describe rotations

[B] Angular position

Commentary

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Commentary

  • We do not reset θ to zero with each complete rotation of the reference line. If the reference line completes two revolutions, then the angular position is θ=4π rad.
  • Avoid the popular misconception that something is moving along the rotation vector. Instead, something is moving around the direction of the vector, i.e. the vector defines the axis of rotation. (Nonetheless, the vector does define the motion through its magnitude and sense).
 

[C] Angular displacement

Key Point 5.2

Angular Displacement: If a rigid body rotates about a fixed axis changing the angular position from θ1 to θ2, the body undergoes an angular displacement

Δθ=θ2-θ1

Note: angular displacement is measured in radians.

[D] Angular velocity

Key Point 5.3

Average angular velocity: If a rigid body rotates around a fixed axis producing an angular displacement Δθ in time Δt, the average angular velocity about the axis is

\[ \omega_{av} = \frac{\Delta \theta}{\Delta t} \]

Key Point 5.4

Instantaneous angular velocity is defined as the average angular velocity over the next infinitesimal time interval

\[ \omega = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d \theta}{d t} \]

Note: the units of angular velocity are rad s-1.

Note

Note

Note: For a rigid body, all lines in it rotate through the same angle in the same time and the angular velocity is characteristic of the body as a whole.
 

Worked example

Worked example

Calculate the angular velocity in rad s-1 of the crankshaft of a car engine that is rotating at 4800 rev min-1.

Solution

The angular velocity is simply the number of radians of rotation per second. In one second the crankshaft rotates

(4800/60)×2π rad
and hence
ω=503 rad s-1

 

[E] Angular acceleration

Key Point 5.5

Average angular acceleration: If a rigid body accelerates about a fixed axis producing a change in angular velocity Δω in time Δt the average angular acceleration is

\[ \alpha_{av} = \frac{\Delta \omega}{\Delta t} \]

Key Point 5.6

Instantaneous angular acceleration is defined as the average angular acceleration over the next infinitesimal time interval

\[ \alpha = \lim_{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t} = \frac{d \omega}{d t} \]

Note: the units of angular acceleration are rad s-2.

Worked examples

Worked examples

A wheel accelerates from rest with constant angular acceleration to an angular velocity of 100 rad s-1 in 20 s. Find the angular acceleration.

Solution

The angular velocity changes by 100 rad s-1 in 20s. Thus

α=100/20=5 rad s-2

Worked Example

During a time interval t the flywheel of a generator turns through the angle

θ=at+bt3-ct4
where a,b and c are constants. Find expressions for the angular velocity and angular acceleration.

Solution

Angular velocity:

\[ \omega=\frac{d\theta}{dt}=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{a+3bt^2-4ct^3} \]

Angular acceleration:

\[ \alpha=\frac{d\omega}{dt}=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{6bt-12ct^2} \]

 

Learning Resources

Textbook: HRW Chapter 10.1-10.3
Course Questions:
Self-Test Questions: