Chapter 7. Circular Motion and Gravitation

7.4.1. Describing Angular Motion

Describing Angular Motion

• Objects that rotate move in a circular path around a center of rotation.

• To gain a better understanding of rotational motion, we begin by considering the position, speed, and acceleration of a rotating object.

• Read Holt Physics p 898-903

© 2014 Pearson Education, Inc.

Describing Angular MotionAs a wheel rotates, every point on the wheel moves in a circular path around the axle, which is the axis of rotation. The angular position of the red dot is the angle 𝜃 that it makes with respect to a reference line 𝜃 = 0, which indicates how far the dot has rotated.

© 2014 Pearson Education, Inc.

The common convention is that positive angles are counterclockwise from the reference line, and negative angles are clockwise.

Arc Length

How can one compute the distance that a rotating particle travels?

The arc length is equal to the radius times the angle moved in radians:

𝑠 = 𝑟 ∙ 𝜃 rad

Angle Unit ComparisionOne revolution = 1 rev = 360 degrees = 360⁰

= 2π radians = 2 π rad

1 rad ≈ 57.3 ⁰

The radian is actually dimensionless, since

it is a ratio of lengths, 𝜃 = 𝑠 𝑟 .nevertheless the

unit “rad” is often specified to indicate it is not deg or rev.

Example 1

A bike wheel rotates 4.50 revolutions.

(a) How many radians has it rotated?

4.50 rev2𝜋 rad

1 rev= 28.3 rad

(b) How many degrees is that?

4.50 rev360°

1 rev= 1620°

Angular Displacement and Velocity

The angular displacement is the change in angular position (i.e. angle), ∆𝜃 = 𝜃𝑓 − 𝜃𝑖.

The angular velocity is

𝜔 =∆𝜃

∆𝑡

SI units: rad/s = s-1

Note every point on the wheel moves at the same 𝜔.

Sign of Angular Velocity

For counterclockwise rotation, 𝜔 = ∆𝜃 ∆𝑡 > 0.

For clockwise rotation, 𝜔 < 0.

The magnitude of the angular velocity is the angular speed.

Every particle in the rotating object has the same 𝜔.

Example 2

An LP phonograph record rotates clockwise at 33⅓ rpm (revolutions per minute). What is its angular velocity in radians per second?

𝜔 = −3313

rev

min

2𝜋 rad

rev

1 min

60 s

= −3.49 rad/s

Tangential Speed

The speed in m/s at which a rotating point is moving is the arc length per unit time:

𝑣𝑡 =𝑠

∆𝑡=𝑟𝜃

∆𝑡= 𝑟𝜔

𝑣𝑡 is called the tangential speed, because at any instant its direction is tangential to the circular path. Thus linear speed 𝑣𝑡 in m/s and angular speed 𝜔 in rad/s are directly related through 𝑟.

Example 3

Do children side-by-side on a merry-go-round have the same angular velocity or tangential speed?

They have the same angular velocity but different tangential speeds (𝑣𝑡 = 𝑟𝜔).

Angular Acceleration

Angular acceleration is defined as the change in angular velocity per unit time:

𝛼 =∆𝜔

∆𝑡SI Units: rad/s2 = s-2.

The sign of 𝛼 may differ from the sign of 𝜔. If they have the same sign, the magnitude of 𝜔 is increasing.

Example 4

As the wind dies, a windmill that was rotating at 2.1 rad/s begins to slow down with a constant angular acceleration of -0.45 rad/s2. How much time does it take for the windmill to come to a complete stop?

𝛼 =∆𝜔

∆𝑡

∆𝑡 =∆𝜔

𝛼=𝜔𝑓 −𝜔𝑖

𝛼=0 − 2.1 rad/s

−0.45 rad/s2= 4.7 s

Tangential Acceleration

𝑎𝑡 =∆𝑣𝑡∆𝑡

=∆(𝑟𝜔)

∆𝑡=𝑟∆𝜔

∆𝑡= 𝑟𝛼

The tangential acceleration is the

change in tangential speed

per unit time.

SI Units: m/s2

Total Acceleration

The total acceleration of a rotating particle is the sum of its centripetal acceleration (due to change in direction) plus its tangential acceleration:

𝑣2

𝑣1

𝑣2

−𝑣1

∆𝑣

𝑟

𝒂𝑡𝑜𝑡𝑎𝑙 = 𝒂𝑡 + 𝒂𝑐𝑝

𝑎𝑡 = 𝑟𝛼𝑎𝑐𝑝 = 𝑣𝑡

2 𝑟

= 𝑟𝜔 2 𝑟 = 𝑟𝜔2

Since 𝑎𝑡 and 𝑎𝑐𝑝 are

perpendicular

𝑎𝑡𝑜𝑡𝑎𝑙2 = 𝑎𝑡

2 + 𝑎𝑐𝑝2

Summary of Variables

Property Linear Rotational Relation

Position 𝑥 = 𝑠 𝜃 𝑠 = 𝑟𝜃

Velocity 𝑣𝑡 𝜔 𝑣𝑡 = 𝑟𝜔

Acceleration 𝑎𝑡 𝛼 𝑎𝑡 = 𝑟𝛼

Linear Equation(a = constant)

Angular Equation(𝜶 = constant)

𝑥𝑓 = 𝑥𝑖 + 𝑣𝑎𝑣𝑡 𝜃𝑓 = 𝜃𝑖 +𝜔𝑎𝑣𝑡

𝑥𝑓 = 𝑥𝑖 + 𝑣𝑖𝑡 +12𝑎𝑡

2 𝜃𝑓 = 𝜃𝑖 + 𝜔𝑖𝑡 +12𝛼𝑡

2

𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡 𝜔𝑓 = 𝜔𝑖 + 𝛼𝑡

𝑣𝑎𝑣 =12 𝑣𝑖 + 𝑣𝑓 𝜔𝑎𝑣 =

12 𝜔𝑖 + 𝜔𝑓

𝑣𝑓2 = 𝑣𝑖

2 + 2𝑎∆𝑥 𝜔𝑓2 = 𝜔𝑖

2 + 2𝛼∆𝜃