Unit 8: Circular Motion

Section A: Angular Units

Corresponding Textbook Sections:– 10.1

PA Assessment Anchors:– S11.C.3.1

Angular Position

Defined as the angle, , that a line from the axle to a spot on the wheel makes with a reference line

Unit: Radian (rad)

[dimensionless]

Sign convention for angular position:

If > 0, counterclockwise rotation

If < 0, clockwise rotation

Converting between degrees and radians

1 revolution = 360 = 2 rad

1 rad = 57.3

Convert the same way you would between any other units.

Section B: Angular / Linear Relationships

Corresponding Textbook Sections:– 10.3

PA Assessment Anchors:– S11.C.3.1

Arc Length

The arc length is the distance from a reference line to a spot of interest on a circle.

Equation:

s = r

Angular Velocity

Symbol:

Units: s-1 or 1/s

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av =Δθ

Δt

Sign Convention for

If > 0 Counterclockwise rotation

If < 0 Clockwise rotation

Practice Problem #1

An old phonograph rotates clockwise at 33⅓ rpm. What is the angular velocity in rad/s?

Practice Problem #2

If a CD rotates at 22 rad/s, what is its angular speed in rpm?

Period

The period is the time it takes to complete one revolution.

Units: seconds (s)

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T = 2π

ω

Practice Problem #3

Find the period of a record that is rotating at 45 rpm.

Angular Acceleration

The change in angular speed of a rotating object per unit of time.

Units: rad/s2

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α =ΔΔt

Practice Problem #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 long does it take for the windmill to come to a complete stop?

Section C: Angular Kinematics

Corresponding Textbook Sections:– 10.2

PA Assessment Anchors:– S11.C.3.1

Relationship between angular and linear quantities

Linear Quantity Angular Quantity

x

v ω

a α

Based on these relationships, we can rewrite thekinematics equations from 1-D and 2-D Kinematics

Angular Kinematics Equations

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v = vo + at ω =ωo +αt

x = xo + vot +1

2at 2 θ =θo +ωot +

1

2αt 2

v 2 = vo2 + 2aΔx ω2 =ωo

2 + 2αΔθ

So, basically…

These are just variations of equations we already know how to use.

They work the same way as the linear equations.

We’ll use the same setup as before:• Data table, equation, picture, etc…

Practice Problem #1

To throw a curveball, a pitcher gives the ball an initial angular speed of 36 rad/s. When the catcher gloves the ball 0.595 s later, its angular speed has decreased to 34.2 rad/s. What is the ball’s angular acceleration?

Practice Problem #2

Based on the last problem, how many revolutions does the ball make before being caught?

Practice Problem #2

Refer to Example 10-2 on page 280

Section D: Torque

Corresponding Textbook Sections:– 11.1, 11.2

PA Assessment Anchors:– S11.C.3.1

What is Torque?

Torque is the rotational equivalent of force

It depends on:– Force applied– Distance from the force to the axis of

rotation

More on Torque…

Equation:

Units: Nm

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τ =rFGreek Letter “tau”

Axis of Rotation(where it turns)

Practice Problem #1

If the minimum required torque to open a door is 3.1 Nm, what force must be applied if:– r = 0.94 m– r = 0.35 m

Section E: Moment of Inertia

Corresponding Textbook Sections:– 10.5

PA Assessment Anchors:– S11.C.3.1

What is “Moment of Inertia”?

The “rotational mass” of an object– Rotational mass depends on actual mass,

radius, and distribution of mass

Useful for determining rotational KE:

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KE =1

2Iω2

Moment of inertia

Practice Problem #1

What is the moment of inertia of a hollow sphere with mass of 40 kg and radius of 3 m?

Practice Problem #2

A grindstone with radius of 0.61 m is being used to sharpen an axe. If the linear speed of the stone relative to the ax is 1.5 m/s, and the stones rotational KE is 13 J, what is its moment of inertia?