Chapter 8: Rotational Motion



Chapter 8:

Rotational Motion

Sec 8-1

Rigid Body – the distance between any two particles remains constant.

In general the motion of a rigid body consists of translation of the center of mass plus rotation about the center of mass (CM).

Angular Movement of a Rigid Body

Consider a particle P at position PI moving along a circular arc (S

Linear Speed of P is

[pic]

[pic]

( All particles in the wheel “system” will sweep out the same angle Δθ in a given amount of time.

( Therefore, the rate of change of the angle Δθ with respect to time, Δθ/Δt, is the same for all particles on the wheel.

Multiply both sides of above equation by (1/Δt)

[pic]

[pic]

[pic]

Units for “omega” ω:

[pic]

The Angular Acceleration, α (alpha) is

[pic] units for alpha are [pic]

Relating Angular Acceleration to Linear Acceleration

[pic]

[pic]

[pic]

[pic]

( Remember !!!!!!!!

[pic]

[pic]

Sec 8-2

CONSTANT ANGULAR ACCELERATION EQUATIONS

RECALL: [pic], [pic], [pic]

THEN:

[pic]

OTHERS:

[pic] [pic]

Direction for the [pic] vectors

Any linear direction in the plane of the wheel does not uniquely define the rotation of the wheel because of symmetry

Right-Hand Rule: curl the fingers of your right hand in the direction of the rotation, and thumb will point in the direction of [pic]

[pic] Vectors

Angular Velocity and Frequency

Period, T = time to complete 1 revolution

Frequency, f = # of revolutions per second.

[pic] i.e. 3 rev/sec = 3 Hertz (Hz) [pic]

Period and Frequency are the reciprocal of each other!

Example:

A frequency of 1 rev/sec equals a period of 1 sec. (or sec/rev)

A frequency of 2 rev/sec equals a period of 0.5 sec. (or sec/rev)

Therefore, [pic] OR [pic]

Since 1 rev = 2π radians

So, [pic] (Wow! this is [pic])

Therefore, [pic]

0r [pic]

Angular Velocity, [pic], is also called Angular Frequency.

EXAMPLE PROBLEMS

A particle moves in a circle of radius 100 m with a constant velocity of 20 m/s. (a) What is the angular velocity in radians per second about the center of the circle? (b) How many revolutions does it make in 30 s? Answer: 0.2 rad/s, 0.955 rev

A 30 cm diameter music record (vinyl) rotates at 33 1/3 rev/min. (a) What is its angular velocity in radians per second? (b) Find the speed and linear acceleration of a point on the rim of the record.

a.) 3.49 rad/s b.) 52.35 cm/s c.) atotal = 0 tan + 182 cm/s2 rad

A turntable rotating at 33 1/3 rev/min is shut off. It brakes with a constant angular acceleration and comes to rest in 2.0 min. (a) Find the angular acceleration. (b) What is the average angular velocity of the turntable? (c) How many revolutions does it make before stopping?

a.) -16.7 rev/min2 b.) 16.63 rev/min c.) 33.26 rev

Comparison of Linear Motion and Rotational Motion

|Linear Motion | | |Rotational Motion | |

| | | | | |

|Displacement |[pic] | |Angular Displacement |[pic] |

|Velocity |[pic] | |Angular Velocity |[pic] |

|Acceleration |[pic] | |Angular Acceleration |[pic] |

|Constant Acceleration |[pic] | | Constant Angular Acceleration Equations |[pic] |

|Equations | | | | |

|Mass |m | |Moment of Inertia |I |

|Momentum |[pic] | |Angular Momentum |[pic] |

|Force |[pic] | |Torque |[pic] |

|Power |[pic] | |Power |[pic] |

|Newton’s 2nd Law |[pic] | |Newton’s 2nd Law |[pic] |

| | | | | |

| | | | | |

| | | | | |

8-3 Rolling Motion

Rolling Without Slipping – Velocity and Acceleration

[pic]

s = distance (x the cm of the wheel moved

[pic]

Then

[pic]

Rotation and Translation of Points on the Wheel

While Rolling Without Slipping

[pic] [pic] [pic]

Two Ways to achieve Rolling without Slipping

I. Chain/Gears Spin Wheel – Static Friction from Ground Translates Wheel

II. Push on Axel Translates Wheel – Static Friction from Ground allows wheel to Rotate

8-4 TORQUE

Archimedes (287-312 BC) – examined the turning “effect” of forces in his analysis of levers.

By Definition:

[pic]

Greek Letter “Tau” is

the symbol for torque.

Lever Arm = The perpendicular distance from the pivot point to the line of action of the force.

Often the distance along the stick itself is the lever arm distance - BUT NOT ALWAYS, only when Fpush is tangent to arc of rotation.

[pic]

LEVER ARM FORCE TANGENT

[pic] [pic]

[pic] [pic]

WHERE WHERE

[pic] [pic]

EXAMPLE: A 30 N and a 15 N force are applied to the rod shown below. Determine the net torque and direction of rotation.

8-5 Torque and Moment of Inertia

A force, F, is acting on a wheel

Fradial, does not effect the rotation of the wheel. The torque exerted by F is

[pic]

BY DEFINITION

The magnitude of the torque, τ, “tau” is equal to the product of Ftangent , and the magnitude of the radius vector.

SINCE: F = ma and [pic]

Now if we sum all of the particles, m, on the wheel.

[pic]

The sum, [pic], is a property of the wheel called the moment of inertia, I. The moment of inertia depends on the distribution of the mass relative to the axis of rotation.

From C and D

[pic] [pic]

[pic] Newton’s 2nd

Law for rotation

How to “think” about Torque

1. Torque must be specified about a pivot point

2. Torque is a product quantity made up of distance and force.

3. Torque causes angular acceleration, (, in the same way that forces cause linear accelerations.

4. The Moment of Inertia, I, is a measure of resistance to rotation analogous to mass as a measure of inertia for linear motion.

5. We can’t “see” or “touch” torque, in the same sense that we can’t “see” or “touch” forces – We see the effects of both.

6. Equilibrium means

[pic]

Object at rest totally

or translating and/or rotating with constant v or (

8-6 Solving Problems in Rotational Dymanics

Example: A uniform cylinder of mass M and radius R is held by a hand that is accelerated upward so that the center of mass of the cylinder does not move. Find (a) the tension in the string, (b) the angular acceleration of the cylinder, and (c) the acceleration of the hand

Answers: (a) Mg (b) 2g/R (c) 2g

Example: A body of mass 1.2 kg is tied to a light string wound around a 2.5 kg wheel of radius 0.2 m. The wheel bearing is frictionless.

Find the tension is the string, the acceleration of the block, and its speed after it has fallen a distance h = 0.25 m from rest. T=6N, a = -4.8 m/s2 , v = 1.55 m/s

8-7 Rotational Kinetic Energy

Consider of piece of mass rotating about a central point

The Kinetic energy of the mass is

[pic]

Consider a wheel made up of many such masses, then

[pic]

[pic]

( Remember that the wheel may also be translating

Then

[pic][pic]

Rolling Without Slipping – Different Shapes

Moment of Inertia Considerations (Same Mass, Same Radius)

Hoop Disk Sphere

[pic] [pic] [pic]

[pic]

Hoop Disk Sphere

[pic] [pic] [pic]

[pic] [pic] [pic]

If started from rest at the top of a hill, which will move the fastest at the bottom of a hill?

[pic]

Hoop Disk Sphere

[pic] [pic] [pic]

[pic] [pic] [pic]

Example: A hoop of radius 0.5 m and mass 0.8 kg is rolling without slipping at a speed of 20 m/s toward an incline of slope 30o . How far along the incline will the hoop roll? Assuming it rolls with out slipping. Answer 81.5 m

Example: Four particles of mass m are connected by massless rods to form a rectangle of sides 2a and 2b, as shown below. The system rotates about an axis in the plane of the figure through the center. Find the moment of inertia about this axis and the kinetic energy of rotation if the angular velocity is (

Example: Find the moment of inertia and Kinetic Energy of rotation of the system shown below.

8-8 Angular Momentum and Its Conservation

Definition of Angular Momentum

Recall Now

Since [pic]

If [pic] then [pic] and [pic]

Torque and Angular Momentum – Formal Definitions

[pic] Torque analogous to Force

[pic] Angular momentum analogous to Linear Momentum

( An object moving in a straight line has an angular momentum about the pivot point O.

Newtons 2nd Law for Torques in momentum form

[pic] [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Conservation of Angular Momentum

If the resultant external torque acting on a system is zero, the total angular momentum of the system is constant.

[pic]

Conservation of Angular Momentum

Example: A playground merry-go-round is at rest, pivoted about a frictionless axis. A ferocious rabbit runs along a path tangential to the rim with initial speed vi and jumps onto the merry-go-round. What is the angular velocity of the merry-go-round and the rabbit?

Example: A disk with a moment of inertia I1 is rotating with angular velocity (1 about a frictionless shaft. It drops onto another disk with moment of inertia I2 initially at rest. Because of surface friction, the two disks eventually attain a common angular velocity. What is it?

Appendix I

Continuous Mass Distributions

( For a continuous body replace [pic] with

[pic]

Example: Calculate the Moment of Inertia of a uniform stick about an axis perpendicular to the stick through one end

Appendix II - Some Rotational Inertias

[pic]

-----------------------

Ffriction

Rotation about CM

The Actual Path of the object’s center of mass

Reference Line

Pi

DðS

Dðqð

qðið

[pic]

Rotating Wheel system made up of many particles P.

[pic]

[pic]

[pic]

And Speeding Up

PleasΔS

Δθ

θι

[pic]

Rotating Wheel “system” made up of many particles P.

[pic]

[pic]

[pic]

And Speeding Up

“Please – Make it Stop”

Ffriction

How can friction point in opposite directions?

O

vcm

vcm

30o

O

Δs

s

A

A

Conditions for Rolling Without Slipping

O

O

O

Spinning and Translating

Translating

Spinning

The gears spin the wheel. When the wheel hits the ground it lurches forward - Translation

Fpush translates the wheel. Static friction at the ground lets the wheel fall over itself – Rotation

Fon wheel

by ground

Fon wheel

by ground

Fpush

Pivot Point

Boulder

FPUSH

Lever Pole

FPULL

θ

θ

[pic]

Line-of-action of force

O

Radial vector from pivot point

[pic]

FPULL

θ

[pic]

F2=15 N

40o

30o

2 m

1 m

F1=30 N

40o

[pic]

m

Fradial

Ftangent

F

[pic]

A

B

[pic]

C

D

[pic]

M

M

(s

R

((

m

r

v

Vcm

(

71 % KEtrans

29 % KErot

67 % KEtrans

33 % KErot

50 % KEtrans

50 % KErot

30o

m

m

m

m

2b

2a

m

2b

2a

m

m

m

[pic]

[pic]

[pic]

with

[pic]

is Angular Momentum

[pic]

[pic]

[pic]

with

[pic]

is Linear Momentum

[pic]

[pic]

[pic]

(r,p

m

r

v

I1

I2

I2

I1

dm

x

dx

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