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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