Post on 30-Dec-2015
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Chapter 8: Rotational Motion
• Topic of Chapter: Objects rotating– First, rotating, without translating.
– Then, rotating AND translating together.
• Assumption: Rigid Body– Definite shape. Does not deform or change shape.
• Rigid Body motion = Translational motion of center of mass (everything done up to now) + Rotational motion about an axis through center of mass. Can treat the two parts of motion separately.
COURSE THEME: NEWTON’S LAWS OF MOTION!
• Chs. 4 - 7: Methods to analyze the dynamics of objects in
TRANSLATIONAL MOTION. Newton’s Laws! – Chs. 4 & 5: Newton’s Laws using Forces
– Ch. 6: Newton’s Laws using Energy & Work
– Ch. 7: Newton’s Laws using Momentum.
NOW• Ch. 8: Methods to analyze dynamics of objects in
ROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language! – First, Rotational Language. Analogues of each translational
concept we already know!
– Then, Newton’s Laws in Rotational Language.
A rigid body is an extended object whose size, shape, & distribution of mass don’t change as the object moves and rotates. Example: a CD
Rigid Body Rotation
Three Basic Types of Rigid Body Motion
Pure Rotational MotionAll points in the object movein circles about the rotation
axis (through the Center of Mass)
Reference Line
The axis of rotation is through O & is
to the picture. All points move in circles about O
r
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time.
r
r
Sect. 8-1: Angular Quantities
• Description of rotational
motion: Need concepts:
Angular Displacement
Angular Velocity, Angular Acceleration
• Defined in direct analogy to linear quantities.
• Obey similar relationships!
Positive Rotation! r
• Rigid object rotation:– Each point (P) moves
in a circle with the
same center!
• Look at OP: When P
(at radius R) travels an
arc length ℓ, OP sweeps
out angle θ.
θ Angular Displacement of the object
Reference Line
r
• θ Angular Displacement• Commonly, measure θ in degrees.• Math of rotation: Easier if
θ is measured in Radians
• 1 Radian Angle swept out
when the arc length = radius
• When R, θ 1 Radian
• θ in Radians is defined as:
θ = ratio of 2 lengths (dimensionless)
θ MUST be in radians for this to be valid!
Reference Line
r
• θ in Radians for a circle of radius r, arc length is defined as: θ (/r)
• Conversion between radians & degrees:
θ for a full circle = 360º = (/r) radians
Arc length for a full circle = 2πr
θ for a full circle = 360º = 2π radians
Or 1 radian (rad) = (360/2π)º 57.3º
Or 1º = (2π/360) rad 0.017 rad– In doing problems in this chapter, put your
calculators in RADIAN MODE!!!!
Example 8-2: θ 310-4 rad = ? º
r = 100 m, = ?
a) θ = (310-4 rad)
[(360/2π)º/rad] = 0.017º
b) = rθ = (100) (310-4)
= 0.03 m = 3 cm
θ MUST be in radians in part b!
Angular Displacement
Average Angular Velocity =
angular displacement θ = θ2 – θ1
(rad) divided by time t:
(Lower case Greek omega, NOT w!)
Instantaneous Angular Velocity
(Units = rad/s) The SAME for all points
in the object! Valid ONLY if θ is in rad!
Angular Velocity(Analogous to linear velocity!)
• Average Angular Acceleration = change in angular velocity ω = ω2 – ω1 divided by time t:
(Lower case Greek alpha!)
• Instantaneous Angular Acceleration = limit of α as t, ω 0
(Units = rad/s2)
The SAME for all points in body! Valid ONLY for θ in rad & ω in rad/s!
Angular Acceleration(Analogous to linear acceleration!)
Ch. 5 (circular motion): A mass moving in a circle
has a linear velocity v & a
linear acceleration a.
We’ve just seen that it also
has an angular velocity &
an angular acceleration.
There MUST be relationships between the linear & the angular quantities!
Relations of Angular & Linear Quantities
Δθ
Δ
r
Connection Between Angular & Linear Quantities
v = (/t), = rθ v = r(θ/t) = rω
Radians!
v = rω Depends on r(ω is the same for all points!)
vB = rBωB, vA = rAωA vB > vA since rB > rA
Summary: Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related as:
Relation Between Angular & Linear Acceleration
In direction of motion:(Tangential acceleration!)
atan= (v/t), v = rω
atan= r (ω/t)
atan= rα
atan : depends on r
α : the same for all points
_____________
Angular & Linear AccelerationFrom Ch. 5: there is also
an acceleration to the
motion direction (radial or
centripetal acceleration)
aR = (v2/r)
But v = rω
aR= rω2
aR: depends on r
ω: the same for all points
_____________
Total Acceleration Two vector components
of acceleration
• Tangential:
atan= rα
• Radial:
aR= rω2
• Total acceleration
= vector sum:
a = aR+ atan
_____________
a ---
Relation Between Angular Velocity & Rotation Frequency
• Rotation frequency:
f = # revolutions / second (rev/s)
1 rev = 2π rad
f = (ω/2π) or ω = 2π f = angular frequency
1 rev/s 1 Hz (Hertz)
• Period: Time for one revolution.
T = (1/f) = (2π/ω)
Translational-Rotational Analogues & ConnectionsANALOGUES
Translation Rotation
Displacement x θ
Velocity v ω
Acceleration a α
CONNECTIONS
= rθ, v = rω
atan= r α
aR = (v2/r) = ω2 r
Correspondence between Linear & Rotational quantities
On a rotating merry-go-round, one child sits on a horse near the outer edge & another child sits on a lion halfway out from the center.
a. Which child has the greater translational velocity v?
b. Which child has the greater angular velocity ω?
Conceptual Example 8-3: Is the lion faster than the horse?
Example 8-4: Angular & Linear Velocities & Accelerations
A merry-go-round is initially at rest (ω0 = 0). At t = 0 it is given a constant angular acceleration α = 0.06 rad/s2. At t = 8 s, calculate the following:
a. The angular velocity ω. b. The linear velocity v of a child located r = 2.5 m from the center.c. The tangential (linear) acceleration atan of that child.
d. The centripetal acceleration aR of the child.
e. The total linear acceleration a of the child.
Example 8-5: Hard Drive
The platter of the hard drive of a computer rotates at frequency f = 7200 rpm (rpm = revolutions per minute = rev/min)
a. Calculate the angular velocity ω (rad/s) of the platter.
b. The reading head of the drive r = 3 cm (= 0.03 m) from the rotation axis. Calculate the linear speed v of the point on the platter just below it.
c. If a single bit requires 0.5 μm of length along the direction of motion, how many bits per second can the writing head write when it is r = 3 cm from the axis?