Download - Chapter 8: Rotational Motion
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Chapter 8: Rotational Motion
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• 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.
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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.
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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
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Three Basic Types of Rigid Body Motion
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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
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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
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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
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• 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
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• 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
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• 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!!!!
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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!
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Angular Displacement
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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!)
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• 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!)
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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
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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
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Summary: Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related as:
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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
_____________
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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
_____________
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Total Acceleration Two vector components
of acceleration
• Tangential:
atan= rα
• Radial:
aR= rω2
• Total acceleration
= vector sum:
a = aR+ atan
_____________
a ---
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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)
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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
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Correspondence between Linear & Rotational quantities
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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?
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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.
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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?