Post on 28-May-2018
© 2010 Pearson Education, Inc. Slide 7-2
7 Rotational Motion
© 2010 Pearson Education, Inc. Slide 7-3
© 2010 Pearson Education, Inc.
Recall from Chapter 6…
Angular displacement = θ
θ= ω t
Angular Velocity = ω (Greek: Omega)
ω = 2 π f and ω = θ/ t
All points on a rotating object rotate through the same angle in the same time, and have the same frequency.
Angular velocity: all points on a rotating object have the same angular velocity, ω, but different speeds, v, and v =ωr.
v =ωr
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ω is positive if object is rotating counterclockwise. (Negative if rotation is clockwise.)
Conversion: 1 revolution = 2 π rad
© 2010 Pearson Education, Inc.
Checking Understanding
Two coins rotate on a turntable. Coin B is twice as
far from the axis as coin A.
A. The angular velocity of A is twice that of B.
B. The angular velocity of A equals that of B.
C. The angular velocity of A is half that of B.
Slide 7-13
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Answer
All points on the turntable rotate through the
same angle in the same time. ω = θ/ tAll points have the same period, therefore, all
points have the same frequency.
ω = 2 π f
Two coins rotate on a turntable. Coin B is
twice as far from the axis as coin A.
A. The angular velocity of A is twice
that of B.
B. The angular velocity of
A equals that of B.
C. The angular velocity of A is half
that of B.
Slide 7-14
© 2010 Pearson Education, Inc.
Two coins rotate on a turntable. Coin B is
twice as far from the axis as coin A.
A. The speed of A is twice that of B.
B. The speed of A equals that of B.
C. The speed of A is half that of B.
Checking Understanding
Slide 7-15
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Answer
Twice the radius means twice the speed, v.
Therefore, B moves with twice the speed of A.
Two coins rotate on a turntable. Coin B is
twice as far from the axis as coin A.
A. The speed of A is twice that of B.
B. The speed of A equals that of B.
C. The speed, v, of A is
half that of B.
Slide 7-16
v rw=
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Angular Acceleration, α (Greek: Alpha)
Angular acceleration α measures how
rapidly the angular velocity is changing:
Units for Angular Acceleration
are rad/s/s or rad/s2
If the turning rate, or angular
velocity, ω, is constant, then there
is no angular acceleration, α, but
if ω is changing, then the object is
NOT undergoing UCM and the
object has angular acceleration…
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Sign of the Angular Acceleration—careful!
+ω, +α
=speeding up
-ω, +α =
slowing down +ω, -α =
slowing down
-ω , -α =
speeding up
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Do NOT confuse angular acceleration, α = Δω/ Δt,
with centripetal acceleration,
ac = v 2/r = ω2r. Angular acceleration, α, indicates how rapidly
the angular velocity, ω, is changing. (No UCM)
Centripetal acceleration, ac, is a vector
quantity, directed toward the center of a particle’s circular path, and is nonzero even if the angular velocity (turning rate) is constant. (ac exists always– whether experiencing UCM or not.)
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Linear and Circular Motion Compared
Slide 7-18
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Linear and Circular Kinematics Compared
Slide 7-19
ωf = ωi +αt
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Linear Motion Graphs related to Rotational Motion Graphs:
Displacement vs. Time: Slope = Velocity
Angular displacement, θ, vs. Time:
Slope = Angular Velocity, ω= Δθ/Δt
θ
t
© 2010 Pearson Education, Inc.
Linear Motion Graphs related to Rotational Motion Graphs:
Velocity vs. Time: Slope = Acceleration
Angular velocity vs. Time,
Slope = Angular Acceleration, α =Δω/Δt
ω
t
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Centripetal and Tangential Acceleration
Slide 7-22
No UCM:
© 2010 Pearson Education, Inc.
Example Problem
A high-speed drill rotating CCW
takes 2.5 s to speed up to 2400 rpm.
A. What is the drill’s angular
acceleration?
B. How many revolutions does it make
as it reaches top speed?
Slide 7-21
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Answer:
Given: ωi = 0, ωf = 2400 rev/min, t = 2.5 seconds
ωf = 2400 rev/min (2πrad/1 rev)(1 min/60 sec) = 251.3 rad/s
Find: α = ?, θ = # revolutions = ?
Plan: Find α using ωf = ωi +αt, then find θusing θ=ωi t + 1/2 αt2. Convert θ to revolutions.
Solve: ωf = ωi +αt
251.3 rad/s = 0 + α (2.5 s)
α= (251.3 rad/s)/(2.5 s) = 100.5 rad/s2 = α
Go to next slide for more…
K
j
k
;
K
j
k
;
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Δθ=ωi t + 1/2 αt2
Δθ= 0 + ½ (100.5 rad/s2)(2.5 s)2
Δθ= 314.16 rad
Δθ = 314.16 rad (1 rev/2πrad) = 50 rev
Δθ= 50 revolutions
© 2010 Pearson Education, Inc.
© 2010 Pearson Education, Inc.
Torque… rhymes with fork…
Every time you open a door, turn on a water faucet, or tighten a nut with a wrench, you exert a turning force.
This turning force produces a torque.
Torque is different from a force.
If you want to make an object move, apply a force. Forces tend to make things accelerate.
If you want to make an object turn or rotate, apply a torque. Torques produce rotation.
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Interpreting Torque, ( (Greek: Tau)
Slide 7-25
Torque is due to the component of the force
perpendicular to the radial line.
t = rF̂ = rFsinf
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A Second Interpretation of Torque
t = r̂ F = rFsinfSlide 7-26
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Signs and Strengths of the Torque
Slide 7-27
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Torque: that which causes rotational velocity to change.
Variable for torque= (tau)
Increasing F causes greater linear acceleration
Increasing causes greater rotational acceleration
Demonstration: pushing door open…When does door open best?
A) from outside edge to axis of rotation
B) pushing toward hinge
C) at an angle
D) from center of door to axis of rotation (see next slide)
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Checking Understanding
The four forces shown have the same
strength. Which force would be most
effective in opening the door?
Slide 7-23
A. Force F1
B. Force F2
C. Force F3
D. Force F4
E. Either F1 or F3
© 2010 Pearson Education, Inc.
Answer
The four forces shown have the same strength. Which force
would be most effective in opening the door?
Slide 7-24
A. Force F1B. Force F2
C. Force F3
D. Force F4
E. Either F1 or F3
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What does torque depend on?
1. The size of the force (how hard you push)
2. The point where the force is applied relative to the rotation point.
3. The direction of the force.
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= F rWhere F and r have to be to each other!
= F r or…
= F r (Units: N·m)
F = the force applied
r = the moment arm or the lever arm = distance between axis of rotation and line of action so that where the lever arm (r) and line of action meet, a 90o angle is formed.
Pivot point = axis of rotation(in this case, the hinge)
Torque is positive when the force tends to produce a counter-clockwise rotation, and negativewhen force tends to produce clockwise rotation.
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Even though the magnitude of the applied force is the same in each case, why does the bottom situation produce the greatest amount of torque?
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For You to Solve:
A person pushes on the edge of a 0.90m wide door with a force of 5.0N acting at an angle of 350 to the plane of the door. What is the torque of the force acting on the door?
© 2010 Pearson Education, Inc.
© 2010 Pearson Education, Inc.
Another problem for you to solve…
Revolutionaries attempt to pull down a
statue of the Great Leader by pulling on
a rope tied to the top of his head. The
statue is 17 m tall, and they pull with a
force of 4200 N at an angle of 65° to
the horizontal. What is the torque they
exert on the statue? If they are standing
to the right of the statue, is the torque
positive or negative?
Slide 7-28
© 2010 Pearson Education, Inc.
And the answer is…
F = F (cosθ)
F = 4200N (cos 65o)
F = 1775 N
= F r
= (1775 N)(17 m)
= 30,175 Nm, clockwise
= - 30,175 Nm
θ=65o
F=?
F = 4200N
r = 17 m
© 2010 Pearson Education, Inc.
Which factor does the torque on an
object not depend on?
A. The magnitude of the applied force.
B. The object’s angular velocity.
C. The angle at which the force is
applied.
D. The distance from the axis to the
point at which the force is applied.
Slide 7-7
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Answer
Which factor does the torque on an object
not depend on?
A. The magnitude of the applied force.
B.The object’s angular velocity.
C. The angle at which the force is
applied.
D. The distance from the axis to the
point at which the force is applied.
Slide 7-8
© 2010 Pearson Education, Inc.
© 2010 Pearson Education, Inc. Slide 7-4
© 2010 Pearson Education, Inc.
A net torque applied to an object
causes…???
A. a linear acceleration of the
object.
B. the object to rotate at a constant
rate.
C. the angular velocity of the object
to change.
D. the moment of inertia of the
object to change.
Slide 7-11
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Answer
A net torque applied to an object
causes…
A. a linear acceleration of the object.
B. the object to rotate at a constant
rate.
C.the angular velocity of the
object to change.
D. the moment of inertia of the object
to change.
Slide 7-12