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Chapters 10 & 11 – Rotational motion, torque, and angular momentum
PHY 113 C General Physics I 11 AM - 12:15 PM MWF Olin 101 Plan for Lecture 12: Chapters 10 & 11 – Rotational motion, torque, and angular momentum Torque Angular momentum Problems 1.1,1.6,1.10,1.11 10/08/2013 PHY 113 C Fall Lecture 12
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PHY 113 C Fall Lecture 12 10/08/2013
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angular “displacement” q(t) angular “velocity”
Angular motion angular “displacement” q(t) angular “velocity” angular “acceleration” “natural” unit == 1 radian Relation to linear variables: sq = r (qf-qi) vq = r w aq = r a s 10/08/2013 PHY 113 C Fall Lecture 12
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angular “displacement” q(t) angular “velocity”
Rotational motion q(t) w(t) angular “displacement” q(t) angular “velocity” angular “acceleration” Special case of constant angular acceleration: a = a0: w(t) = wi + a0 t q(t) = qi + wi t + ½ a0 t2 ( w(t))2 = wi2 + 2 a0 (q(t) - qi ) 10/08/2013 PHY 113 C Fall Lecture 12
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Webassign Assignment #10:
A centrifuge in a medical laboratory rotates at an angular speed of 3800 rev/min. When switched off, it rotates 48.0 times before coming to rest. Find the constant angular acceleration of the centrifuge. Special case of constant angular acceleration: a = a0: w(t) = wi + a0 t q(t) = qi + wi t + ½ a0 t2 ( w(t))2 = wi2 + 2 a0 (q(t) - qi ) Recall: 10/08/2013 PHY 113 C Fall Lecture 12
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Review of rotational energy associated with a rigid body
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Which moment of inertia is the smallest? (A) i (B) j (C) k
Note that for a given center of rotation, any solid object has 3 moments of inertia; some times two or more can be equal j d d m m i k iclicker exercise: Which moment of inertia is the smallest? (A) i (B) j (C) k IB=2md2 IC=2md2 IA=0 10/08/2013 PHY 113 C Fall Lecture 12
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From Webassign: 10/08/2013 PHY 113 C Fall Lecture 12
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Digression – use of rotational energy in energy storage
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Beacon Power company (went bankrupt in 2011) Continuing efforts –
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Physics of rolling -- CM CM 10/08/2013
PHY 113 C Fall Lecture 12
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Rolling motion reconsidered:
Kinetic energy associated with rotation: Distance to axis of rotation Rolling: 10/08/2013 PHY 113 C Fall Lecture 12
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Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first? C B A 10/08/2013 PHY 113 C Fall Lecture 12
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How can you make objects rotate?
q How can you make objects rotate? Define torque: t = r x F t = rF sin q r F sin q q F 10/08/2013 PHY 113 C Fall Lecture 12
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Another example of torque:
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10/08/2013 PHY 113 C Fall Lecture 12
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X Example form Webassign #11 iclicker exercise t3
When the pivot point is O, which torque is zero? A. t1? B. t2? C. t3? t3 X t2 t1 10/08/2013 PHY 113 C Fall Lecture 12
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Newton’s second law applied to center-of-mass motion
Newton’s second law applied to rotational motion ri mi di Fi 10/08/2013 PHY 113 C Fall Lecture 12
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In this case I = ½ m R2 and t = FR
An example: A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s. R F K = ½ I w t = I a w = wi + at = at In this case I = ½ m R and t = FR 10/08/2013 PHY 113 C Fall Lecture 12
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Re-examination of “Atwood’s” machine
T1-m1g = m1a T2-m2g = -m2a t =T2R – T1R = I a = I a/R R I T2 T1 T1 T2 10/08/2013 PHY 113 C Fall Lecture 12
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Another example: Two masses connect by a frictionless pulley having moment of inertia I and radius R, are initially separated by h=3m. What is the velocity v=v2= -v1 when the masses are at the same height? m1=2kg; m2=1kg; I=1kg m2 ; R=0.2m . h m1 m2 v1 v2 h/2 10/08/2013 PHY 113 C Fall Lecture 12
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Example from Webassign #10
Tl 10/08/2013 PHY 113 C Fall Lecture 12
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Note that rolling motion is caused by the torque of friction:
Newton’s law for torque: F fs 10/08/2013 PHY 113 C Fall Lecture 12
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Bicycle or automobile wheel:
fs 10/08/2013 PHY 113 C Fall Lecture 12
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What happens when the bicycle skids? Too much torque is applied
iclicker exercise: What happens when the bicycle skids? Too much torque is applied Too little torque is applied The coefficient of kinetic friction is too small The coefficient of static friction is too small More than one of these 10/08/2013 PHY 113 C Fall Lecture 12
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Vector cross product; right hand rule
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For unit vectors: k j i 10/08/2013 PHY 113 C Fall Lecture 12
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More details of vector cross products:
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What is the point of vector products To terrify physics students
iclicker exercise: What is the point of vector products To terrify physics students To exercise your right hand To define an axial vector To keep track of the direction of rotation 10/08/2013 PHY 113 C Fall Lecture 12
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From Newton’s second law:
iclicker exercise: Is this Wrong? Approximately right? Exactly right? 10/08/2013 PHY 113 C Fall Lecture 12
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From Newton’s second law – continued – conservation of angular momentum:
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Torque and angular momentum Define angular momentum:
For composite object: L = Iw Newton’s law for torque: In the absence of a net torque on a system, angular momentum is conserved. 10/08/2013 PHY 113 C Fall Lecture 12
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The student will remain at rest.
iclicker exercise: A student sits on a rotatable stool holding a spinning bicycle wheel with angular momentum Li. What happens when the wheel is inverted? counterclockwise The student will remain at rest. The student will rotate counterclockwise. The student will rotate clockwise. 10/08/2013 PHY 113 C Fall Lecture 12
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More details: 10/08/2013 PHY 113 C Fall Lecture 12
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Other examples of conservation of angular momentum
w1 w2 m m m m d1 d2 d1 d2 I2=2md22 I1=2md12 I1w1=I2w2 w2=w1 I1/I2 10/08/2013 PHY 113 C Fall Lecture 12
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What about centripetal acceleration?
ar = v2/R = w2 R 10/08/2013 PHY 113 C Fall Lecture 12
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Webassign problem: A disk with moment of inertia I1 is initially rotating at angular velocity wi. A second disk having angular momentum I2, initially is not rotating, but suddenly drops and sticks to the second disk. Assuming angular moment to be conserved, what would be the final angular velocity wf? 10/08/2013 PHY 113 C Fall Lecture 12
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Conserved quantity Necessary condition
Summary – conservation laws we have studied so far Conserved quantity Necessary condition Linear momentum p Fnet = 0 Angular momentum L tnet = 0 Mechanical energy E No dissipative forces 10/08/2013 PHY 113 C Fall Lecture 12
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