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 2013-- Lecture 12

PHY 113 C Fall 2013-- Lecture 12 10/08/2013

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 2013-- Lecture 12

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 2013-- Lecture 12

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 2013-- Lecture 12

Review of rotational energy associated with a rigid body 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

From Webassign: 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

Digression – use of rotational energy in energy storage http://www.beaconpower.com/products/about-flywheels.asp 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

Beacon Power company (went bankrupt in 2011) Continuing efforts – http://www.scientificamerican.com/article.cfm?id=energy-storage-role-in-electric-grid 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

Physics of rolling -- CM CM 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

Rolling motion reconsidered: Kinetic energy associated with rotation: Distance to axis of rotation Rolling: 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

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 2013-- Lecture 12

Another example of torque: 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

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 2013-- Lecture 12

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 w2 t = I a w = wi + at = at In this case I = ½ m R2 and t = FR 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

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 2013-- Lecture 12

Example from Webassign #10 Tl 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

Bicycle or automobile wheel: fs 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

Vector cross product; right hand rule 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

For unit vectors: k j i 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

More details of vector cross products: 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

From Newton’s second law: iclicker exercise: Is this Wrong? Approximately right? Exactly right? 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

From Newton’s second law – continued – conservation of angular momentum: 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

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 2013-- Lecture 12

More details: 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

What about centripetal acceleration? ar = v2/R = w2 R 10/08/2013 PHY 113 C Fall 2013-- Lecture 12

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 2013-- Lecture 12

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 2013-- Lecture 12