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Welcome back to Physics 211

Welcome back to Physics 211. Today’s agenda: Rotational Dynamics Kinetic Energy Angular Momentum. Reminder. Tutorial HW10 (equilibrium rigid bodies) due Tue/Wed Exam 3 in class Thursday Nov 13

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Welcome back to Physics 211

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  1. Welcome back to Physics 211 Today’s agenda: Rotational Dynamics Kinetic Energy Angular Momentum

  2. Reminder • Tutorial HW10 (equilibrium rigid bodies) due Tue/Wed • Exam 3 in class Thursday Nov 13 • linear momentum, center of mass, equilibrium of rigid bodies, torque, rotational dynamics, angular momentum, periodic motion

  3. Recap • Torque  tendency of force to cause rotation • Angular velocity, acceleration for rigid body rotating about axis • Equation of rotational dynamics • Ia=t • Moment of inertia I

  4. Computing torque F |t|=|F|d =|F||r|sinq =(|F| sinq)|r| component force at 900 to position vector times distance q r d O

  5. Discussion Dw/Dt (Smiri2)=tnet a - angular acceleration Moment of inertia I I a =tnet cf Newton’s 2nd law Ma=F

  6. Demo • Spinning a weighted bar – moments of inertia

  7. Conditions for equilibrium of an extended object For an extended object that remains at rest and does not rotate: • The net force on the object has to be zero. • The net torque on the object has to be zero.

  8. A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center as point P. The angular velocity of Q is • twice as big as P • the same as P • half as big as P • none of the above

  9. A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center as point P. The linear velocity of Q is • twice as big as P • the same as P • half as big as P • none of the above

  10. Beam problem N r r CM m M? MP=2m Vertical equilibrium ? Rotational equilibrium ?

  11. Suppose M replaced by M/2? vertical equilibrium ? rotational dynamics ? net torque ? which way rotates ? initial angular acceleration ?

  12. Moment of Inertia ? I=Smiri2: replace plank by point mass situated at CM depends on pivot position! I= Hence a=I/t=

  13. Constant angular acceleration Assume a is constant • Dw/Dt= a i.e (wF- wI)/t= a • wF = wI + at • Now (wF+wI)/2=wav if constant a • And qF-qI= wavt  • qF= qI+ wIt +1/2 a t2

  14. Problem – slowing a DVD wI=27.5 rad/s, a=-10.0 rad/s2 how many revolutions per second ? linear speed of point on rim ? angular velocity at t=0.3s ? when will it stop ? 10 cm

  15. Rotational Kinetic Energy K=Si1/2mivi2=1/2w2Simiri2 Hence K= 1/2Iw2 Energy rigid body possesses by virtue of rotation

  16. Simple problem cable wrapped around cylinder. Pull off with constant force F. Suppose unwind a distance d of cable 2R F what is final angular speed of cylinder ?

  17. cylinder+cable problemenergy method • Use work-KE theorem • work W=? • Moment of inertia of cylinder ? • all mass is at distance R from center (axis of rotation)

  18. cylinder+cable problemconst acceleration method F N extended free body diagram no torque due to N or W why direction of N ? torque due to t=FR hence a=FR/(1/2MR2) =2F/(MR)  w W radius R

  19. Angular Momentum • can define rotational analog of linear momentum called angular momentum • in absence of external torque it will be conserved in time • True even in situations where Newton’s laws fail ….

  20. Definition of Angular Momentum Back to slide on rotational dynamics: miri2Dw/Dt = ti Rewrite: using li=miri2w Dli/ Dt= ti Summing over all particles in body DL/ Dt=text L – angular momentum=Iw

  21. Demos • Rotating stand plus dumbells – spin faster when arms drawn in • Rotating plastic ball

  22. Rotational Motion w Particle i: |vi|=ri w at 900 to r ri Newton’s 2nd law: pivot miDvi/Dt=FiT component at 900 to r substitute for vi and multiply by ri  Fi mi miri2Dw/Dt= FiT ri = ti Finally, sum over all masses Dw/Dt Smiri2 =Sti=tnet

  23. Points to note • If text=0 L=Iw is constant in time • conservation of angular momentum • Internal forces/torques do not contribute to external torque. • L=mvr if v is at 900 to r for single particle • L=r x p general result (x= vector cross product)

  24. The angular momentum L of a particle • is independent of the specific choice of origin • is zero when its position and momentum vectors are parallel • is zero when its position and momentum vectors are perpendicular • is zero if the speed is constant

  25. An ice skater spins about a vertical axis through her body with here arms held out. As she drwas her arms in, her angular velocity • 1. increases • 2. decreases • 3. remains the same • 4 need more information

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