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Understanding Rotational Dynamics: Kinetic Energy and Angular Momentum in Physics 211

Welcome back to Physics 211! Today, we'll dive into Rotational Dynamics, discussing key concepts like kinetic energy, angular momentum, torque, and the behavior of rigid bodies. We'll recap crucial topics from previous sessions and prepare for our upcoming Exam 3 on November 13 and HW10 (due Tue/Wed). We'll explore moments of inertia, conditions for equilibrium, and key equations governing rotational motion. Engage in demos and problem-solving activities to solidify your understanding of these essential physics concepts. Let’s rotate through the intricacies of motion!

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Understanding Rotational Dynamics: Kinetic Energy and Angular Momentum in Physics 211

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Presentation Transcript

  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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