Test 2 Thursday, March 25 8:20 pm Covers: Ch. 6, Ch. 7, Secs. 8.1-8.5 Bring: calculator, ID, formula sheet (no phones, formula sheet 2-sided, handwritten) Last year’s Exam 2 is now posted on our website A-G Room TUR L007 H-M CLB C130 N-U WEIM 1064 V-Z LIT 0113 1st letter last name Kinetic Energy and Moment of Inertia 1 1 1 2 2 KEi mi vi = mi ri = mi ri 2 2 2 2 2 v Moment of Inertia I - measures rotational inertia of object - depends on the axis 1 2 KE KEi I 2 i r piece i of rigid body, mass mi Kinetic Energy and Moment of Inertia I mi ri 2 Compare to linear motion: M mi i 1 2 KE I 2 (pure rotation) 1 2 KE Mv 2 Rotational Kinetic Energy 1 2 KEr I 2 Conservation of Mechanical Energy (KEt KEr PE )i (KEt KEr PE )f Rolling and Kinetic Energy P 1 2 KE I 2 ... 1 1 2 I cm Mv 2 2 2 Instantaneous pure rotation about P or Combination of rotation about cm and translation of cm KEt KEr Ball rolling down an incline How fast does it leave the bottom of the incline? h Conservation of Mechanical Energy (KEt KE r PE )i (KEt KE r PE )f 0 + 0 + mgh = mv2/2+I2/2 + 0 I=2mr2/5 and r=v so I2/2= mv2/5 Giving: mgh = mv2/2+ mv2/5 = 7 mv2/10 Note m cancels out as usual or v = 10gh/7 angular momentum REMINDERS Linear Motion Rotations coordinate velocity acceleration x v a mass force 1st Newton’s Law 2nd Newton’s Law m F F=0: v=const F = ma Kinetic Energy Momentum K = ½mv2 p = ma angle angular velocity angular acceleration =0: =const = I moment of inertia torque K = ½I2 ??? Kinetic Energy Angular Momentum Torque and Angular Acceleration Newton’s Second Law for a Rotating Object I analogous to ∑F = ma I = moment of inertia 2 2 I m r MR For Uniform Ring i i moment of inertia depends on quantity of matter and its distribution and location of axis of rotation Other Moments of Inertia Angular Momentum of rigid body L=Iω Just like p = mv L always conserved unless there is external torque! Impulse L t p Just like F t Isolated system Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. 0, Li Lf or Ii i If f Conservation of Angular Momentum– Example 1 How does a skater spin faster in the air? L is conserved As arms come in-I decreases increases http://www.youtube.com/watch?v=AQLtcEAG9v0 Applying Conservation Rules In an isolated system, the following three quantities are conserved: Mechanical energy Linear momentum Angular momentum Example 2: Angular momentum conservation Gyroscope (for fun) r F rF , direction: L pivot point dL dt dL dt d L L F looking from above ) L(t+dt L(t) dL this gyroscope precesses counterclockwise d Precession angular velocity: dt rF L I