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Physics 1710 Chapter 12 Rolling Motion and Angular Momentum

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Title: Physics 1710 Chapter 12 Rolling Motion and Angular Momentum


1
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Quiz
  • What torque must be applied to a discus (assume
    a short cylindrical shape) that has a diameter
    of 20 cm and a mass of 3.0 kg, if it leaves the
    hand of the discus thrower spinning at a rate of
    180 rpm if he takes 0.5 sec to throw the disc?

? 2??(180)/60 18.7 rad/s I ½ MR2 ½ (3.0)
(0.10)2 0.015 kg m
2
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Quiz
  • What torque must be applied to a discus (assume
    a short cylindrical shape) that has a diameter
    of 20 cm and a mass of 3.0 kg, if it leaves the
    hand of the discus thrower spinning at a rate of
    180 rpm if he takes 0.5 sec to throw the disc?

? 18.7 rad/s I ½ MR2 0.015 kg m ? 18.7
rad/s/(0.5 sec) 37.4 rad/s/s T I? (0.015)
(37.4) N?m 0.56 N?m F T/R 0.56/0.1 5.6 N
3
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • 1' Lesson
  • The total Kinetic energy of a rotating system is
    the sum of the rotational energy about the Center
    of Mass and the translational KE of the CM.
  • T r x F
  • Angular momentum L is the vector product of the
    moment arm and the linear momentum.
  • The net externally applied torque is equal to
    the time rate of change in the angular momentum.

4
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • 1' Lesson (contd)
  • Angular momentum about an axis z is equal to
    the product of the moment of inertia of the body
    about that axis and the angular velocity about z.
  • In the absence of torques, the angular momentum
    is conserved.

5
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Rolling Motion
  • The compound motion of a rolling object is a
    translation plus a rotation.
  • vCM R? aCM R?
  • Cylinder rolling down a ramp
  • F Mg sin ? T RF I ?
  • aCM R 2 (Mg sin ?)/I x ½ aCM t 2
  • Which will win? A large disk or a small one?
  • A heavy one or a light one?

6
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Total Energy of Motion
  • K ½ Ip ? 2 ½ ICM ? 2 ½ MR 2 ? 2
  • By parallel axis theorem.
  • Thus, the total kinetic energy is the energy of
    motion of the center of mass plus the energy of
    rotation about the CM

7
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Vector Product
  • C A x B
  • Cx Ay Bz Az By
  • Cyclically permute (xyz), (yzx), (zxy)
  • C vCx2 Cy2 Cz2
  • AB sin ?
  • Directed by RH Rule.

8
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Vector Product
  • A x B - B x A
  • A x ( B C ) A x B A x C
  • d/dt ( A x B ) d A /dt x B A x d B/dt
  • i x i j x j k x k 0
  • i x j - j x i k
  • j x k - k x j i
  • k x i - i x k j

9
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Angular Momentum
  • L r x p
  • The angular momentum is the vector product of
    the moment arm and the linear momentum.
  • ? T d L/dt
  • The net torque is equal to the time rate of
    change in the angular momentum.

10
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Proof
  • ? T r x ?F r x d p/dt
  • And
  • d L/dt d( r x p) /dt
  • d r/dt x p r x d p/dt.
  • But p m d r/dt , therefore d r/dt x p 0
  • d L/dt r x d p/dt
  • And thus
  • ? T d L/dt.

11
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Rotating Platform Demonstration

12
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Analysis
  • Why does an ice skater increase her angular
    velocity without the benefit of a torque?
  • L r x p
  • r x ( m v)
  • r x ( m r x ?)
  • Li mi ri 2 ?
  • Lz (?i mi ri 2 ) ?
  • Lz I ? ? Lz / I
  • Therefore, a decrease in I ( by reducing r) will
    result in an increase in ?.

13
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Torque and Angular Momentum (redux)
  • L I ?
  • The quantity I is a tensor .
  • Changing I changes the magnitude and alignment
    of L and ?.

14
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • If I is constant then for rotation about an axis
    z
  • ? Tz d Lz /dt Iz ?
  • In the absence of net external torque the angular
    momentum is conserved.

15
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Spinning Bicycle Wheel
  • Demonstration

16
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Gyroscope Demonstration

17
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Fundamental Angular Momentum
  • Fundamental unit of angular momentum ?
  • ? 1.054 x 10 -34 kg?m/s2
  • ICM? ?
  • ? ? / ICM
  • 1.054 x 10 -34 kg?m/s2 / (1.95 x 10 -46 kg?m)
  • 5.41 x 10 11 rad/s

18
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Summary
  • The total Kinetic energy of a rotating system is
    the sum of the rotational energy about the Center
    of Mass and the translational KE of the CM.
  • K ½ ICM ? 2 ½ MR 2 ? 2
  • T r x F

19
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Summary Question
  • Which will win a race down an incline?
  • A ball rolling without slipping?
  • A ball sliding with no friction?
  • Why?

K ½ ICM ? 2 ½ MR 2 ? 2
20
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Summary
  • Angular momentum L is the vector product of the
    moment arm and the linear momentum.
  • L r x p
  • The net externally applied torque is equal to
    the time rate of change in the angular momentum.
  • ? Tz d Lz /dt Iz ?

21
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
  • Summary
  • Angular momentum about an axis z is equal to
    the product of the moment of inertia of the body
    about that axis and the angular velocity about z.
  • L I ?
  • Lz Iz ?
  • In the absence of torques, the angular momentum
    is conserved.

22
Physics 1710Chapter 12 Rolling Motion and
Angular Momentum
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