Angular Impulse and Momentum

Lecture 28

ME 231: Dynamics Question of the Day Determine the magnitude HO of the angular momentum of the 2-kg sphere about point O.

Hint: HO is the moment of linear momentum about point O.

HO  r  mv

ME 231: Dynamics Outline for Today

Question ofof the dayday Angular momentum Rate of change of angular momentum Angular impulse-momentum principle Plane-motion applications Conservation of angular momentum Answer your questions!

Exam 2 Survey ME 231: Dynamics Angular Momentum HO  r  mv

Particle of mass m is located by position vector r Velocity v and linear momentum G = mv are tangent to its path The moment of the linear momentum vector mv about point O is the angular momentum HO of P about O Perpendicular to plane A defined by r and v

ME 231: Dynamics Angular Momentum HO  r  mv

i j k

HO  x y z mvx mvy mvz

H  m(v y  v z) O x z y H  m(v z  v x) O y x z H  m(v x  v y) O z y x

ME 231: Dynamics Rate of Change of H  r  mv B M  H Angular Momentum O O O

Differentiate HO with respect to time H  r  mv  r  mv O 0   HO  v mv  r  mv   HO  r  mv M M  H Resultant O of all moments O x O x of forces on m is the vector M  H O y O y cross product of r and F  M  H MO  r F  r  mv O z O z ME 231: Dynamics Angular Impulse- H  r  mv B M  H Momentum Principle O O O

t t 2 M dt  2 H dt Integrate to describe O O O O t1 t1 the effect of the angular t impulse MO*t on H  2 M dt  H  O 1 O O O 2 angular momentum HO t1 of m about O over a finite period of time t m(v y  v z)  2 M dt  m(v y  v z) z y 1 O O x z y 2 t1 t m(v z  v x)  2 M dt  m(v z  v x) x z 1 O O y x z 2 t1 t m(v x  v y)  2 M dt  m(v x  v y) y x 1 O O z y x 2 t1

ME 231: Dynamics Plane-Motion H  r  mv B M  H Applications O O O

t H  2 M dt  H  O 1 O O O 2 t1

Moments taken about a single axis normal to the plane of motion Angular momentum may change magnitude and t sense, but the mv d  2 Fr sin dt  mv d 1 1 O 2 2 direction is constant t1

ME 231: Dynamics Conservation of H  r  mv B M  H Angular Momentum O O O

HO  0 or t H  2 M dt  H  O 1 O O O 2 H  H t1 O 1 O 2

If the resultant moment about a fixed point O is zero, then angular momentum remains constant, or is said to be conserved Angular momentum may be conserved in one coordinate (e.g., x), but not necessarily in others (e.g., y or z)

ME 231: Dynamics Angular Impulse-Momentum: Exercise

The particle of mass m is launched from point O with a horizontal velocity u at time t = 0.

Determine its angular momentum HO relative to point O as a function of time.

ME 231: Dynamics Angular Impulse-Momentum: Another Exercise

The assembly starts from rest and reaches an angular speed of 150 rev/min under the action of a 20-N force T applied to the string for t seconds. Neglect friction and all masses except those of the four 3-kg spheres.

Determine t. ME 231: Dynamics Angular Impulse-Momentum: Yet Another Exercise

A 0.1-kg particle with a velocity of 2 m/s in the x-direction at A is guided by a curved rail. The radius of curvature of the rail at B is 500 mm.

Determine the time rate of change of the angular momentum HO about the z-axis through O at both A and B. ME 231: Dynamics Outline for Today

Question ofof the dayday AngularAngular momentummomentum Rate ooff chanchangege ooff anangulargular momentumomentumm AngularAngular impulse-momentumimpulse-momentum principleprinciple Plane-motionPlane-motion applicationsapplications Conservation ofof anangulargular momentumomentumm Answer your questions!

Exam 2 Survey ME 231: Dynamics For Next Time…

Begin Homework #10 due on Wednesday (11/7) Read Chapter 5, Section 5.3 Read Chapter 8, Section 8.2

ME 231: Dynamics