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PHYS 1443 – Section 003 Lecture #16 Monday, Nov. 11, 2002 Dr. Jaehoon Yu 1. 2. 3. 4. 5. Angular Momentum Angular Momentum and Torque Angular Momentum of a System of Particles Angular Momentum of a Rotating Rigid Body Angular Momentum Conservation Today’s homework is homework #16 due 12:00pm, Monday, Nov. 18!! Monday, Nov. 11, 2002 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu 1 Similarity Between Linear and Rotational Motions All physical quantities in linear and rotational motions show striking similarity. Similar Quantity Mass Length of motion Speed Acceleration Force Work Power Momentum Kinetic Energy Monday, Nov. 11, 2002 Linear Mass Rotational Moment of Inertia M I r 2 dm Displacement r Angle (Radian) v dr dt d dt a dv dt d dt Force F ma Work W Fdx xf xi Kinetic Torque Work I f W d i P F v P p mv L I K 1 mv 2 2 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu Rotational KR 1 I 2 2 2 Angular Momentum of a Particle If you grab onto a pole while running, your body will rotate about the pole, gaining angular momentum. We’ve used linear momentum to solve physical problems with linear motions, angular momentum will do the same for rotational motions. z Let’s consider a point-like object ( particle) with mass m located at the vector location r and moving with linear velocity v L=rxp The instantaneous angular momentum L r p L of this particle relative to origin O is O y m r What is the unit and dimension of angular momentum? kg m2 / s 2 p Note that L depends on origin O. Why? Because r changes x What else do you learn? The direction of L is +z Since p is mv, the magnitude of L becomes L mvr sin What do you learn from this? The point O has to be inertial. Monday, Nov. 11, 2002 If the direction of linear velocity points to the origin of rotation, the particle does not have any angular momentum. If the linear velocity is perpendicular to position vector, the particle moves exactly the same way as a point on a 3rim. PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu Angular Momentum and Torque Can you remember how net force exerting on a particle and the change of its linear momentum are related? F dp dt Total external forces exerting on a particle is the same as the change of its linear momentum. The same analogy works in rotational motion between torque and angular momentum. Net torque acting on a particle is F r d p r dt z dL d r p dr dp dp pr 0r L=rxp dt dt dt dt dt O x r y m Why does this work? p Thus the torque-angular momentum relationship Because v is parallel to the linear momentum dL dt The net torque acting on a particle is the same as the time rate change of its angular momentum Monday, Nov. 11, 2002 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu 4 Angular Momentum of a System of Particles The total angular momentum of a system of particles about some point is the vector sum of the angular momenta of the individual particles L L1 L2 ...... Ln L Since the individual angular momentum can change, the total angular momentum of the system can change. Both internal and external forces can provide torque to individual particles. However, the internal forces do not generate net torque due to Newton’s third law. Let’s consider a two particle system where the two exert forces on each other. Since these forces are action and reaction forces with directions lie on the line connecting the two particles, the vector sum of the torque from these two becomes 0. Thus the time rate change of the angular momentum of a system of particles is equal to the net external torque acting on the system Monday, Nov. 11, 2002 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu ext dL dt 5 Example 11.4 A particle of mass m is moving in the xy plane in a circular path of radius r and linear velocity v about the origin O. Find the magnitude and direction of angular momentum with respect to O. Using the definition of angular momentum y v L r p r mv mr v r O x Since both the vectors, r and v, are on x-y plane and using right-hand rule, the direction of the angular momentum vector is +z (coming out of the screen) mrrvv mrv sin mrv sin 90 mrv The magnitude of the angular momentum is LL m So the angular momentum vector can be expressed as L mrvk Find the angular momentum in terms of angular velocity . Using the relationship between linear and angular speed 2 2 L mrvk mr k mr I Monday, Nov. 11, 2002 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu 6 Angular Momentum of a Rotating Rigid Body z Let’s consider a rigid body rotating about a fixed axis Each particle of the object rotates in the xy plane about the zaxis at the same angular speed, L=rxp O y m r x Magnitude of the angular momentum of a particle of mass mi about origin O is miviri 2 Li mi ri vi mi ri p Summing over all particle’s angular momentum about z axis Lz Li mi ri 2 i i Since I is constant for a rigid body Thus the torque-angular momentum relationship becomes i dL z d I dt dt ext I Lz mi ri 2 What do you see? I is angular acceleration dLz I dt Thus the net external torque acting on a rigid body rotating about a fixed axis is equal to the moment of inertia about that axis multiplied by the object’s angular acceleration with respect to that axis. Monday, Nov. 11, 2002 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu 7 Example 11.6 A rigid rod of mass M and length l pivoted without friction at its center. Two particles of mass m1 and m2 are connected to its ends. The combination rotates in a vertical plane with an angular speed of . Find an expression for the magnitude of the angular momentum. y m2 l m2 g O m1 m1 g x The moment of inertia of this system is 1 1 1 2 2 I I I Ml m l m2l 2 I rod m1 m2 1 12 4 4 2 l2 1 l 1 M m1 m2 L I M m1 m2 4 3 4 3 Find an expression for the magnitude of the angular acceleration of the system when the rod makes an angle with the horizon. If m1 = m2, no angular momentum because net torque is 0. If /-p/2, at equilibrium so no angular momentum. Monday, Nov. 11, 2002 First compute net external torque m1 g l cos 2 ext 2 2 -m2 g gl cos m1 - m2 2 1 m1 - m1 gl cos 2 Thus ext 2 l 1 I M m1 m2 becomesPHYS 1443-003, Fall 2002 4 3 Dr. Jaehoon Yu l cos 2 2m1 - m1 cos g /l 1 M m1 m2 3 8 Conservation of Angular Momentum Remember under what condition the linear momentum is conserved? Linear momentum is conserved when the net external force is 0. F 0 dp dt p const By the same token, the angular momentum of a system is constant in both magnitude and direction, if the resultant external torque acting on the system is 0. What does this mean? ext dL 0 dt L const Angular momentum of the system before and after a certain change is the same. Li L f constant Three important conservation laws for isolated system that does not get affected by external forces Monday, Nov. 11, 2002 K i U i K f U f Mechanical Energy Linear Momentum pi p f Angular Momentum Li L f PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu 9 Example 11.8 A start rotates with a period of 30days about an axis through its center. After the star undergoes a supernova explosion, the stellar core, which had a radius of 1.0x104km, collapses into a neutron start of radius 3.0km. Determine the period of rotation of the neutron star. What is your guess about the answer? Let’s make some assumptions: Using angular momentum conservation The period will be significantly shorter, because its radius got smaller. 1. There is no torque acting on it 2. The shape remains spherical 3. Its mass remains constant Li L f I i I f f The angular speed of the star with the period T is Thus Tf I i mri 2 2p f If mrf2 Ti 2p f Monday, Nov. 11, 2002 r f2 2 r i 2p T 2 3 . 0 -6 Ti 2 . 7 10 days 0.23s 30 days 4 1 . 0 10 PHYS 1443-003, Fall 2002 Dr. Jaehoon Yu 10