Physics 121 Newtonian Mechanics

1. Physics 121 Newtonian Mechanics Instructor Karine Chesnel April 2, 2009 Review 3 Lecture notes are posted on www.physics.byu.edu/faculty/chesnel/physics121.aspx

2. Mid-term exam 3 • Friday April 3 through Tuesday April 7 • At the testing center : 8 am – 9 pm • Closed Book and closed Notes • Only bring: -Math reference sheet • Pen / pencil • Calculator • your CID • No time limit (typically 2 – 3 hours)

3. Midterm exam 3 Review: ch 9 – ch 13 • Ch. 9 Linear Momentum & collision • Center of mass • Linear momentum • Impulse • Collisions 1D and 2D • Ch. 10 Rotation of solid • Moment of inertia • Rotational kinematics • Rolling motion • Torque • Ch. 11 Angular momentum • Angular momentum • Newton’s law for rotation • Isolated system • Precession motion • Ch. 12 Static equilibrium and elasticity • Rigid object in equilibrium • Elastic properties of solid • Ch. 13 Universal gravitation • Newton’s law of Universal gravitation • Gravitational Field & potential energy • Kepler’s laws and motion of planets

4. 4/2/09 Review 3 • The linear momentum of a particle is the product of its mass by its velocity Units: kg.m/s or N.s • Newton’s second law • For an isolated system • The impulse is the integral of the net force, during • an abrupt interaction • in a short time Modeling of an impulse t t1 t2 • According to Newton’s 2nd law: Linear Momentum & Impulse

5. 4/2/09 Review 3 V1,i V1,f V2,f V2,i 1. Conservation of linear momentum (1) 2. Conservation of kinetic energy (2) Collisions Elastic collision

6. 4/2/09 Review 3 • Combining (1) and (2), we get expression for final speeds: y V1,i V2,i V1,f V1,i q1 q2 x V1,f V2,f V2,f • If one of the objects is initially at rest: Collisions 2D • 3 equations • 4 unknow parameters Collisions 1D Inelastic collision: the kinetic energy K is NOT conserved

7. 03/05/09 Ch.9 Momentum and collision m5 m2 m4 dm m1 m3 P C r1 O r m6 y O r6 x Solid object C M OC M OC Center of Mass Ensemble of particles z

8. 4/2/09 Review 3 • The center of mass is defined as: r r å = M a F C C dm O r • The moment of inertia of the solid about one axis: M OC Solid characteristics

9. 4/2/09 Review 3 • Linear/angular relationship For any point in the solid Velocity • Acceleration • Tangential • Centripetal • Rotational kinetic energy Rotational kinematics • Solid’s rotation • Angular position q • Angular speed w • Angular acceleration a

10. 4/2/09 Review 3 w C R • The kinetic energy of the solid is given by the sum of the translational and rotational components: P Ksolid = Kc + Krot Motion of rolling solid Non- sliding situation If all the forces are conservative:

11. 4/2/09 Review 3 The torque is defined as angular momentum r r r = ´ L r p Deriving Newton’s second law in rotation Linear momentum For an object in pure rotation Torque & angular momentum When a force is inducing the rotation of a solid about a specific axis: F q t The angular momentum is defined as

12. 4/2/09 Review 3 Side view w mg df q L Top view t is the projection of the angular momentum in the horizontal plane The precession speed is L dL Precession If an object spinning at very high speedw is experiencing a torque in a direction different than its angular momentum L, then it will precess about a second axis The angular momentum moves along a cone

13. 4/2/09 Review 3 • Apply the equality • Apply the equality Solving a problem Static equilibrium • Define the system • Identify and list all the forces • Locate the center of mass (where gravity is applied) • Choose a convenient point to calculate the torque • (you may choose the point at which most • of the forces are applied, so their torque is zero) • List all the torques applied on the same point.

14. 3/24/09 Ch.10 Rotation of Solids P Mg C Example 3 Beam and cable tension • Find the tension on the cable T R q Q • We do not know the force R that the hinge applies to the beam. • P is a convenient point to calculate the torque

15. 3/24/09 Ch.10 Rotation of Solids P Mg C Example 3 Beam and cable tension • Find the magnitude and direction • for the force R • exerted by the wall on the beam T R a q Q • in the horizontal direction • in the vertical direction

16. 4/2/09 Review 3 Any material object is producing a gravitational field M ur Fg r m Any object placed in that field experiences a gravitational force The gravitational field created by a spherical object is centripetal (field line is directed toward the center) Ug The gravitational potential energy is r 0 Gravitational laws

17. 4/2/09 Review 3 First Law “The orbit of each planet in the solar system is an ellipse with the Sun as one focus ” Second Law “The line joininga planet to the sun sweeps out equal areas during equal time intervals as the planet travels along its orbit.” L dA = = 0 cst dt 2 m Third Law “The square or the orbital period of any planet is proportional to the cube of the semimajor axis of the orbit” Kepler’s Laws

18. 4/2/09 Review 3 L0 L0 r L0 V The motion of a body orbiting around another body under the only influence gravitational force must be in a plane Solar system Kepler’s laws First law • Physical observations (Brahe and Kepler, early 17th ) • showed that orbits are elliptical • This phenomenon could be demonstrated later (late 17th ) • using the Newton’s laws of motion

19. 4/2/09 Review 3 V r The area swept by the radius during the time interval dt is “The line joininga planet to the sun sweeps out equal areas during equal time intervals as the planet travels along its orbit.” Kepler’s laws Second law

20. 4/2/09 Review 3 Applying Newton’s law of motion With gravitational force Case of circular orbit Also The proportionality constant is Kepler’s laws Third law Fg Solar system Ks =2.97 10-19 s2/m3 “The square or the orbital period of any planet is proportional to the cube of the semi-major axis of the orbit”