Physics 121 Newtonian Mechanics Lecture notes are posted on Instructor Karine Chesnel April 2, 2009 Review 3

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)

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

Linear Momentum & Impulse Review 34/2/09 Newton’s second law The linear momentum of a particle is the product of its mass by its velocity Units: kg.m/s or N.s 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 t1t1 t2t2 t According to Newton’s 2nd law:

Collisions 1. Conservation of linear momentum 2. Conservation of kinetic energy (2) Elastic collision (1) V 1,i V 2,i V 1,f V 2,f Review 34/2/09

Collisions 1D If one of the objects is initially at rest: Combining (1) and (2), we get expression for final speeds: V 1,i V 1,f V 2,f V 2,i Collisions 2D 3 equations 4 unknow parameters V 1,i x y V 1,f V 2,f   Inelastic collision: the kinetic energy K is NOT conserved Review 34/2/09

Center of Mass m1m1 m2m2 m3m3 m4m4 m5m5 m6m6 M OC O C y x z Ensemble of particles r1r1 r6r6 Ch.9 Momentum and collision03/05/09 M OC Solid object C O dm r P

Solid characteristics M OC The center of mass is defined as: C O dm r   FaM C   The moment of inertia of the solid about one axis: Review 34/2/09

Rotational kinematics Solid’s rotation Angular position  Angular speed  Angular acceleration  Linear/angular relationship Velocity Acceleration Tangential Centripetal For any point in the solid Rotational kinetic energy Review 34/2/09

Motion of rolling solid P  C R Non- sliding situation The kinetic energy of the solid is given by the sum of the translational and rotational components: K solid = K c + K rot If all the forces are conservative: Review 34/2/09

Torque & angular momentum The torque is defined as F   The angular momentum is defined as Deriving Newton’s second law in rotation angular momentum Linear momentum prL    When a force is inducing the rotation of a solid about a specific axis: For an object in pure rotation Review 34/2/09

Precession Top view L dL dd is the projection of the angular momentum in the horizontal plane The angular momentum moves along a cone The precession speed is mgmg  Side view  L  If an object spinning at very high speed  is experiencing a torque in a direction different than its angular momentum L, then it will precess about a second axis Review 34/2/09

Solving a problem Static equilibrium Define the system Locate the center of mass (where gravity is applied) Identify and list all the forces Apply the equality 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. Apply the equality Review 34/2/09

Example 3 Beam and cable tension We do not know the force R that the hinge applies to the beam. P is a convenient point to calculate the torque R P Find the tension on the cable T MgMg C Q  Ch.10 Rotation of Solids3/24/09

Example 3 Beam and cable tension Find the magnitude and direction for the force R exerted by the wall on the beam in the horizontal direction in the vertical direction R P T MgMg C Q  Ch.10 Rotation of Solids3/24/09

Gravitational laws Any object placed in that field experiences a gravitational force Any material object is producing a gravitational field M r urur m FgFg The gravitational field created by a spherical object is centripetal (field line is directed toward the center) The gravitational potential energy is UgUg 0 r Review 34/2/09

Kepler’s Laws “The orbit of each planet in the solar system is an ellipse with the Sun as one focus ” First Law “The line joining a planet to the sun sweeps out equal areas during equal time intervals as the planet travels along its orbit.” Second Law cst m L dt dA  2 0 “The square or the orbital period of any planet is proportional to the cube of the semimajor axis of the orbit” Third Law Review 34/2/09

Kepler’s laws First law Physical observations (Brahe and Kepler, early 17 th ) showed that orbits are elliptical This phenomenon could be demonstrated later (late 17 th ) using the Newton’s laws of motion The motion of a body orbiting around another body under the only influence gravitational force must be in a plane r V L0L0 L0L0 L0L0 Solar system Review 34/2/09

Kepler’s laws Second law V r The area swept by the radius during the time interval dt is “The line joining a planet to the sun sweeps out equal areas during equal time intervals as the planet travels along its orbit.” Review 34/2/09

Case of circular orbit Kepler’s laws Third law FgFg Applying Newton’s law of motion With gravitational force Also “The square or the orbital period of any planet is proportional to the cube of the semi-major axis of the orbit” The proportionality constant is Solar system K s = s 2 /m 3 Review 34/2/09