Comprehensive Review Comprehensive Review a) Exam information b) What kind of questions? c) Review
Final Exam • In-class only!!! • Review the following material • homework problems. • video pre-lectures/textbook. • Lecture slides • Unit Main Points • Multiple choice…but show your work and justification. • Mostly Calculations…”step by step” • Some Conceptual questions…like checkpoint problems. • Bring calculators and up to ten sheets of notes. • It is best to prepare your own hand-written notes! • Derived Equations…(e.g. projectile motion)
General Topics • Kinematics • Description of Motion • Force • Dynamics-how objects change velocity • Energy • Kinetic and Potential • Conservation Laws • Momentum and Energy • Collisions • Elastic and In-elastic • Rotations • Torque/ Angular Momentum/Statics
Problem Solving Techniques • Visualize/Diagram • “Sketch” problem • Identify variables, input and what we are trying to solve • Free-body diagrams • Express in Mathematical Equations • Scalars-1d • Vectors-2d,3d Break into components • System of n-equations with n-unknowns • Use Mathematical tools to solve: • Quadratic Equation • Vector operations • Trigonometry • Conceptual Understanding • Does answer make sense?
Potential Problem Topics • Projectile Motion • Relative Motion - 2d • Uniform Circular Motion • Forces • Weight (near earth) • Gravitational (satellite) • Springs • Normal Force • Tension • Friction • Free-Body Diagrams • Work-Kinetic Energy • Potential Energy • Center of Mass • Conservation of Momentum • Collisions • In-elastic • Elastic • Rotations • Kinematics • Dynamics • Statics • Moment of Inertia • Torque • Angular Momentum
HyperphysicsMotion Displacement vs timet Velocity vs timet Acceleration vs timet
1d-Kinematic Equations for constant acceleration • Basic Equations to be used for 1d – kinematic problems. • Need to apply to each object separately sometimes with time offset • When acceleration changes from one constant value to another say a=0 The problem needs to be broken down into segments
Ballistic Projectile Motion Quantities Initial velocity speed,angle Maximum Height of trajectory, h=ymax “Hang Time” Time of Flight, tf Range of trajectory, D Height of trajectory at arbitrary x,t
Derived Projectile Trajectory Equations Maximum height Time of Flight (“Hang Time”) Range of trajectory Height of trajectory as f(t) , y(t) Height of trajectory as f(x), y(x)
Relative Motion in 2 Dimensions Direction w.r.t shoreline Speedrelative to shore
Uniform Circular Motion • Constant speed in circular path • Acceleration directed toward center of circle • What is the magnitude of acceleration? • Proportional to: • Speed • time rate of change of angle or angular velocity v = wR
Inventory of Forces • Weight • Normal Force • Tension • Gravitational • Springs • …Friction
Be careful what value you use for r !!! Should be distance between centers of mass of the two objects
m2 1) FBD N T m2 g f T m1 m1 m2g m1g
m2 1) FBD 2) SF=ma N T m2 g f T m1 m1 m2g N = m2g m1g m1g – T = m1a T – mm2g = m2a add m1g – mm2g = m1a + m2a m1g – mm2g a = m1 + m2
m2 1) FBD 2) SF=ma N T m2 g f T m1 m1 m2g m1g m1g – T = m1a m1g – mm2g T = m1g – m1a a = m1 + m2 T is smaller when a is bigger
Work-Kinetic Energy Theorem But again…!!! The work done by force F as it acts on an object that moves between positions r1 andr2is equal to the change in the object’s kinetic energy:
Energy Conservation Problems in general For systems with only conservative forces acting Emechanical is a constant
Determining Motion • Energy • Total Energy Motion, Location • Work • Conservative forces • Motion from Energy conservation • Force • Unbalanced Forces acceleration • (otherwise objects velocity is constant) • Determine Net Force acting on object • Use kinematic equations to determine resulting motion
Friction “It is what it has to be.”
Example: Pendulum h h Conserve Energy from initial to final position.
Gravitational Potential Problems • conservation of mechanical energy can be used to “easily” solve problems. • Define coordinates: where is U=0? as • Add potential energy from each source.
Collisions • Center of Mass • Conservation of Momentum • Inelastic collisions • Elastic Collisions • Impulse and Reference Frames Multiple particles, Solid Objects Isolated system, No external force Non-conservative internal force Conservative internal force Individual Particle changes momentum due to Force acting over a given duration Favg = DP/Dt
Rotations • Rotational Kinematics • Moment of Inertia • Torque • Rotational Dynamics • Rotational Statics • Angular Momentum Description of motion about a center of mass Resistance to changes in angular velocity Force applied at a lever arm resulting in angular acceleration Newton’s 2nd law for rotations How to ensure stability Vector Quantity describing object(s) rotation about an axis