Vectors & Matrices

1. Vectors & Matrices UA Summer Engineering Program 2012

2. What is a vector? • Scalar vs. Vector • A scalar is a regular number with just magnitude: • 1 • 737 • 8 apples • It is 75°F. • A vector is a quantity with magnitude and direction: • 50 miles east • The force of gravity on my dog is 50 pounds downward. • (1, 1) UA Summer Engineering Program 2012

3. Vectors • Important for physics & engineering: • force • acceleration • momentum • velocity • Coordinate System as reference frame • No matter what the problem, it is important to assign a reference. • Vectors are used to define the dimensions of the problem. UA Summer Engineering Program 2012

4. Vectors • Compare these 3 vectors: both magnitude & direction • Since engineers prefer using math…use vector algebra and trigonometry. UA Summer Engineering Program 2012

5. Vectors • Vectors are broken down to fit into easy directions: UA Summer Engineering Program 2012

6. Vector • Vectors can be added & subtracted using their scalar components... UA Summer Engineering Program 2012

7. What is a Matrix? • Vectors of Vectors • Used in programming, physical examples such as rotation. (Rocket to Earth) • Referenced by their row, then column. • They can easily be added & subtracted if they come from matrices of the same dimensions: UA Summer Engineering Program 2012

8. Forces & Moments UA Summer Engineering Program 2012

9. What is Force? • Push or pull in a specific direction • Vector quantity: Must include magnitude and direction • Measured in Newtons or pounds UA Summer Engineering Program 2012

10. Types of Force • Elastic forces • Friction force • Static • Kinetic • Fluid resistance • Contact • Force between two surfaces • Divide into normal and frictional • Field forces • Electric & Magnetic • Gravitational UA Summer Engineering Program 2012

11. Modeling Forces • FREE BODY DIAGRAMS • 1) Draw a box • 2) Draw all the forces coming in and out of the box. • 3) Decompose each force vector into directional components. • 4) Write equations to reflect the diagram. UA Summer Engineering Program 2012

12. Newton’s Law of Motion #1 • Every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it. • Defines inertia. • If there is no net force acting on an object then the object will maintain a constant velocity. UA Summer Engineering Program 2012

13. Newton’s Law of Motion #2 • Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration. F = m (dmv/dt) F = ma _ Force mass acceleration • The law defines a force to be equal to change in momentum per change in time. • For an external applied force, the change in velocity depends on the mass of the object. • Equation works both ways • A force will cause a change in velocity; • A change in velocity will generate a force. Change in momentum per change in time UA Summer Engineering Program 2012

14. Newton’s Law of Motion #3 • For every action there is an equal and opposite re-action. F1,2 = - F2,1 • Examples: • Lift on a wing • Thrust by a jet engine • Recoil of a cannon UA Summer Engineering Program 2012

15. Example Problems: Force • A horse pulls a sleigh through the snow with a net force of 500 N. It accelerates the sleigh as 2.85 m/s2. What is the mass of the sleigh? • A 3.5 kg. mass is travelling at a constant velocity of 25 m/s. What force is required to slow it to a stop in 5 seconds? • 42 kg. mass is resting on the Earth's surface. What force does it exert on the Earth? UA Summer Engineering Program 2012

16. Example Problems: Force cont. • A man wants to lift a crate full of pineapples. The crate weighs 230 kg. What force is required to lift it (at a constant velocity) straight up? • You are trying to move a piano. The coefficient of static friction between the carpet and the piano is 0.63. The piano weighs 405 kg. How much force is required to accelerate the piano? UA Summer Engineering Program 2012

17. Torques & Moments • Question: • What happens when you apply a force to a pinned object? UA Summer Engineering Program 2012

18. What is torque? • Torque or moment is the tendency of a force to rotate an object about an axis, fulcrum, pivot • Vector! • Magnitude: T = L * F Torque = Perpendicular Distance to the force xForce • Direction: Right-hand rule • Measured in Newton-meters (Nm) or Inch pounds UA Summer Engineering Program 2012

19. Newton’s Rotational Laws of Motion • All of Newton’s Laws of Motion apply to torques on an object with a few physical changes: • #1: Every object persists in its state of rest or uniform motion in arotation unless it is compelled to change that state bytorques impressed on it. • #2: Torque is equal to the change in momentum per change in time. For a constant mass,torque equals rotational inertia times rotational acceleration. • #3: For every action there is an equal and opposite re-action. UA Summer Engineering Program 2012

20. Example Problems: Torque • Where must Mickey sit to balance Dumbo? UA Summer Engineering Program 2012

21. Real-life Applications • Finding the force to turn a lug wrench if you need to change a tire. • Door handles • Rotation UA Summer Engineering Program 2012

22. References • Vector: • http://www.grc.nasa.gov/WWW/K-12/airplane/vectcomp.html • http://www.grc.nasa.gov/WWW/K-12/rocket/vectors.html • http://physics.bu.edu/~erikl/Subject_Forces.pdf • http://www.mathwarehouse.com/algebra/matrix/ • Forces: • http://www.grc.nasa.gov/WWW/K-12/airplane/forces.html • http://library.thinkquest.org/C003546/sample_problems.html • Torque: • http://demonstrations.wolfram.com/TorqueExertedOpeningADoor/ • http://www.physics.drexel.edu/~jelena/post/Week2_a.pdf UA Summer Engineering Program 2012