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Vectors More math concepts

Objectives Distinguish between vector and scalar quantities. Carry out addition and scalar multiplication of vectors. Understand forces as vectors.

What’s the Point? How can we specify quantities that depend on direction? How do forces combine?

Vectors and Scalars Vector: quantity needing a direction to fully specify (direction + magnitude) Scalar: directionless quantity

Arrows for Vectors direction: obvious magnitude: length location is irrelevant these are identical

Represent as Components Components: projections in (x, y) directions B A A = (4, 3) B = (0, –2) x y

Magnitude from Components Components: lengths of sides of right triangle Magnitude: length of hypotenuse A A = (4, 3) ||A ||= A =

Physics Vectors and Scalars Position, displacement, velocity, acceleration, and force are vector quantities. Mass and time are scalar quantities. (Yes, there are many others)

Combine Displacement Vectors (CR to HA) + (HA to Union) = (CR to Union)

Add Vectors A C B A + B = C Head-to-tail (not in your book) A B

How to Add Vectors Place following vector’s tail at preceding vector’s head Resultant starts where the first vector starts and ends where the last vector ends Add any number of vectors, one after another

Sum by Components Vector sum: Add (x, y) components individua lly C B A A = (4, 3) B = (0, –2) C = A + B = (4+0, 3–2) = (4, 1)

Poll Question Which vector is the sum of vectors A and B? A DC B A B

Group Work 1.Draw two vectors A and B. Graphically find: A + B

Poll Question Is vector addition commutative? A.Yes. B.No.

Vector Addition is Commutative A + B = CA + B = C A B B + A = CB + A = C A + B = B + A

Add Vectors Book uses parallelogram rule emphasizes commutativity

Respect the Units For a vector sum to be meaningful, the vectors you add must have the same units! 5 s + 10 s = 15 s 5 kg + 10 m = 15 ? Or, algebra in general: 5 a + 10 a = 15 a 5 b + 10 c = 15 ? good! Bad! good! Bad! Just as with scalars:

Subtract Vectors A B Add the negative of the vector being subtracted. –B–B A – B = A + (–B) = D D –B–B A (Negative = same magnitude, opposite direction: what you must add to get zero)

Group Work 2.Make up three vectors A, B, and C. Graphically show: A – B A + B + C C + A + B

Multiplication by a Scalar Product of (scalar)(vector) is a vector The scalar multiplies the magnitude of the vector; direction does not change Direction reverses if scalar is negative A 2 A 1/2 A –2 A

Scalar Multiplication Example Velocity (a vector)  time (a scalar) v  t =  r Result is displacement (a vector). The vectors are in the same direction, but have different units!

Net Force Forces on an object add together. Forces can oppose each other. Net force is the vector sum of all forces acting on a body. The net force on a body at rest is zero.

Poll: Hammock Example A hammock slung between trees 8 m apart sags 1 m when a person lies in it. The net force acting on the person is A.Equal to the weight of the person. B.Equal to the tension in one cable. C.Zero. D.There is not enough information to answer. 8 m 1 m weight F F

Working with Commonly- Encountered Forces Tension

Tension Forces In cables, threads, chains, etc. Direction: along the cable, inward

Poll: Hammock Tension A hammock slung between trees 8 m apart sags 1 m when a person lies in it. The tension in a cable is A.Equal to the weight of the person. B.About half the weight of the person. C.Zero. D.Much more than the weight of the person. E.There is not enough information to answer. 8 m 1 m weight F F

Hammock Forces weight tension forces add to zero Tension exceeds weight for a shallow angle!

Application: Lumbar Forces Spinal curvature standingsitting

Application: Lumbar Forces Reaching with a load standingsitting weight

Application: Lumbar Forces Standing torque support tension

Application: Lumbar Forces Sitting torque support tension huge!

Reading for Next Time Force, mass, and acceleration: how and why motion changes Keep in mind how this applies in everyday experience.