Download Vectors - barransclass

Announcements
• Please complete the survey on Moodle
• Twitter feed is on the class website
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
y
A = (4, 3)
B = (0, –2)
x
Magnitude from Components
Components: lengths of sides of right triangle
Magnitude: length of hypotenuse
A = (4, 3)
A
||A ||= A = 42 + 32
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
Head-to-tail (not in your book)
A
A
C
B
A+B=C
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 individually
A = (4, 3)
B
A
C
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
B
C
D
B
A
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
A+B=C
B
B+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!
• Just as with scalars:
5 s + 10 s = 15 s
good!
5 kg + 10 m = 15 ?
Bad!
• Or, algebra in general:
5 a + 10 a = 15 a
5 b + 10 c = 15 ?
good!
Bad!
Subtract Vectors
Add the negative of the vector being
subtracted.
(Negative = same magnitude, opposite
direction: what you must add to get zero)
D
A
B
A
–B
–B
A – B = A + (–B) = D
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
2A
–2 A
1/2 A
Scalar Multiplication Example
Velocity (a vector)  time (a scalar)
v Dt = Dr
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
weight
1m
F
8m
F
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.
B.
C.
D.
Equal to the weight of the person.
Equal to the tension in one cable.
Zero.
There is not enough information to answer.
Working with CommonlyEncountered Forces
Tension
Tension Forces
• In cables, threads, chains, etc.
• Direction: along the cable, inward
Poll: Hammock Tension
weight
1m
F
8m
F
A hammock slung between trees 8 m apart sags
1 m when a person lies in it.
The tension in a cable is
A.
B.
C.
D.
E.
Equal to the weight of the person.
About half the weight of the person.
Zero.
Much more than the weight of the person.
There is not enough information to answer.
Hammock Forces
tension
tension
weight
forces add to zero
Tension exceeds weight
for a shallow angle!
Application: Lumbar Forces
Spinal curvature
standing
sitting
Application: Lumbar Forces
Reaching with a load
weight
standing
weight
sitting
Application: Lumbar Forces
Standing
torque
support
Application: Lumbar Forces
Sitting
torque
support
huge!
Reading for Next Time
• Force, mass, and acceleration: how and
why motion changes
• Keep in mind how this applies in everyday
experience.