Transcript Chapter 2 - Motion in One Dimension

Vectors
• Vectors vs. Scalars
• Vector Addition
– Commutative Law
– Associative Law
– Subtraction
• Vector Components
– Component Addition
• Unit Vectors
Vector addition – Tip to tail method
A
B
AB
B
A
AB
Vector addition – Commutative Law
A
B
B A
B A
A
B
A
B
AB
BA  AB
Vector addition – Parallelogram method
A
AB
A
AB
B
B
Vector addition – Associative Law

 

V1  V2  V3  V1  V2  V3
Vector subtraction - Add the negative
A
 
A  B  A  B
B
AB
A
B
Multiplication of a vector by a scalar
B  mA
A
mA
In this case is m greater
than or less than 1?
Components of a vector
y
A x  A cos 
A  A 2x  A 2y
Ay
A y  A sin 
A

Ax
Using unit vectors:
tan  
x
A  A x ˆi  A y ˆj
Ay
Ax
Vector Addition using components
A
B
AB
A  A x ˆi  A y ˆj
B  Bx ˆi  By ˆj
A  B   A x  Bx  ˆi   A y  By  ˆj
Steps for vector addition
y
A x  A cos 
A
• Select a coordinate system
A
A y  A sin 
• Draw the vectors

A
• Find the x and y coordinates of all
x
vectors
• Find the resultant components with
A  B   A x  Bx  ˆi   A y  By  ˆj
addition and subtraction
• Use the Pythagorean theorem to find
A  A A
the magnitude of the resulting vector
A
• Use a suitable trig function to find the
tan  
A
angle wrt to x axis
y
x
2
x
2
y
y
x
Example – Force
• Given:
• Find:
F1  7.0N
30o with the x-axis
F2  8.0N
105o with the x-axis
F1  F2