10/22/2009
Work
Work
How to calculate a dot product
A dot product is the multiplication of two vectors to form a scalar.
There are two ways to calculate the dot product.




A B  Ax Bx  Ay By  Az Bz
A B  AB cos 
where θ is the angle between A and B
Component of A along B
A
θ
B
A cos 
1
Work
Work
Examples of Dot Products




A  (5m) i  (3m) j (3m) k



B  (6m) i  (4m) k


A B  (5m)(6m)  (3m)(0m)  (3m)(4m)  18m2
4m
110o
10m


A B  (4m)(10m) cos(110o )  13.68m2
2
1
10/22/2009
Work
Work
Definition of Work
Work is defined as the amount of force acting over a distance.
The unit of work is either Joules (J) or electron-volts (eV)
Dr
F


W  F D r




D r  Dx i  Dy j Dz k
3
Work
Work

Example of Work

W  F D r
The following is an example of the work done by gravity on a
mass.
y
Δw
Δh1
x
m
Δh2
W  mgDh1  0  mgDh2   mgDh1   0  mgDh2
4
2
10/22/2009
Work
Work

Example of Work

W  F D r
This is an example using the the fact that integrals are areas
under a curve. F (N)
1
A1
0
-1
A2
1
5
X (m)
2
0
1
1
A1  bh  5m2 N   5 J
2
2
1
1
A2  bh  3m 4 N   6 J
2
2
W  A1  A2  1J
3