WorkWork

1

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.

cosAB

BA where θ is the angle between A and B

zzyyxx BABABA

BA

A

θcosA

B

Component of A along B

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Examples of Dot Products

268.13)110cos()10)(4( mmm o

BA

4m 110o

10m

kjiA )3()3()5( mmm

kiB )4()6( mm

218)4)(3()0)(3()6)(5( mmmmmmm

BA

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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

rFW

kjir zyx

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Example of Work

The following is an example of the work done by gravity on a mass.

Δh2

Δh1

Δw

y

xm

2121 00 hmghmghmghmgW

rFW

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A2

A1

Example of Work

This is an example using the the fact that integrals are areas under a curve.

JNmbhA 5252

1

2

11

JNmbhA 6432

1

2

12

JAAW 121

F (N)

0

1

-1 X (m)

0 1 2

rFW