7.2 Work Done by a Constant Force

7.1 Work done by a constant force

The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude F of the force, the magnitude r of the displacement of the point of application of the force, and cos where is the angle between the force and the displacement vectors:

W = F r cos

7.3 The Scalar Product of Two Vectors

7.2 Scalar product of any two

vectors and

7.3

7.4

7.5

7.6

Aur

Bur

rA⋅

rB ≡ABcos

W =Fr cos =rF ⋅

rr

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

⋅ = ⋅ = ⋅ =

⋅ = ⋅ = ⋅ =

i i j j k k

i j i k j k

1

0

rAg

rB =AxBx + AyBy + AzBz

7.4 Work Done by a Varying Force

7.7

7.8

7.9 Spring force

7.10

7.11

7.12

7.13

=∫f

i

x

xxW F dx

W =Wnet =

rF∑( )⋅d

rr

x i

xf∫∑

Fr

s =Fsi = −kxi

W

s=

rFs∫ ⋅d

rr = −kxi( )⋅dxi( )

xi

xf∫ = −kx( )dx=12−xmax

0

∫ kxmax2

( )= − = −∫ 2 21 1

2 2f

i

x

s i fxW kx dx kx kx

( )= = −∫ 2 21 1

2 2f

i

x

app f ixW kx dx kx kx

Fs = - kx

7.5 Kinetic Energy and the Work–KineticEnergy Theorem

7.14

7.15

7.16

7.17

W

net= F dx∑xi

xf∫

W

net=

12

mvf2 −

12

mvi2

W

net=Kf −K i =K

K = ½ mv2

7.6 Potential Energy of a System

7.18

7.19

7.20

7.21

7.22

W

net=

rFapp( )⋅

rr =(mgj)⋅ yf −yi( ) j⎡

⎣⎤⎦=mgyf −mgyi

Us ≡

12

kx2

Ug = mgy

Wnet = Ug

W app= ½ kxf2 – ½ kxi

2

7.7 Conservative andNonconservative Forces

7.23

7.24 Emech ≡K + U

WC = Ui - Uf - U

7.8 Relationship Between ConservativeForces and Potential Energy

7.25

7.26

7.27 dU = ¯ Fxdx

7.28

f

i

x

C xxW F dx U= =−∫

U =Uf −Ui =− Fx dx

xi

xf∫

F

x=−

dUdx