Download - Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Transcript
Page 1: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Where are we?

Forces of the form F (v)

Example: F (v) = −m Γ v

Another example: F (v) = −mβ v2

Forces of the form F (x)

Review of the harmonic oscillator

Linearity and Time Translation Invariance

Back to F (v) = −m Γ v

Page 2: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 3: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 4: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 5: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 6: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 7: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 8: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (v)

Formal solution of F = ma

F (v) = ma

F (v) = mdv

dt

dt = dvm

F (v)

t− t0 =∫ t

t0dt′ =

∫ v(t)

v(t0)dv′

m

F (v′)solve this for v(t) as a function of t

Page 9: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Example: F (v) = −β v = −m Γ v

Frictional force appropriate for smallvelocities in viscous fluids.

Page 10: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 11: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 12: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 13: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 14: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 15: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 16: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −β v = −m Γ v

dt = − dv

Γ vΓ dt = −dv

v∫ t

t0Γ dt′ = −

∫ v(t)

v(t0)dv′

1

v′

Γ (t− t0) = − ln v(t) + ln v(t0)

eΓ (t−t0) = v(t0)/v(t)

v(t) = v(t0) e−Γ (t−t0)

Page 17: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

1 2 3 4 5

0.2

0.4

0.6

0.8

1

↑v(t)v(t0)

Γ (t−t0) →

Page 18: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = x(t0) +∫ t

t0dt′ x(t′)

x(t) = x(t0) +∫ t

t0dt′ v(t′)

= x(t0) +∫ t

t0dt′ v(t0) e−Γ (t′−t0)

u = e−Γ (t′−t0) du = −dt′ Γ e−Γ (t′−t0)

x(t) = x(t0)− v(t0)

Γ

∫ e−Γ (t−t0)

1du

x(t) = x(t0) +v(t0)

Γ

(1− e−Γ (t−t0)

)

Page 19: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = x(t0) +∫ t

t0dt′ x(t′)

x(t) = x(t0) +∫ t

t0dt′ v(t′)

= x(t0) +∫ t

t0dt′ v(t0) e−Γ (t′−t0)

u = e−Γ (t′−t0) du = −dt′ Γ e−Γ (t′−t0)

x(t) = x(t0)− v(t0)

Γ

∫ e−Γ (t−t0)

1du

x(t) = x(t0) +v(t0)

Γ

(1− e−Γ (t−t0)

)

Page 20: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

1 2 3 4 5

0.2

0.4

0.6

0.8

1

↑x(t)−x(t0)

v(t0)/Γ

Γ (t−t0) →

Page 21: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

More general but still very simple case

F (v) = F0 −m Γ v

Page 22: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

More general but still very simple case

F (v) = F0 −m Γ v

v(t) = v(t0) e−Γ (t−t0) +F0

m Γ

(1− e−Γ (t−t0)

)

Page 23: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

More general but still very simple case

F (v) = F0 −m Γ v

v(t) = v(t0) e−Γ (t−t0) +F0

m Γ

(1− e−Γ (t−t0)

)

v(∞) =F0

m Γindependent of v0

Arisotelian physics if Γ is large - v ∝ F0

Page 24: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

1 2 3 4 5

0.5

1

1.5

2

2.5

3

↑v(t)

F0/mΓ

Γ (t−t0) →

v0 > F0

v0 < F0

Page 25: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Another example: F (v) = −mβ v2

Frictional force for rapid movement througha thin medium like a gas — force arisesbecause the body is bumping into moleculesand knocking them out of the way. Theforce from each collision is ∝ v, and the #per unit time is also ∝ v

Page 26: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

1

v(t0)+ β (t− t0) =

1

v(t)− 1

v(t0)− 1

v(t0)

v(t) =v(t0)

1 + β v(t0) (t− t0)

Page 27: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

1

v(t0)+ β (t− t0) =

1

v(t)− 1

v(t0)− 1

v(t0)

v(t) =v(t0)

1 + β v(t0) (t− t0)

Page 28: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

1

v(t0)+ β (t− t0) =

1

v(t)− 1

v(t0)− 1

v(t0)

v(t) =v(t0)

1 + β v(t0) (t− t0)

Page 29: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

β (t− t0) =1

v(t)− 1

v(t0)

v(t) =v(t0)

1 + β v(t0) (t− t0)

Page 30: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

β (t− t0) =1

v(t)− 1

v(t0)

1

v(t0)+ β (t− t0) =

1

v(t)

Page 31: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

1 + v(t0)β (t− t0)

v(t0)=

1

v(t)

1

v(t0)+ β (t− t0) =

1

v(t)

Page 32: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

ma = mdv

dt= F (v) = −mβ v2

β dt = −dv

v2

∫ t

t0β dt′ = −

∫ v(t)

v(t0)

dv′

v′ 2

1 + v(t0)β (t− t0)

v(t0)=

1

v(t)

v(t) =v(t0)

1 + β v(t0) (t− t0)

Page 33: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

F (v) = −β v = −m Γ v

⇒ v(t) = v(t0) e−Γ (t−t0)

F (v) = −mβ v2

⇒ v(t) =v(t0)

1 + β v(t0) (t− t0)

Page 34: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

F x(t)

F0 ⇒ x(t0) + v(t0) (t− t0) +F0

2m(t− t0)

2

−m Γ v ⇒ x(t0) +v(t0)

Γ

(1− e−Γ (t−t0)

)

−mβ v2 ⇒ x(t0) +1

βlog

(1 + β v(t0) (t− t0)

)

Page 35: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Forces of the form F (x)

Lots of examples (if you don’t look tooclosely) - gravity - the electric force. Manysituations in which there is v dependence,but it is small and can be ignored.

Page 36: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

mdv

dt= F (x)

mdv

dtv = F (x)

dx

dt1

2m

d

dtv2 = F (x)

dx

dt1

2m

∫ t

t0dt′

d

dt′v(t′)2 =

∫ t

t0dt′ F (x(t′))

dx

dt′1

2m

(v(x)2 − v2

0

)=

∫ x

x0

dx′ F (x′)

Page 37: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

We use a dirty trick - that we will eventuallysee is related to conservation of energy

Page 38: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

mdv

dt= F (x)

mdv

dtv = F (x)

dx

dt1

2m

d

dtv2 = F (x)

dx

dt1

2m

∫ t

t0dt′

d

dt′v(t′)2 =

∫ t

t0dt′ F (x(t′))

dx

dt′1

2m

(v(x)2 − v2

0

)=

∫ x

x0

dx′ F (x′)

Page 39: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

mdv

dt= F (x)

mdv

dtv = F (x)

dx

dt1

2m

d

dtv2 = F (x)

dx

dt1

2m

∫ t

t0dt′

d

dt′v(t′)2 =

∫ t

t0dt′ F (x(t′))

dx

dt′1

2m

(v(x)2 − v2

0

)=

∫ x

x0

dx′ F (x′)

Page 40: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

mdv

dt= F (x)

mdv

dtv = F (x)

dx

dt1

2m

d

dtv2 = F (x)

dx

dt1

2m

∫ t

t0dt′

d

dt′v(t′)2 =

∫ t

t0dt′ F (x(t′))

dx

dt′1

2m

(v(x)2 − v2

0

)=

∫ x

x0

dx′ F (x′)

Page 41: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

mdv

dt= F (x)

mdv

dtv = F (x)

dx

dt1

2m

d

dtv2 = F (x)

dx

dt1

2m

∫ t

t0dt′

d

dt′v(t′)2 =

∫ t

t0dt′ F (x(t′))

dx

dt′1

2m

(v(x)2 − v2

0

)=

∫ x

x0

dx′ F (x′)

Page 42: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

mdv

dt= F (x)

mdv

dtv = F (x)

dx

dt1

2m

d

dtv2 = F (x)

dx

dt1

2m

∫ t

t0dt′

d

dt′v(t′)2 =

∫ t

t0dt′ F (x(t′))

dx

dt′1

2m

(v(x)2 − v2

0

)=

∫ x

x0

dx′ F (x′)

Page 43: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Once you know v(x), you can find x(t) byintegration

dx

dt= v(x)

Page 44: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Once you know v(x), you can find x(t) byintegration

dx

dt= v(x)

dt =dx

v(x)

Page 45: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Once you know v(x), you can find x(t) byintegration

dx

dt= v(x)

dt =dx

v(x)∫ t

t0dt′ = t− t0 =

∫ x(t)

x0

dx′

v(x′)

Page 46: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Once you know v(x), you can find x(t) byintegration

dx

dt= v(x)

dt =dx

v(x)∫ t

t0dt′ = t− t0 =

∫ x(t)

x0

dx′

v(x′)This can now be solved IN PRINCIPLE forx(t)

Page 47: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

ªªªªªªªª­­­­­­­­¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡

Review of the harmonic oscillator

Page 48: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

ªªªªªªªª­­­­­­­­¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡

x →|

equilibrium at x = 0

F = −K x

K is the spring constant the trajectories arethe solutions of this

Page 49: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

ªªªªªªªª­­­­­­­­¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡

x →|

equilibrium at x = 0

F = −K x

K is the spring constant the trajectories arethe solutions of this

Page 50: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

¡¡¡

ªªªªªªªª­­­­­­­­¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡ ¤¡

x →|

equiilbrium at x = 0

F = −K x

F = ma = md2

dt2x = m x = −K x

the trajectories are the solutions of this

Page 51: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

Page 52: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

check that this is right

d

dzcos z = − sin z

d

dzsin z = cos z

chain rule

d

dtcos ωt = −ω sin ωt

d

dtsin ωt = ω cos ωt

Page 53: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

check that this is right

d

dzcos z = − sin z

d

dzsin z = cos z

chain rule

d

dtcos ωt = −ω sin ωt

d

dtsin ωt = ω cos ωt

Page 54: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

check that this is right

d

dzcos z = − sin z

d

dzsin z = cos z

chain rule

d

dtcos ωt = −ω sin ωt

d

dtsin ωt = ω cos ωt

Page 55: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

x(t) = −aω sin ωt + bω cos ωt

Page 56: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

x(t) = −aω sin ωt + bω cos ωt

x(t) = −aω2 cos ωt− bω2 sin ωt

Page 57: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

x(t) = −aω sin ωt + bω cos ωt

x(t) = −aω2 cos ωt− bω2 sin ωt

= −ω2(a cos ωt + b sin ωt) = −ω2 x(t)

m x(t) = −ω2 mx(t) = −K x(t)

Page 58: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

x(t) = −aω sin ωt + bω cos ωt

x(t) = −aω2 cos ωt− bω2 sin ωt

= −ω2(a cos ωt + b sin ωt) = −ω2 x(t)

m x(t) = −ω2 mx(t) = −K x(t)

Page 59: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

constant ω is called the Angular Frequency

Page 60: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt for ω =√

K/m

constant ω is called the Angular Frequency

Two constants (a and b) label thetrajectories - as expected - two initialconditions for one degree of freedom -related to initial x and v

Page 61: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt

setting t = 0 gives

x(0) = a cos 0 + b sin 0 = a

⇒ a is the position of the mass at t = 0

Page 62: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m x = −K x

x(t) = a cos ωt + b sin ωt

v(t) = x(t) = −aω sin ωt + bω cos ωt

setting t = 0 gives

v(0) = x(0) = −aω sin 0 + bω cos 0 = bω

⇒ bω is the velocity of the mass at t = 0

Page 63: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = x(0) cos ωt +v(0)

ωsin ωt

for arbitrary initial time t = t0

= x(t0) cos[ω(t− t0)] +v(t0)

ωsin[ω(t− t0)]

Page 64: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

F x(t)

F0 ⇒ x(t0) + v(t0) (t− t0) +F0

2m(t− t0)

2

−m Γ v ⇒ x(t0) +v(t0)

Γ

(1− e−Γ (t−t0)

)

−mβ v2 ⇒ x(t0) +1

βlog

(1 + β v(t0) (t− t0)

)

−mω2 x ⇒ x(t0) cos[ω(t− t0)

]+

v(t0)

ωsin

[ω(t− t0)

]

Page 65: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = a cos ωt + b sin ωt = c cos(ωt− φ)

Page 66: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = a cos ωt + b sin ωt = c cos(ωt− φ)

= c (cos ωt cos φ + sin ωt sin φ)

= c cos φ cos ωt + c sin φ sin ωt

Page 67: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = a cos ωt + b sin ωt = c cos(ωt− φ)

= c (cos ωt cos φ + sin ωt sin φ)

= c cos φ cos ωt + c sin φ sin ωt

a = c cos φ b = c sin φ

Page 68: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = a cos ωt + b sin ωt = c cos(ωt− φ)

= c (cos ωt cos φ + sin ωt sin φ)

= c cos φ cos ωt + c sin φ sin ωt

a = c cos φ b = c sin φ

Page 69: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = a cos ωt + b sin ωt = c cos(ωt− φ)

= c (cos ωt cos φ + sin ωt sin φ)

= c cos φ cos ωt + c sin φ sin ωt

a = c cos φ b = c sin φ

c =√

a2 + b2 φ = arctanb

a

Page 70: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

x(t) = a cos ωt + b sin ωt = c cos(ωt− φ)

..........................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................

........................................................................................................................................................................................................................................................

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c

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.. slope is b ω

0 2πω

10ω

φω

0

a

t →

↑x

Page 71: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

sin and cos periodic with period 2π

..........................................................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................

......................................................................................................................................................................................................................................................

..............................................................................

........

.....

........

.....

........

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c

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.. slope is b ω

0 2πω

10ω

φω

0

a

t →

↑x

Page 72: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

sin and cos periodic with period 2π

x(t + 2π/ω) = x(t)

motion repeats after a time τ = 2π/ω called“the period of the oscillation” — frequency(rather than angular frequency) is

ν =ω

2π=

1

τ

Page 73: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Remembering where the 2πs go —

ωt — argument of sin and cos is an angle

ω has unitsradians

sec

Page 74: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Remembering where the 2πs go —

ωt — argument of sin and cos is an angle

ω has unitsradians

secνt is the number of repeats or “cycles”

ν has unitscycles

sec

Page 75: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Remembering where the 2πs go —

ωt — argument of sin and cos is an angle

ω has unitsradians

secνt is the number of repeats or “cycles”

ν has unitscycles

sec

ωradians

sec= 2π ν

cycles

sec

Page 76: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Linearity and Time TranslationInvariance

equations of motion of the formm x + K x = 0 are ubiquitous

appears in many different mechanicalsystems, and even in electrical systems

What is so special about this???

Page 77: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Linearity and Time TranslationInvariance

equations of motion of the formm x + K x = 0 are ubiquitous

appears in many different mechanicalsystems, and even in electrical systems

What is so special about this???

Page 78: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Linearity and Time TranslationInvariance

equations of motion of the formm x + K x = 0 are ubiquitous

appears in many different mechanicalsystems, and even in electrical systems

What is so special about this???

Page 79: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m d2

dt2x + K x = 0

1: Time translation invariance — noexplicit dependence on t — d/dt but no t

Page 80: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m d2

dt2x + K x = 0

1: Time translation invariance — noexplicit dependence on t — d/dt but no t

if x(t) is a solution then so is x(t + a)

Page 81: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m d2

dt2x + K x = 0

1: Time translation invariance — noexplicit dependence on t — d/dt but no t

if x(t) is a solution then so is x(t + a)

2: Linearity — all terms ∝ one power of x

or its derivatives

Page 82: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

m d2

dt2x + K x = 0

1: Time translation invariance — noexplicit dependence on t — d/dt but no t

if x(t) is a solution then so is x(t + a)

2: Linearity — all terms ∝ one power of x

or its derivatives

new solutions as linear combinations of oldones — if x1(t) and x2(t) are solutionsthen so is Ax1(t) + B x2(t)

Page 83: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Time translation invariance — laws ofphysics don’t change with time or do so veryslowly

Page 84: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Linearity — what does it have to do withphysics? Why should many systems beapproximately linear?

Page 85: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Oscillations about equilibrium

d2

dt2x = F(x)

Page 86: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Oscillations about equilibrium

d2

dt2x = F(x)

equilibrium ⇒ F(0) = 0

Page 87: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Oscillations about equilibrium

d2

dt2x = F(x)

equilibrium ⇒ F(0) = 0

F(x) = F(0) + xF ′(0) +1

2x2F ′′(0) + · · ·

Page 88: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Oscillations about equilibrium

d2

dt2x = F(x)

equilibrium ⇒ F(0) = 0

F(x) = F(0) + xF ′(0) +1

2x2F ′′(0) + · · ·

= xF ′(0) +1

2x2F ′′(0) + · · ·

Page 89: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Oscillations about equilibrium

d2

dt2x = F(x)

equilibrium ⇒ F(0) = 0

F(x) = F(0) + xF ′(0) +1

2x2F ′′(0) + · · ·

= xF ′(0) +1

2x2F ′′(0) + · · ·

small for sufficiently small x – unless F ′(0)

Page 90: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Oscillations about equilibrium

d2

dt2x = F(x)

equilibrium ⇒ F(0) = 0

F(x) = F(0) + xF ′(0) +1

2x2F ′′(0) + · · ·

= xF ′(0) +1

2x2F ′′(0) + · · ·

Only the linear term relevant for sufficientlysmall oscillations

Page 91: Where are we?phys16/lectures/sl02.pdfWhere are we? Forces of the form F(v) Example: F(v) = ¡m¡v Another example: F(v) = ¡mflv2 Forces of the form F(x) Review of the harmonic oscillator

Back to F (v) = −m Γ v

It’s Linear!

Smoothness - reversibility - unlike −β v2

friction