Class 12: Coefficient of restitution and Class 12: Coefficient of - - PowerPoint PPT Presentation

Class 12: Coefficient of restitution and Class 12: Coefficient of restitution and elastic collision

Center of mass

∑

⎫

Many particles m1, m2, m3, …. located at (x1, y1), (x2, y2), (x3, y3),…. respectively.

i i i 3 3 2 2 1 1 cm i i i i i 3 2 1 3 3 2 2 1 1 cm

r m r m r m r m r m x m m m m x m x m x m x

∑ ∑ ∑ ∑ ∑

= + + + = ⇒ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ = + + + + + + = v L v v v v L L

i i 3 2 1 cm i i i i i 3 2 1 3 3 2 2 1 1 cm

m m m m m y m m m m y m y m y m y

∑ ∑ ∑

+ + + ⎪ ⎪ ⎪ ⎭ ⎬ = + + + + + + = L L L

i x i i 3 2 1 x 3 x 2 x 1 x

v m m v m m m m v m v m v m v

i 3 2 1 cm

∑ ∑ ∑

⎪ ⎪ ⎪ ⎫ = + + + + + + = v L L

i i i i i 3 2 1 3 3 2 2 1 1 cm y i i y 3 y 2 y 1 y i i 3 2 1

m v m m m m v m v m v m v m v m m m m v m v m v m v m m m m

i 3 2 1 cm

∑ ∑ ∑ ∑ ∑

= + + + + + + = ⇒ ⎪ ⎪ ⎪ ⎪ ⎬ = + + + + + + = + + + L L v v v v L L

i i 3 2 1

m m m m

∑

⎪ ⎭ + + + L

Physics of the center of mass

1. Momentum: Total momentum of all particles observed in the center of mass frame = 0. 2. Kinetic energy:

2

frame CM in the

• bserved

energy Kinetic Mv 2 1 Energy Kinetic Total

CM +

=

2 CM i i i

) v v ( m 2 1 frame CM in the

• bserved

energy Kinetic − =∑

M = Total mass of all particles, vCM = velocity of the center of mass 3.

a M F

CM Total =

v v particles. all

• f

mass Total M particles. all

• n

acting force external total the is F

Total

= v

C ffi i t f tit ti Coefficient of restitution ' v ' v

1 2 1 2

v v ' v ' v

• e

− − =

rel rel

v v'

• collision

before velocity relative collision after velocity relative

• =

=

rel

y 0 ≤ e ≤ 1: 0 ≤ e ≤ 1: e=1 for elastic collision e=0 for completely inelastic collision e 0 for completely inelastic collision

El ti lli i Elastic collision

2 2 1 1 2 2 1 1

' v m ' v m v m v m + = + 1 1 1 1

2 2 2 2 1 1 2 2 2 2 1 1

' v m 2 1 ' v m 2 1 v m 2 1 v m 2 1 + = + Replace v v ' v ' v

• 1

1 2 1 2

− − =

General collision problem General collision problem (1) ' '+ + (1) ' v m ' v m v m v m

2 2 1 1 2 2 1 1

+ = + (2) v v ' v ' v

• e

1 2 1 2

− − = v v

1 2

Two equations two unknown (v ’ and v ’) Two equations, two unknown (v1 and v2 )

Energy loss in a collision Energy loss in a collision m m 1

2 2 1 2 2 1 2 1

) e

• (1

) v

• (v

m m m m 2 1 Loss Energy + =

2 2 rel

) e

• (1

v 2 1 = μ

2 1m

m

2 1 2 1

m m m m mass reduced + = μ