{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Ch11 - Chapter 11 Rotational Vectors and Angular Momentum...

This preview shows pages 1–12. Sign up to view the full content.

Chapter 11: Rotational Vectors and Angular Momentum Vector (cross) products ± Definition of vector product and its properties Axis of rotation a r b r b a r r × θ sin ˆ b a n b a r r r r = × n ˆ unit vector normal to the plane defined by a r and b r a b b a r r r r × = × 0 r r r r r = × = × a a a a

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Vector (cross) products (cont’d) ± Properties of vector product k j i ˆ ˆ ˆ = × x y z i ˆ j ˆ k ˆ k j i ˆ , ˆ , ˆ i k j ˆ ˆ ˆ = × j i k ˆ ˆ ˆ = × Consider three vectors: ) , , ( z y x a a a a : unit vector in x,y,z direction = r ) , , ( z y x b b b b = r ) , , ( z y x c c c c = r k b a b a j b a b a i b a b a k b j b i b k a j a i a b a c x y y x z x x z y z z y z y x z y x ˆ ) ( ˆ ) ( ˆ ) ( ) ˆ ˆ ˆ ( ) ˆ ˆ ˆ ( + + = + + × + + = × = r r r Then
Vector (cross) products (cont’d) ± Properties of vector product (cont’d) z j ˆ k ˆ y z z y x b a b a c = z x x z y b a b a c = y x y y x z b a b a c = x i ˆ z y x z y x b b b a a a k j i ˆ ˆ ˆ = b a c r r r × =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Torque Torque is a quantitative measure of the tendency of a force to cause or change the rotational motion. ± Case for 1 point-like object of mass m with 1 force x y r r θ m F r r ˆ ˆ massless rigid rod φ • r is constant only tangential component of causes rotation F r ˆ = F F t r α ω mr r r r m a m F = = = ˆ ) ˆ ˆ ( ˆ ˆ 2 r r 0 τ I mr r F t = = 2 t F moment of inertia torque unit Nm
Torque ± Case for 1 point-like object of mass m with 1 force x y r r F r r ˆ φ sin r r t = lever arm F r Fr Fr r F t t r r × = = = = τ sin F r t F Define r v r × = r is aligned with the rotation axis

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Torque (cont’d) ± Case for 2 point-like objects with 2 forces 2 1 θ = = 1 r r 2 r r t F 2 t F 1 2 2 1 1 r F r F t t net + = τ > 0 < 0 x y increases decreases Example: A see-saw in balance Mm r R Mg 0 2 2 1 1 = = + = mgr MgR r F r F t t net r R M m / / = mg
Work & energy (I) ± A massive body on massless rigid rod y r r θ m t F r Work done by the force: ∫∫ ∫ = = = τ d rd F ds F W t t Also ∫∫ = = ω α d dt d I d I W ) / ( = d dt d d d I ) / )( / ( 1 0 1 0 | ) 2 / 1 ( 2 = = I d I K I I = = 2 0 2 1 ) 2 / 1 ( ) 2 / 1 ( x W= K

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Work & energy (I) (cont’d) ± Power in rotational motion Work done by the force: ∫∫ ∫ = = = θ τ d rd F ds F W t t d dW = dt d dt dW / / = τω = P Power :
Correspondence between linear & angular quantities linear angular displacement velocity acceleration mass force Newton’s law kinetic energy work x θ dt dx v / = dt d / ω= dt dv a / = dt d / ω α= m = 2 i i r m I F r α τ I = ma F = F r r r r × = 2 ) 2 / 1 ( mv K = 2 ) 2 / 1 ( I K = = Fdx W = d W

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Work & energy (II) ± A massive body in rotational & translational motion Kinetic energy: = i i i v v m K r r ) 2 / 1 ( V v v i i r r r + = ' where ' i v r Now is the velocity with respect to V r the center of mass (COM) and ∑∑ + = + + = + + = + + = + + = = 2 2 ' ' 2 2 ' ' 2 2 ' 2 ' 2 ' ' ' ) 2 / 1 ( ) 2 / 1 ( / ) ( ) 2 / 1 ( ) 2 / 1 ( / ) 2 / 1 ( ) 2 / 1 ( ) 2 / 1 ( ) 2 / 1 ( ) ( ) ( ) 2 / 1 ( ) 2 / 1 ( MV v m dt r m d V MV v m dt r d m V m V v m V m v V m v m V v V v m v v m K i i i i i i i i i i i i i i i i i i i i i i r r r r r r r r r r the velocity of COM. Then r r 0 com ' i v r V r
Work & energy (II) cont’d ±

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 49

Ch11 - Chapter 11 Rotational Vectors and Angular Momentum...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online