Class 7 relative motion and fixed axis rotation

2 velocity vector of point on body in pure rotation

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Unformatted text preview: Distribute the cross-product, but keep the order the same ˆ×ˆ = 0 ii ˆ×ˆ = k ijˆ ˆ ˆ = −k ˆ j× i Fixed Axis Rotation vω/ A = r B × y’ rB / A B/ A vB / A = rB / Aω aα/ A = × ω/ A +ω ×r × r ( B B at = rB / Aα an = rB / Aω 2 A B/ A ) vB B x’ Figure 7.2 Velocity vector of point on body in pure rotation about fixed axis θ Fixed Axis Rotation v B/ A =ω× r B/ A aB/ A d d = ( vωA )r= ( B/ × dt dt & aB / A =ω× r B/ A ) +ω× r &B/ A B/ A aB/ A =α× r B/ A+ω× v aB/ A =α× r B/ A+ω×ω× r ( B/ A B/ A ) Planar simplification aB/ A =α× r B/ A+ω×ω× r ( B/ A ) Fixed Axis Rotation in 2­D y’ v B/ A =ω× r B/ A rB / A vB / A = rB / Aω A aB/ A =α× r B/ A- ω r B/ A an = rB / Aω 2 B x’ Figure 7.2 Velocity vector of point on body in pure rotation about fixed axis 2 at = rB / Aα vB θ Special Case If the angular acceleration of the body is constant, α = αC, the equations for angular velocity and acceleration can be integrated to yield the set of algebraic equations below. ω = ω0 + α C t θ = θ 0 + ω0 t + 1 α C t 2 2 ω 2 = ω0 2 + 2α C ( θ − θ 0 ) θ0 and ω0 are...
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