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Analytical Mech Homework Solutions 159

Analytical Mech Homework Solutions 159 - mx = q v × B x...

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Unformatted text preview: ( mx = q v × B ) x ˆjˆ Due to the cyclic nature of the Cartesian coordinates, i.e., i × ˆ = k … () mz = q ( v × B ) Altogether, mr = q ( v × B ) my = q v × B y z 10.26 V = mgz 1 m ( x2 + y 2 + z 2 ) 2 p p p x= x similarly, y = y and z = z m m m T= ∂T = mx , ∂x 1 H = T +V = ( px2 + p y2 + pz2 ) + mgz 2m ∂H px = =x ∂px m ∂H px = constant = 0 = − px , ∂x d px = ( mx ) = mx = 0 similarly, p y = constant , or my = 0 dt ∂H p y = =y ∂p y m px = ∂H pz = =z ∂pz m ∂H d pz = ( mz ) = mz = − mg = mg = − pz , dt ∂z These agree with the differential equations for projectile motion in Section 4.3. 10.27 (a) Simple pendulum … V = −mgl cos θ 1 22 ml θ 2 p θ = θ2 ml T= ∂T = ml 2θ , ∂θ pθ2 H = T +V = − mgl cos θ 2ml 2 p ∂H = θ2 = θ ∂pθ ml ∂H = mgl sin θ = − pθ ∂θ pθ = 21 ...
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