Page 212 - Intro to Tensor Calculus

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2
3
I 11. Express the generalized velocity and acceleration in spherical coordinates (x ,x ,x )= (ρ, θ, φ)and
show 1 2 2 2
dx 1 dρ f = δV = d ρ − ρ dθ − ρ sin θ dφ
1
2
1
V = = δt dt 2 dt dt
dt dt
2
dx 2 dθ 2 δV 2 d θ dφ 2 2 dρ dθ
2
V = = f = = − sin θ cos θ +
dt dt δt dt 2 dt ρ dt dt
dx 3 dφ 3 2
3
V = = 3 δV d φ 2 dρ dφ dθ dφ
dt dt f = = 2 + +2 cot θ
δt dt ρ dt dt dt dt
Find the physical components of velocity and acceleration in spherical coordinates and show
2
dρ d ρ dθ 2 2 dφ 2
V ρ = f ρ = 2 − ρ − ρ sin θ
dt dt dt dt
dθ d θ dφ 2 dρ dθ
2
V θ =ρ f θ =ρ − ρ sin θ cos θ +2
dt dt 2 dt dt dt
dφ 2
V φ =ρ sin θ d φ dρ dφ dθ dφ
dt f φ =ρ sin θ 2 +2 sin θ +2ρ cos θ
dt dt dt dt dt
I 12. Expand equation (2.2.39) and write out all the components of the moment of inertia tensor I ij .
I 13. For ρ the density of a continuous material and dτ an element of volume inside a region R where the
material is situated, we write ρdτ as an element of mass inside R. Find an equation which describes the
center of mass of the region R.

I 14. Use the equation (2.2.68) to derive the equation (2.2.69).

I 15. Drop the bar notation and expand the equation (2.2.70) and derive the equations (2.2.71).

I 16. Verify the Euler transformation, given in example 2.2-5, is orthogonal.

I 17. For the pulley and mass system illustrated in the figure 2.2-7 let

a = the radius of each pulley.

` 1 = the length of the upper chord.
` 2 = the length of the lower chord.

Neglect the weight of the pulley and find the equations of motion for the pulley mass system.