If a wire with linear density ρ ( x , y , z ) lies along a space curve C , its moments of inertia about the x -, y -, and z -axes are defined as I x = ∫ C ( y 2 + z 2 ) ρ ( x , y , z ) ds I y = ∫ C ( x 2 + z 2 ) ρ ( x , y , z ) ds I z = ∫ C ( x 2 + y 2 ) ρ ( x , y , z ) ds Find the moments of inertia for the wire in Exercise 35. 35. (a) Write the formulas similar to Equations 4 for the center of mass ( x ¯ , y ¯ , z ¯ ) of a thin wire in the shape of a space curve C if the wire has density function ρ ( x , y , z ). (b) Find the center of mass of a wire in the shape of the helix x = 2 sin t , y = 2 cos t , z = 3 t , 0 ⩽ t ⩽ 2 π , if the density is a constant k .
If a wire with linear density ρ ( x , y , z ) lies along a space curve C , its moments of inertia about the x -, y -, and z -axes are defined as I x = ∫ C ( y 2 + z 2 ) ρ ( x , y , z ) ds I y = ∫ C ( x 2 + z 2 ) ρ ( x , y , z ) ds I z = ∫ C ( x 2 + y 2 ) ρ ( x , y , z ) ds Find the moments of inertia for the wire in Exercise 35. 35. (a) Write the formulas similar to Equations 4 for the center of mass ( x ¯ , y ¯ , z ¯ ) of a thin wire in the shape of a space curve C if the wire has density function ρ ( x , y , z ). (b) Find the center of mass of a wire in the shape of the helix x = 2 sin t , y = 2 cos t , z = 3 t , 0 ⩽ t ⩽ 2 π , if the density is a constant k .
Solution Summary: The author explains the moment of inertia for a thin wire about the x, y, and z axes.
If a wire with linear density ρ(x, y, z) lies along a space curve C, its moments of inertia about the x-, y-, and z-axes are defined as
Ix = ∫C (y2 + z2) ρ(x, y, z) ds
Iy = ∫C (x2 + z2) ρ(x, y, z) ds
Iz = ∫C (x2 + y2) ρ(x, y, z) ds
Find the moments of inertia for the wire in Exercise 35.
35. (a) Write the formulas similar to Equations 4 for the center of mass (
x
¯
,
y
¯
,
z
¯
) of a thin wire in the shape of a space curve C if the wire has density function ρ(x, y, z).
(b) Find the center of mass of a wire in the shape of the helix x = 2 sin t, y = 2 cos t, z = 3t, 0 ⩽ t ⩽ 2π, if the density is a constant k.
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