Find the moment of inertia of a non-uniform rod having mass per unit length λ = λ 0 ( 1 + x L ) about an axis ⊥ to the length of the rod and passing through an end point L. Length of the rod is L and m is mass of the rod. A. m L 2 3 B. 7 m L 2 12 C. 7 m L 2 18 D. m L 2 18

Question

Find the moment of inertia of a non-uniform rod having mass per unit length       λ    =            λ        0    (  1  +      x    L    ) about an axis ⊥ to the length of the rod and passing through an end point L. Length of the rod is L and m is mass of the rod. A.             m              L        2              3   B.             7      m              L        2              12   C.             7      m              L        2              18   D.             m              L        2              18

Armeebildzku · Accepted Answer

Total mass of the rod,  m  =  ∫  d  m  m  =  ∫      λ    d  x  =      ∫    0    L              λ        o    (  1  +      x    L    )  d  x  =            λ        o    [      ∫    0    L    d  x  +      1    L        ∫    0    L    x  d  x  ]  =            λ        o    [  x  +            x      2              2      L            ]    0    L    =            λ        o    [  L  +      L    2    ]  ⇒  m  =            3                        λ                o            L        2   Currently, rod inertia moment  I  =  ∫  d  m         x    2    =  ∫            λ        o    (  1  +      x    L    )  d  x         x    2    =            λ        o    [      ∫    0    L        x    2    d  x  +      1    L        ∫    0    L        x    3    d  x  ]  =            λ        o    [            L      3        3    +            L      4              4      L        ]  I  =            7                        λ                o                    L        3              12    =      7    12    [            3                        λ                o            L        2    ]  ×      2    3        L    2    =      7    18    m      L    2

Find the moment of inertia of a non-uniform rod having mass per unit length lambda=lambda_0(1+x/L) about an axis ⊥ to the length of the rod and passing through an end point L. Length of the rod is L and m is mass of the rod. A. (mL^2)/3 B. (7mL^2)/12 C. (7mL^2)/18 D. (mL^2)/18

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