Question

The rotational inertia of a thin rod about one end is 1/3 ML2. What is the rotational inertia of the same rod about a point located 0.40 L from the end

Answer 1

The value is
`I = 0.0932 ML ^2`

Step-by-step explanation:

From the question we are told that

The rotational inertia about one end is
`I_R = (1)/(3) ML^2`

The location of the axis of rotation considered is
`d = 0.4 L`

Generally the mass of the portion of the rod from the axis of rotation considered to the end of the rod is
`0.4 M`

Generally the length of the rod from the its beginning to the axis of rotation consider is

`k = 1 - 0.4 L = 0.6L`

Generally the mass of the portion of the rod from the its beginning to the axis of rotation consider is

`m = 1- 0.4 M = 0.6 M`

Generally the rotational inertia about the axis of rotation consider for the first portion of the rod is

`I_(R1) = (1)/(3) (0.6 M )(0.6L)^2`

`I_(R1) = (1)/(3) (0.6 M )L^2 0.6^2`

Generally the rotational inertia about the axis of rotation consider for the second portion of the rod is

`I_(R2) = (1)/(3) (0.6 M )(0.6L)^2`

=>
`I_(R2) = (1)/(3) (0.4 M )(0.4L)^2`

=>
`I_(R2) = (1)/(3) (0.4 M )L^2 0.4^2`

Generally by the principle of superposition that rotational inertia of the rod at the considered axis of rotation is

`I = (1)/(3) (0.6 M )L^2 0.6^2 + (1)/(3) (0.4 M )L^2 0.4^2`

=>
`I = (1)/(3) ML ^2 [0.6 * (0.6)^2 + 0.4 * (0.4)^2 ]`

=>
`I = 0.0932 ML ^2`