Question

Inertia and kinetic energy?

In the figure, two particles, each with mass m = 0.76 kg, are fastened to each other, and to a rotation axis at O, by two thin rods, each with length d = 5.9 cm and mass M = 1.0 kg. The combination rotates around the rotation axis with angular speed ω = 0.28 rad/s. Measured about O, what is the combination's (a) rotational inertia and (b) kinetic energy?

Answer 1

Rotational inertia `=2.25times 10^-2 " Kg"*"m"^2`

Kinetic energy `=0.0177" J"`

Explanation:

One can think of the two rods together as a single rod of mass `2M` and length `2d`. Then the moment of inertia of the two rod system about `O` is `1/3times 2M times (2d)^2 =8/3Md^2`

The net moment of inertia of the system is

`8/3Md^2+md^2+m(2d)^2 = (8/3M+5m)d^2`

Substituting the values, we get

`I = (8/3times 1" Kg"+5 times 0.76" Kg") times (0.059" m")^2~~2.25times 10^-2 " Kg"*"m"^2`

The kinetic energy is

`1/2 I omega^2 = 1/2 times 2.25times 10^-2 " Kg"*"m"^2times (0.28" s"^-1)^2 ~~0.0177" J"`