Question

The moment of inertial of the hoop yo-yo when it is: (a) rotating about its center of mass, and (b) rotating about the point where the tension force is applied

mass=332g
diamater=35.9cm
thickness.95cm

Answer 1

a) The moment of inertia of the hoop yo-yo rotating about its center of mass is
`I_(g) = 0.0108\,kg\cdot m^(2)`.

b) The moment of inertial of the hoop yo-yo rotating about its center of mass is
`I_(O) = 0.0216\,kg\cdot m^(2)`.

Step-by-step explanation:

a) The hoop yo-yo can be modelled as a tours with a minor radius
`a`, related with the thickness, and with a major radius
`b`, related with the diameter, and with an uniform mass. The momentum of inertia about its center of mass (
`I_(g)`), measured in kilogram-square meters, which is located at the geometrical center of the element, is determined by the following formula:

`I_(g) = (1)/(4)\cdot m \cdot (4\cdot b^(2)+3\cdot a^(2))` (1)

`b = 0.5\cdot D` (2)

`a = 0.5\cdot t` (3)

Where:

`D` - Diameter, measured in meters.

`t` - Thickness, measured in meters.

`m` - Mass, measured in kilograms.

If we know that
`m = 0.332\,kg`,
`D = 0.359\,m` and
`t = 9.5* 10^(-3)\,m`, then the moment of inertia of the hoop yo-yo is:

`a = 0.5\cdot (9.5* 10^(-3)\,m)`

`a = 4.75* 10^(-3)\,m`

`b = 0.5\cdot (0.359\,m)`

`b = 0.180\,m`

`I_(g) = (1)/(4)\cdot (0.332\,kg)\cdot [4\cdot (0.180\,m)^(2)+3\cdot (4.75* 10^(-3)\,m)^(2)]`

`I_(g) = 0.0108\,kg\cdot m^(2)`

The moment of inertia of the hoop yo-yo rotating about its center of mass is
`I_(g) = 0.0108\,kg\cdot m^(2)`.

b) The hoop yo-yo rotate at a point located at a distance of half diameter from the center of mass of the element, whose moment of inertia is determined by the Theorem of Parallel Axes:

`I_(O) = I_(g) +m\cdot r^(2)` (4)

Where:

`r` - Distance between parallel axes, measured in meters.

If we know that
`I_(g) = 0.0108\,kg\cdot m^(2)`,
`m = 0.332\,kg` and
`r = 0.180\,m`, then the moment of inertial of the hoop yo-yo rotating about the point where the tension force is applied is:

`I_(O) = 0.0108\,kg\cdot m^(2)+(0.332\,kg)\cdot (0.180\,m)^(2)`

`I_(O) = 0.0216\,kg\cdot m^(2)`

The moment of inertial of the hoop yo-yo rotating about its center of mass is
`I_(O) = 0.0216\,kg\cdot m^(2)`.