Question

Students run an experiment to determine the rotational inertia of a large spherically shaped object around its center. Through experimental data, the students determine that the mass of the object is distributed radially. They determine that the radius of the object as a function of its mass is given by the equation r = km², where k = 3.

Which of the following is a correct expression for the rotational inertia of the object?

(A) m3
(B) 1.8 m3
(C) 3.6 m3
(D) 6 m3
(E) 9 m3

Answer 1

The provided question seems to have a discrepancy as the calculated value of rotational inertia for a spherical object with a given mass-radius relationship is 4.5M³, which does not match any of the supplied answer choices.

Step-by-step explanation:

The question is asking for the correct expression for the rotational inertia of a spherically shaped object with mass distribution given by the radius as a function of mass (r = km² where k = 3). The rotational inertia, or moment of inertia, for a solid sphere is given by the formula ⅒MR², where M is the mass of the sphere, and R is its radius. Considering that R is defined by r = km², we substitute R with km² in the formula:

I = ⅒M(km²)² = ⅒Mk²m⁴ = ⅒Mk²M²

Since k = 3, we further simplify the expression:

I = ⅒M(3M)² = ⅒(3²)M³ = ⅒ × 9M³ = 4.5M³

However, none of the options (A) to (E) match the value 4.5M³, which indicates there may be an error in the supplied options or an error within the initial assumptions or question parameters. It's important to recheck the given data and the calculation steps to ensure accuracy. If the question and the parameters are indeed accurate as stated, additional information or clarification would be necessary.

Answer 2

(C)
`3.6m^(5)`

Step-by-step explanation:

`r=km^2\\` =
`3m^2`

Since the object is a solid sphere, the equation for rotational inertia is:

`I = (2)/(5)mr^2`

`I=(2)/(5)m(3m^2)^2=(2)/(5)*9m^5=3.6m^5`