Question about rotational energy.

[physicsbro]

Homework Statement

Five objects of equal mass are shown below together with the axis about which they are rotating. Select the objects in order of increasing rotational energy

Solid Sphere, about any diameter, with R = 3 m and ω=5 rad/s


Thin rod, about axis through center, perpendicular to the length with l = 8 m and ω=6 rad/s


Solid Cylinder, about cylinder axis with R = 3 m and ω=5 rad/s


Thin Spherical shell, about any diameter, with R = 2 m and ω=7 rad/s


Thin cylindrical shell, about cylinder axis with R = 1 m and ω=7 rad/s

Homework Equations

(1/2)Iw^2

The Attempt at a Solution

Most of these have no mass or moment of intertia included and I am not sure how to deal with that.

 

[kuruman]

You are supposed to calculate the moments of inertia first. Your textbook must have a table where these are listed. Note that all the masses are the same, therefore you can rank the energies because they will always be

E = (some number)*m

 

[kerimek]

Based on the given information, the order of increasing rotational energy would be:

1. Thin cylindrical shell, about cylinder axis with R = 1 m and ω=7 rad/s (lowest rotational energy)
2. Solid Sphere, about any diameter, with R = 3 m and ω=5 rad/s
3. Solid Cylinder, about cylinder axis with R = 3 m and ω=5 rad/s
4. Thin rod, about axis through center, perpendicular to the length with l = 8 m and ω=6 rad/s
5. Thin Spherical shell, about any diameter, with R = 2 m and ω=7 rad/s (highest rotational energy)

The formula for rotational energy, (1/2)Iω^2, takes into account the moment of inertia (I) and the angular velocity (ω). For objects with the same mass and angular velocity, the one with the larger moment of inertia will have a higher rotational energy. In this case, the thin cylindrical shell has the smallest moment of inertia due to its smaller radius, followed by the solid sphere and solid cylinder with the same radius but different shapes. The thin rod has a larger moment of inertia due to its longer length, and the thin spherical shell has the largest moment of inertia due to its larger radius. Therefore, the objects can be ordered from lowest to highest rotational energy based on their moment of inertia.