Four objects - a hoop, a solid cylinder, a solid sphere, and a thin, spherical shell - each has a mass of 4.59 kg and a radius of 0.252 m.

(a) Find the moment of inertia for each object as it rotates about the axes shown in the table above.
hoop____ kg·m2
solid cylinder____ kg·m2
solid sphere____kg·m2
thin, spherical shell_____kg·m2

(b) Suppose each object is rolled down a ramp. Rank the translational speed of each object from highest to lowest.

A)solid cylinder > thin spherical > solid sphere > hoop

B)solid sphere > solid cylinder > thin spherical > hoop

C)hoop > solid cylinder > solid sphere > thin spherical

D)thin spherical > solid sphere > solid cylinder > hoop

(c) Rank the objects' rotational kinetic energies from highest to lowest as the objects roll down the ramp.

A)solid cylinder > thin spherical > solid sphere > hoop

B)hoop > thin spherical > solid cylinder > solid sphere

C)hoop > solid cylinder > solid sphere > thin spherical

D)thin spherical > solid sphere > solid cylinder > hoop

(a)

hoop: I = MR^2 = (4.59 kg)(0.252 m)^2 = 0.291 kg·m^2
solid cylinder: I = 1/2 MR^2 = 1/2 (4.59 kg)(0.252 m)^2 = 0.146 kg·m^2
solid sphere: I = 2/5 MR^2 = 2/5 (4.59 kg)(0.252 m)^2 = 0.233 kg·m^2
thin spherical shell: I = 2/3 MR^2 = 2/3 (4.59 kg)(0.252 m)^2 = 0.389 kg·m^2

(b) The moment of inertia does not directly affect the translational speed of the objects when rolled down a ramp, so the answer to this question is not determined by the moment of inertia values given in (a). Therefore, the correct answer to this question is not provided.

(c) The rotational kinetic energy of an object is given by the equation KE_rotational = 1/2 I ω^2, where I is the moment of inertia and ω is the angular velocity.

Using the moment of inertia values from part (a) and assuming all objects have the same angular velocity, we can rank the objects' rotational kinetic energies:

solid cylinder: KE_rotational = 1/2 (0.146 kg·m^2) ω^2
thin spherical shell: KE_rotational = 1/2 (0.389 kg·m^2) ω^2
solid sphere: KE_rotational = 1/2 (0.233 kg·m^2) ω^2
hoop: KE_rotational = 1/2 (0.291 kg·m^2) ω^2

Since all objects have the same mass and angular velocity, their rotational kinetic energies are proportional to their respective moments of inertia. Therefore, the objects can be ranked from highest to lowest rotational kinetic energy as follows:

A) solid cylinder > thin spherical shell > solid sphere > hoop

To answer these questions, we need to use the formulas for moment of inertia and translational speed.

(a) The moment of inertia for different objects rotating about different axes can be calculated using the following formulas:

- Hoop: The moment of inertia of a hoop rotating about its central axis is given by I = MR^2, where M is the mass and R is the radius. In this case, I = 4.59 kg * (0.252 m)^2.

- Solid Cylinder: The moment of inertia for a solid cylinder rotating about its central axis is given by I = (1/2)MR^2. In this case, I = (1/2) * 4.59 kg * (0.252 m)^2.

- Solid Sphere: The moment of inertia for a solid sphere rotating about its central axis is given by I = (2/5)MR^2. In this case, I = (2/5) * 4.59 kg * (0.252 m)^2.

- Thin, Spherical Shell: The moment of inertia for a thin, spherical shell rotating about its central axis is given by I = (2/3)MR^2. In this case, I = (2/3) * 4.59 kg * (0.252 m)^2.

Now you can calculate the moment of inertia for each object.

(b) The translational speed of each object rolling down a ramp can be determined using the principle of conservation of energy. The potential energy is converted into both rotational kinetic energy and translational kinetic energy. However, we can approximate that the rotational kinetic energy is much smaller compared to the translational kinetic energy and can even be neglected.

Since the objects have the same mass and roll down the same ramp, the object with the highest rotational inertia will have the lowest translational speed. Therefore, the ranking from highest to lowest translational speed is:

D) thin spherical > solid sphere > solid cylinder > hoop

(c) The rotational kinetic energy of each object can be calculated using the formula:

Rotational Kinetic Energy = (1/2)Iω^2

where I is the moment of inertia and ω is the angular velocity. However, in this case, we can omit the ω^2 terms as it is not given, and we only need to rank the objects based on their moment of inertia.

Since rotational kinetic energy is proportional to the moment of inertia, we can simply rank the objects based on their moment of inertia. Therefore, the ranking from highest to lowest rotational kinetic energy is:

A) solid cylinder > thin spherical > solid sphere > hoop

I figure it out

(a) You should take the time to look these up.

I(hoop) = M R^2
I(solid sphere) = (2/5) M R^2
I(solid cylnder) = (1/2) MR^2
I(spherical shell) = (2/3) MR^2

http://hyperphysics.phy-astr.gsu.edu/HBASE/isph.html

(b) The higher the value of (I/MR^2), the slower it rolls, because more of the potential energy is used up making it spin. You do the ranking

(c) The order will be opposite from (b)