Moment of Inertia? Potential and Kinetic Energy? Anybody awake?

[riseofphoenix]

I don't know how to answer these - please please please help!

Consider the following objects of mass m rolling down an incline of height h.

(a) A hoop has a moment of inertia I = (1/2)mr². What is the equation for the velocity v_hoop of the hoop at the bottom of the incline? (Use the following as necessary: m, h, r, and g.)

v_hoop =

(b) A solid cylinder has a moment of inertia I = (1/2)mr². What is the equation for the velocity v_cylinder of the cylinder at the bottom of the incline? (Use the following as necessary: m, h, r, and g.)

v_cylinder =

(c) We know that the velocity of the sphere at the bottom of the ramp is √ [ (10gh)/7 ] from which we can conclude that the mass of the sphere does not affect the velocity of the sphere. Which of the following statements help to explain why the equations for the velocity in the case of the rolling cylinder and rolling hoop should be different from each other and from that of the sphere? (Select all that apply.)

a) Even if all three objects had the same mass and the same radius, they would all have different moments of inertia and therefore different rotational kinetic energies, which will affect the velocity of the object as it rolls down the incline.
b) The rotational kinetic energy of a solid depends on its moment of inertia I, which will affect the velocity of the object as it rolls down the incline.
c) The moment of inertial of a sphere, cylinder and hoop are different because of how the mass is distributed in each of these objects. This would affect the velocity of the object as it rolls down the incline.

 

[SammyS]

I don't know how to answer these - please please please help!

Consider the following objects of mass m rolling down an incline of height h.

(a) A hoop has a moment of inertia I = (1/2)mr². What is the equation for the velocity v_hoop of the hoop at the bottom of the incline? (Use the following as necessary: m, h, r, and g.)

v_hoop =

(b) A solid cylinder has a moment of inertia I = (1/2)mr². What is the equation for the velocity v_cylinder of the cylinder at the bottom of the incline? (Use the following as necessary: m, h, r, and g.)

v_cylinder =

(c) We know that the velocity of the sphere at the bottom of the ramp is √ [ (10gh)/7 ] from which we can conclude that the mass of the sphere does not affect the velocity of the sphere. Which of the following statements help to explain why the equations for the velocity in the case of the rolling cylinder and rolling hoop should be different from each other and from that of the sphere? (Select all that apply.)

a) Even if all three objects had the same mass and the same radius, they would all have different moments of inertia and therefore different rotational kinetic energies, which will affect the velocity of the object as it rolls down the incline.
b) The rotational kinetic energy of a solid depends on its moment of inertia I, which will affect the velocity of the object as it rolls down the incline.
c) The moment of inertial of a sphere, cylinder and hoop are different because of how the mass is distributed in each of these objects. This would affect the velocity of the object as it rolls down the incline.

The moment of inertia for the hoop is mr², not 1/2 there.

 

[riseofphoenix]

The moment of inertia for the hoop is mr², no 1/2 there.

Oh oops, you're right..
Do you happen to know the answer to part a, b, and c though?

 

[riseofphoenix]

The moment of inertia for the hoop is mr², no 1/2 there.

also, isn't part c all three of them (choices a, b, and c)?

 

[Nickie]

I can provide some insight into these concepts. Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is affected by the mass and distribution of mass of an object. In the case of the hoop and cylinder, although they have the same moment of inertia, their mass and distribution of mass is different, which will affect their rotational kinetic energy and subsequently their velocity as they roll down the incline. This is why the equations for their velocities are different.

Furthermore, potential and kinetic energy are related concepts in the study of energy. Potential energy is the energy an object has due to its position or configuration, while kinetic energy is the energy an object has due to its motion. In the case of the objects rolling down the incline, they all start with the same potential energy (due to their height) but their distribution of mass and moment of inertia will determine how much of that energy is converted into kinetic energy, resulting in different velocities at the bottom of the incline.

Lastly, to answer the question "anybody awake?", I can assure you that there are scientists and individuals who are awake and actively studying and researching these concepts. I hope this helps to clarify the concepts of moment of inertia, potential and kinetic energy, and their relation to the velocity of objects rolling down an incline.