A cylinder (I = 1/2 MR2), a sphere (I = 2/5MR2), and a hoop (I = MR2) roll without slipping down the same incline, beginning from rest at the same height. All three objects share the same radius and total mass. Which object has the greatest kinetic energy at the bottom of the incline?

Question

A cylinder (I = 1/2 MR2), a sphere (I = 2/5MR2), and a hoop (I = MR2) roll without slipping down the same incline, beginning from rest at the same height. All three objects share the same radius and total mass. Which object has the greatest kinetic energy at the bottom of the incline?

Answer 1

To determine which object has the greatest kinetic energy at the bottom of the incline, we can use the principle of conservation of energy. The potential energy at the top of the incline is converted into both rotational and translational kinetic energy as the objects roll down without slipping.

Let's calculate the potential energy at the top of the incline:

Potential Energy (PE) = mgh

Since all three objects have the same mass and starting height, the potential energy is the same for all of them.

Now let's calculate the total kinetic energy at the bottom of the incline for each object. The total kinetic energy is the sum of rotational and translational kinetic energy.

For a rolling object, the relationship between rotational and translational kinetic energy is given by:

Rotational Kinetic Energy (K_rot) = (1/2) I ω^2
Translational Kinetic Energy (K_trans) = (1/2) mv^2

where I is the moment of inertia, ω is the angular velocity, m is the mass, and v is the linear velocity.

Let's calculate the rotational and translational kinetic energy for each object:

For the cylinder:
I = (1/2) MR^2
K_rot_cylinder = (1/2) (1/2) MR^2 (ω^2)
K_trans_cylinder = (1/2) mv^2

For the sphere:
I = (2/5) MR^2
K_rot_sphere = (1/2) (2/5) MR^2 (ω^2)
K_trans_sphere = (1/2) mv^2

For the hoop:
I = MR^2
K_rot_hoop = (1/2) MR^2 (ω^2)
K_trans_hoop = (1/2) mv^2

Now, we need to compare the total kinetic energy (K_total = K_rot + K_trans) for each object at the bottom of the incline.

K_total_cylinder = K_rot_cylinder + K_trans_cylinder
K_total_sphere = K_rot_sphere + K_trans_sphere
K_total_hoop = K_rot_hoop + K_trans_hoop

By substituting the respective values for each object, we can compare the total kinetic energies and determine which object has the greatest value.

After evaluating the expressions, you will find that the cylinder has the greatest kinetic energy at the bottom of the incline.

Answer 2

A cylinder (I = 1/2 MR2), a sphere (I = 2/5MR2), and a hoop (I = MR2) roll without slipping down the same incline, beginning from rest at the same height. All three objects share the same radius and total mass. Which object has the greatest kinetic energy at the bottom of the incline?

To determine which object has the greatest kinetic energy at the bottom of the incline, we can use the principle of conservation of energy. The potential energy at the top of the incline is converted into both rotational and translational kinetic energy as the objects roll down without slipping.

All the same.