An advertising firm has contacted your engineering firm to create a new advertisement for a local ice-cream stand. The owner of this stand wants to add rotating solid cones (painted to look like ice-cream cones, of course) to catch the eye of travelers. Each cone will rotate about an axis parallel to its base and passing through its apex. The actual size of the cones is to be decided upon, and the owner wonders if it would be more energy-efficient to rotate smaller cones than larger ones. He asks your firm to write a report showing the determination of the moment of inertia of a homogeneous right circular cone of height H, base radius R, and mass M. What is the result of your report? (Use any variable or symbol stated above as necessary.)

Question

An advertising firm has contacted your engineering firm to create a new advertisement for a local ice-cream stand. The owner of this stand wants to add rotating solid cones (painted to look like ice-cream cones, of course) to catch the eye of travelers. Each cone will rotate about an axis parallel to its base and passing through its apex. The actual size of the cones is to be decided upon, and the owner wonders if it would be more energy-efficient to rotate smaller cones than larger ones. He asks your firm to write a report showing the determination of the moment of inertia of a homogeneous right circular cone of height H, base radius R, and mass M. What is the result of your report? (Use any variable or symbol stated above as necessary.)

Answer 1

The moment of inertia of a solid object measures its resistance to changes in rotational motion. To determine the moment of inertia of a homogeneous right circular cone, we need to calculate its moment of inertia about the axis of rotation.

The moment of inertia (I) of a solid cone can be calculated using the formula:

I = (3/10) * M * R^2

Where:
- I is the moment of inertia
- M is the mass of the cone
- R is the base radius of the cone

In this case, the owner wants to determine if it would be more energy-efficient to rotate smaller cones than larger ones. To compare the energy efficiency, we need to consider the rotational kinetic energy (KE) of the cones. The rotational kinetic energy is given by the formula:

KE = (1/2) * I * ω^2

Where:
- KE is the rotational kinetic energy
- I is the moment of inertia
- ω is the angular velocity of the rotating cone

Since the angular velocity (ω) would be the same for both smaller and larger cones, we can compare the rotational kinetic energy by dividing the energy of the larger cone by the energy of the smaller cone:

KE_larger / KE_smaller = (1/2 * I_larger * ω^2) / (1/2 * I_smaller * ω^2)

The ω^2 terms cancel out, simplifying the equation to:

KE_larger / KE_smaller = I_larger / I_smaller

This means that the ratio of the rotational kinetic energies is equal to the ratio of the moments of inertia.

Therefore, in your report, you would calculate the moment of inertia of a homogeneous cone using the given formula (I = (3/10) * M * R^2) and then explain that the energy efficiency depends on the ratio of the moments of inertia. If the moment of inertia of the larger cone is greater than that of the smaller cone, then it would be less energy-efficient to rotate the larger cones.