# How Does the Rotational Inertia Affect Rolling Motion on an Inclined Plane?

• ILOVEPHYSIC
In summary, a uniform wheel with a mass of 14.0 kg and a radius of 0.200 m is mounted on a massless axle and released from the top of a 30.0° inclined surface. The wheel rolls down the surface without slipping, and its rotational and translational kinetic energies can be determined using the equations Mgh= 1/2Iw^2 and v=ωr. The rotational motion of the wheel is affected by the contact between the axle and the surface, not the fact that part of the wheel is in a groove.
ILOVEPHYSIC
A uniform wheel of mass 14.0 kg is mounted rigidly on a massless axle through its center, as shown in the figure below. The radius of the axle is 0.200 m, and the rotational inertia of the wheel-axle combination about its central axis is 0.600 kg·m2. The wheel is initially at rest at the top of a surface that is inclined at angle θ = 30.0° with the horizontal; the axle rests on the surface while the wheel extends into a groove in the surface without touching the surface. Once released, the axle rolls down along the surface smoothly and without slipping. The wheel-axle combination moves down the surface by 3.00 m.

(a)Determine its rotational kinetic energy at this point? J

(b) Determine its translational kinetic energy at this point? J

Mgh= 1/2mw^2 + 1/2mv^2 --(1)
v=wR --(2)
The problem i feel confused is that what is R for since part of the wheel is inside the groove. I am not sure the question is just simply sub (2) into (1) and find the answer.

ILOVEPHYSIC said:
Mgh= 1/2mw^2 + 1/2mv^2 --(1)
Fix the first term on the right hand side, for rotational KE. (Should be I, not m.)

ILOVEPHYSIC said:
The problem i feel confused is that what is R for since part of the wheel is inside the groove.
What matters is what surfaces are in contact with each other. It's the axle that is touching the surface of the incline. The fact that the part of the wheel is in the groove is irrelevant.

Doc Al said:
Fix the first term on the right hand side, for rotational KE. (Should be I, not m.)What matters is what surfaces are in contact with each other. It's the axle that is touching the surface of the incline. The fact that the part of the wheel is in the groove is irrelevant.
Sorry for typing wrong in rotational KE. But the axle and wheel should move with same angular speed. Why don't wheel slow down the rotational motion?

ILOVEPHYSIC said:
But the axle and wheel should move with same angular speed.
They do!

ILOVEPHYSIC said:
Why don't wheel slow down the rotational motion?
The wheel does slow down the rotational motion, but that is reflected in the rotational inertia. The relationship v = ωr is the condition for rolling without slipping; the "r" is the distance from the surface to the center, which is the radius of the axle, not the wheel.

## 1. What is a rotational wheel and axle?

A rotational wheel and axle is a simple machine consisting of a circular object (the wheel) and a cylindrical object (the axle) that are connected and rotate together. The wheel and axle work together to transmit force and motion, making tasks like lifting, pulling, and turning easier.

## 2. How does a rotational wheel and axle work?

The wheel and axle work together by utilizing the principle of rotational motion. When a force is applied to the wheel, the axle rotates along with it, allowing for the transfer of energy and motion. This makes it easier to lift or move heavy objects, as the force applied to the wheel is amplified at the axle.

## 3. What are some examples of rotational wheel and axle in everyday life?

Rotational wheel and axle can be found in many everyday objects, such as bicycles, car wheels, screwdrivers, and steering wheels. They can also be found in larger machines, such as cranes and wind turbines, where they are used to lift and move heavy loads.

## 4. How is the mechanical advantage of a rotational wheel and axle calculated?

The mechanical advantage of a rotational wheel and axle is calculated by dividing the radius of the wheel by the radius of the axle. This ratio represents how much the force applied to the wheel is amplified at the axle. The larger the ratio, the greater the mechanical advantage.

## 5. What are the benefits of using a rotational wheel and axle?

The use of a rotational wheel and axle can provide several benefits, such as making tasks easier and more efficient, reducing the amount of force needed to lift or move objects, and increasing the speed at which tasks can be completed. They are also simple and durable machines, making them cost-effective and widely used in various industries.

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