# Possible to use work-energy theorem from a non-inertial frame?

zenterix
Homework Statement:
Consider a wheel of mass m and radius R rolling down a plane inclined at an angle ϕrelative to the horizontal ground. The wheel starts at rest, and does not slip. What is the velocity of the wheel after it has rolled down a height of h?
Relevant Equations:
Here is a solution:
$$E_{m_i}=mgh$$

$$E_{m_f}=\frac{mv_{cm,f}^2}{2}+\frac{I_{cm}\omega^2}{2}$$

$$v_{cm,f}=R\omega$$

No non-conservative forces do any work on the system (in particular static friction) so mechanical energy is conserved.

$$E_{m_i}=E_{m_f}$$

$$\implies mgh= \frac{mv_{cm,f}^2}{2}+\frac{I_{cm}\omega^2}{2}$$

$$\implies v_{cm_f}=\sqrt{\frac{2mgh}{m+\frac{I_{cm}}{R^2}}}$$
In learning about translational and rotational motion, I solved a problem involving a wheel rolling down an inclined plane without slipping.

There are multiple ways to solve this problem, but I want to focus on solutions using energy.

Now to my questions. The reference frame in the posted solution is the ground, correct?

Is it possible to perform this calculation from the frame of the center of mass?

I'm going to guess no because the center of mass is accelerating due to gravity. It is not an inertial reference frame. Is this correct? I just wanted to be sure.

Homework Statement:: Consider a wheel of mass m and radius R rolling down a plane inclined at an angle ϕrelative to the horizontal ground. The wheel starts at rest, and does not slip. What is the velocity of the wheel after it has rolled down a height of h?
Relevant Equations:: Here is a solution:
$$E_{m_i}=mgh$$

$$E_{m_f}=\frac{mv_{cm,f}^2}{2}+\frac{I_{cm}\omega^2}{2}$$

$$v_{cm,f}=R\omega$$

No non-conservative forces do any work on the system (in particular static friction) so mechanical energy is conserved.

$$E_{m_i}=E_{m_f}$$

$$\implies mgh= \frac{mv_{cm,f}^2}{2}+\frac{I_{cm}\omega^2}{2}$$

$$\implies v_{cm_f}=\sqrt{\frac{2mgh}{m+\frac{I_{cm}}{R^2}}}$$

Is it possible to perform this calculation from the frame of the center of mass?
Yes it is possible but you must take into account a work done by the inertia forces and moreover a work done by friction. The friction does not work relative the ground frame but it do work relative moving frame. Even if the frame moves with constant velocity

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