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

• zenterix
In summary, the conversation was about solving a problem involving a wheel rolling down an inclined plane without slipping. The focus was on using energy to find the velocity of the wheel after rolling down a certain height. The solution involved considering the reference frame of the ground and taking into account non-conservative forces such as friction. It was also mentioned that it is possible to perform the calculation from the frame of the center of mass, but the calculation would need to take into account work done by inertia forces and friction in the moving 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.

zenterix said:
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

Lnewqban

## 1. Is the work-energy theorem valid in non-inertial frames?

Yes, the work-energy theorem is valid in both inertial and non-inertial frames of reference. This is because the theorem is derived from the fundamental laws of physics, which are applicable in all frames of reference.

## 2. How is the work-energy theorem applied in non-inertial frames?

In non-inertial frames, the work-energy theorem can still be applied by considering the pseudo forces that arise due to the acceleration of the frame. These pseudo forces are included in the total work done on an object and the resulting change in its kinetic energy.

## 3. Can the work-energy theorem be used to analyze motion in accelerating frames?

Yes, the work-energy theorem can be used to analyze motion in accelerating frames. It is a useful tool for understanding the energy changes that occur in a system, regardless of the frame of reference.

## 4. Are there any limitations to using the work-energy theorem in non-inertial frames?

One limitation of using the work-energy theorem in non-inertial frames is that it assumes the frame is accelerating at a constant rate. If the acceleration is changing, the theorem may not accurately predict the energy changes in the system.

## 5. Can the work-energy theorem be used to analyze rotational motion in non-inertial frames?

Yes, the work-energy theorem can be applied to rotational motion in non-inertial frames by considering the torque and angular velocity of the rotating object. This allows for the calculation of the work done by rotational forces and the resulting change in rotational kinetic energy.

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