Writing an expression for work done by frictional force

[imbadatphysics212]

Homework Statement

Suppose that the cart (mass m) oscillates so that the maximum speed attained is v. The amplitude of the cart’s motion is A and the force constant of the spring is k. Suddenly, an axle supporting the wheels breaks so that one of the wheels can no longer rotate and is locked in its position. This would cause the cart to begin to experience friction. In terms of m, v, A and k, write an expression for the maximum amount of work that the force of friction can perform on the cart after the axle breaks.

Relevant Equations

work, kinetic energy, potential energy

I took a stab at the question, but I don't think I did it right.

I know that Work = Change in Energy
thus, Work = final energy - initial energy

Because there is no energy at the final position, then final energy = 0 (I'm not sure if this is correct).
So I got the equation W = 0 - (1/2kA^2 + 1/2mv^2)

Is this correct at all?

 

[Mister T]

(1/2kA^2 + 1/2mv^2)

You are adding two terms together here to get the total initial energy. When the system has this much energy, what is the cart doing? Is it moving with its maximum or minimum speed? What about its position? Is it in the central (equilibrium) position or is one of the furthest positions (±A)?

 

[imbadatphysics212]

You are adding two terms together here to get the total initial energy. When the system has this much energy, what is the cart doing? Is it moving with its maximum or minimum speed? What about its position? Is it in the central (equilibrium) position or is one of the furthest positions (±A)?

It would be moving at maximum speed, and the position should be the farthest away...?

 

[haruspex]

It would be moving at maximum speed, and the position should be the farthest away...?

But at no point in the oscillation is that the case. When speed is maximum displacement is zero, and when displacement is maximum magnitude speed is zero.

When you have understood that error in your work, the next thing to consider is whether it might or must stop in such a way that some elastic potential energy persists.

I hope this is not a trick question. Since the work done by friction is negative, the maximum would be the value of least magnitude. So do they want the maximum magnitude or the least magnitude?

 

[Mister T]

It would be moving at maximum speed, and the position should be the farthest away...?

Have you watched an oscillator in motion? When it's furthest from the equilibrium position the velocity is zero. And when it passes through the equilibrium position it has its maximum speed.

 

[imbadatphysics212]

But at no point in the oscillation is that the case. When speed is maximum displacement is zero, and when displacement is maximum magnitude speed is zero.

When you have understood that error in your work, the next thing to consider is whether it might or must stop in such a way that some elastic potential energy persists.

I hope this is not a trick question. Since the work done by friction is negative, the maximum would be the value of least magnitude. So do they want the maximum magnitude or the least magnitude?

They want the maximum magnitude.

 

[haruspex]

They want the maximum magnitude.

Ok.
Have you understood the main thing that was wrong with your attempt, or do we need to explain more?

 

[imbadatphysics212]

Ok.
Have you understood the main thing that was wrong with your attempt, or do we need to explain more?

Can you please explain it more?

Thank you for being patient with me..

 

[Mister T]

Can you please explain it more?

The total energy of a harmonic oscillator is $\frac{1}{2}kx^2 + \frac{1}{2}mv^2$. If you set x=A then you have to set v=0, because if x=A then v=0. Likewise, if you set v=v_max then you have to set x=0, because if v=v_max then x=0.

I suggest you review the section in your textbook (or any college-level introductory algebra or calculus-based physics textbook) on the energy of a simple harmonic oscillator.