Spring potential energy and mass

[kolua]

Homework Statement

One end of a vertical spring of spring constant k = 1900 N/m is attached to the floor. You compress the spring so that it is 2.50 m shorter than its relaxed length, place a 1.00-kg ball on top of the free end, and then release the system att = 0. (All values are measured in the Earth reference frame.)

A. By how much does the mass of the spring change during the time interval from t = 0 to the instant the ball leaves the spring?

B. By how much does the mass of the Earth-ball-spring system change during the time interval from t = 0to the instant the ball reaches its maximum height?

Homework Equations

ΔE=Δmc²
U_s=1/2 kx²

The Attempt at a Solution

ΔE=ΔU=1900/2×2.5²=5937.5
Δm=5937.5/9/18⁸=6.597×10^-6
is this correct for question A?
I'm not sure if the mass changes because the instant the ball leaves the potential energy is converted to the ball's kinetic energy. And there should be no change in mass?

For question B, how should I approach?

 

[haruspex]

ΔE=ΔU=1900/2×2.52=5937.5
Δm=5937.5/9/188=6.597×10-6
is this correct for question A?

That looks right for a).
For b), what energy has entered or left the Earth-spring-mass system?

 

[Herman Trivilino]

I'm not sure if the mass changes because the instant the ball leaves the potential energy is converted to the ball's kinetic energy.

Well, it's not an instantaneous process. As the spring relaxes the ball picks up speed, so the process takes as long as it takes for the spring to go from it's compressed state to its relaxed state. And during the process the ball rises, so there's also an increase in the gravitational potential energy of the ball-Earth system.

And there should be no change in mass?

Change in mass of the spring, or change in mass of the whole system?

 

[kolua]

That looks right for a).
For b), what energy has entered or left the Earth-spring-mass system?

potential energy and kinetic energy?

 

[haruspex]

potential energy and kinetic energy?

Those are both internal to the Earth-spring-mass system.

 

[kolua]

Those are both internal to the Earth-spring-mass system.

So should I count the mass of the ball and the Earth intot the calculation for this one?

 

[Herman Trivilino]

What haruspex is trying to get you to do is determine which of these energies is leaving or exiting the system. Note that the choice of system is arbitrary. It's different in Parts A and B of this problem.

 

[kolua]

What haruspex is trying to get you to do is determine which of these energies is leaving or exiting the system. Note that the choice of system is arbitrary. It's different in Parts A and B of this problem.

For question A, since there is only the spring in the system, there is only the change in the spring potential energy, which is U=1/2 kx² = E=mc², is this correct?
For question B, there are potential and kinetic energy for this system, so U=1/2 kx²=E-K? is this right?

 

[Herman Trivilino]

Best to think of it in terms of states of the system. The initial state has energy $E_i$ and the final state has energy $E_f$. In this way you can account for kinetic energy and both types of potential energy. Something you're not doing now is including both types of potential energy.

 

[kolua]

Best to think of it in terms of states of the system. The initial state has energy $E_i$ and the final state has energy $E_f$. In this way you can account for kinetic energy and both types of potential energy. Something you're not doing now is including both types of potential energy.

Uspring=1/2 kx²=E-K-U_gravity?

 

[Herman Trivilino]

Ok, but you're not solving for $U_{spring}$.

 

[kolua]

Ok, but you're not solving for $U_{spring}$.

Δmc²=K+U_gravity?

 

[Herman Trivilino]

I think you should explain briefly what you're doing to get these expressions. In particular, go back and read the statement of the problem, then read Post #9 and follow the scheme outlined there.
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