Basic Thermodynamics ~ J. Pohl ©

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Box Slides Between Springs

The sketch (left part) shows a box shoved against the strong spring (ss) compressing it 4 centimeters and restrained there by a "trigger" mechanism. When the trigger is pulled (our catalyst) the box is projected to slide across the floor, ultimately to encounters the weaker spring (ws)...

Calculate the greatest length of compression the weaker spring will experience.

As our system we take both springs and the box. The energy equation applies. Since all forces are internal to the system, the event has no work. There is thermal equilibrium initially. We assume the event happens quickly such that there is no heat. We conclude: the energy change of the system is zero.

(1)(1)

The system energy has three components. The potential energy and kinetic energy of a spring relate to the elevation and speed of its center of mass, respectively. The elevations of the center of mass of the springs do not change, hence each ΔPE is zero. We assume both springs are motionless before and after the event. Therefore each spring ΔKE is zero. Finally, the box has zero initial and final speed and the same elevation throughout the event. Its ΔKE and ΔPE are zero. ΔU of the box can happen if it has a change of density or temperature; it has neither. The internal energy change is zero.

(2)(2)

The internal energy change of an adiabatic spring equals the work it experiences. At the end of the event, the length of the stronger spring equals its free length. At the initiation of the event, the length of the weaker spring equals its free length. Apply these facts to the equation.

(3)(3)

Some algebra...

(4)(4)

Apply the numeric information...

(5)(5)

This is it!

(6)(6)