The first law of thermodynamics for an elastic rod of length L and subjected to a tension f is dU = TdS +fdL. By consideration of free energy F = U ?TS show that
The first law of thermodynamics for an elastic rod of length L and subjected to a...
A=U-TS is Helmholtz free energy, which is commonly used in statistical thermodynamics. Starting from dU=TdS-pdV, show that dA=-SdT-pdV
Problem 1: Axial vibrations of a rod The rod of length L is fixed at ends x = 0 and x = L. The density of the rod is ρ(x), stiffness k(x) being subjected to a force f(x, t). Let's derive the equations for axial vibrations of a rod using almped model. We express the rod niy mol 41 in as a chain of masses m,mm, connected to each other through springs as shown in the figure. Let's say each...
Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form du - hu- az ar 0<x<L, t > 0, ha constant. Find the temperature uix, t) if the initial temperature is fx) throughout and the ends 0 and XL are insulated. See the figure u(x, t) *)-(wax) ). 2 [(? I'moscoap 90.cr)()+(-*...
(b) We would like to demonstrate that the appropriate free energy is minimized in thermodynamic equilibrium by considering a system at constant temperature and pressure, for which the Gibbs free energy G-U-TS+ PV is appropriate o Write down the first law of thermodynamics linking heat, q, to AU and work Au-grw i) du tfestdv ) du dsreu Tds-Pdv-) sPdV dus Write down an inequality relating the temperature-weighted entropy change, TAS to q ds ii) Use the result of (i) and...
A long steel bar (length L-100 m, elastic modulus E = 2x 1011 N/m2, density ρ 7850 kgm3) of unit cross section is held at one end while hanging freely inside a deep pit and you are asked to estimate the longitudinal stress and displacement along the length of the bar Assuming linear elastic behavior, the internal elastic energy U of the hanging bar is given by the expression where z is the distance from the top of the pit...
A 1.34 kg ball is connected by means of two massless strings, each of length L =1.70 m, to a vertical rotating rod. The strings are tied to the rod with separation d =1.70 m and are taut. The tension in the upper string is 35 N. (a) Draw the free body diagram for the ball, and label the r ˆ-direction in the diagram. (Please use either a polar or cartessian coordinate system & please show r direction as well)...
A conducting rod with length L, is to slide without friction on horizontal infinitely long metal rails connected through resistance R as show in Fig. uniform magnetic field B is directed out of the plane of the figure. What is the applied force F required to move the bar to the right with a constant speed At what rate is energy dissipated in the resistor R?
QUESTION 15 Which statement A-D about the first law of thermodynamics (AEsystem-q+w) is not correct? O q is the energy added to or removed from the system by a temperature difference. O w is the energy added to or removed from the system by a deformation: that is, a force moves something. AEo stem is the change in the Gibbs free energy of the system. O If the only work that is done is P-V work, then q -the change...
A compound pendulum is made up of a rod of length L, with mass M and a solid sphere of radius r, with mass m (see figure below). The pendulum is pivoted about one end and released from rest from and angle of 0, (angle with the vertical). (a) Find the distance, dom, of center of mass of this pendulum from its pivot. (b) Draw a free body diagram and write down Newton's 2nd Law (for rotation) for the pendulum...
Question 7 The conducting rod shown in the figure has length L and is being pulled along horlizontal, frictionless, conducting rails at a constant metal strip. A uniform magnetic field, directed of the magnetic fieid is 8-1.0 T. (a) What is the magnitude Assume that L15 cm, the speed of the rod is v -5.9 m/s, and the magnitude of emf induced in voits in the rod? (b) What is the current in amperes in the conducting loop? Assume that...