Prove how the heat capacity at constant length of an elastic rod changes with L and...
Homework Answers
The energy absorbed by the rod during the heating is
where, m = mass
C = heat capacity
= change in temperature
m but. mass = density * volume = density * area * length
From this equation, it is clear that for a material rod with the same area of cross-section, and if the amount of energy provided is kept constant; the value of heat capacity can vary with the length of the rod or the change in the tmperature.
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Prove how the heat capacity at constant length of an elastic rod
changes with L and...
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal...
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal properties (assume k=1): - xe" (0 < x <l, t>0) 04 (0,1) = 0 (1,1)= 1 a) Determine the equilibrium temperature distribution, and plot it on the interval [0, 1] b) Where does the energy enter, and where does it leave the rod? Explain your answers.
The conductive heat transfer in a rod of length L is described by the equation au...
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
Suppose heat is lost from the lateral surface of a thin rod of length L into...
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)()+(-*...
how do I prove Lf = L/2*mv? A thin rod of mass M and length L...
how do I prove Lf = L/2*mv?
A thin rod of mass M and length L is struck at one end by a ball of clay of mass in, moving with speed v as shown in the figure. The ball sticks to the rod. Determine the angular momentum of the clay-rod system about A (the midpoint of the rod) after the collision.
The first law of thermodynamics for an elastic rod of length L and subjected to a...
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
OST OL, of OTT
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endp...
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<
Problem #6: A rod of length I coincides with the interval [0, L] on the x-axis....
Problem #6: A rod of length I coincides with the interval [0, L] on the x-axis. Let u(x, t) be the temperature. Consider the following conditions. (A) There is heat transfer from the lateral surface of the rod into the surrounding medium, which is held at temperature 0° (B) There is heat transfer from the left end into the surrounding medium, which is held at a constant temperature of 0° (C) The left end is insulated. (D) The right end...
1. When considering the heat conduction in a rod (of length L) with zero temperature at...
1. When considering the heat conduction in a rod (of length L) with zero temperature at both ends, we encounter eigenvalue problem ψ't λψ = 0, ψ(0) = ψ(L) = 0. Show that in this problem, all eigenvalues λ are real and positive [Remember: eigenfunction (x) can be complex when eigenvalue λ is complex.]
In an experiment to determine the spring constant of an elastic cord of length 0.60m, a...
In an experiment to determine the spring constant of an elastic cord of length 0.60m, a student hangs the cord form a rod as represented above and then attaches a variety of weights to the cord. For each weight, the student allows the weights to hang in equilibrium and then measures the entire length of the cord. The data are recorded in the table below.Use the data to plot a graph of weight versus length on the axes below. Sketch...
33 Heat capacity at a constant volume of a system with average energy <e 37e dT...
33 Heat capacity at a constant volume of a system with average energy <e 37e dT /N,V Cv Prove that Cv(E-CEY)) kT2
33 Heat capacity at a constant volume of a system with average energy
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal properties (assume k=1): - xe" (0 < x <l, t>0) 04 (0,1) = 0 (1,1)= 1 a) Determine the equilibrium temperature distribution, and plot it on the interval [0, 1] b) Where does the energy enter, and where does it leave the rod? Explain your answers.
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
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)()+(-*...
how do I prove Lf = L/2*mv?
A thin rod of mass M and length L is struck at one end by a ball of clay of mass in, moving with speed v as shown in the figure. The ball sticks to the rod. Determine the angular momentum of the clay-rod system about A (the midpoint of the rod) after the collision.
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
OST OL, of OTT
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<
Problem #6: A rod of length I coincides with the interval [0, L] on the x-axis. Let u(x, t) be the temperature. Consider the following conditions. (A) There is heat transfer from the lateral surface of the rod into the surrounding medium, which is held at temperature 0° (B) There is heat transfer from the left end into the surrounding medium, which is held at a constant temperature of 0° (C) The left end is insulated. (D) The right end...
1. When considering the heat conduction in a rod (of length L) with zero temperature at both ends, we encounter eigenvalue problem ψ't λψ = 0, ψ(0) = ψ(L) = 0. Show that in this problem, all eigenvalues λ are real and positive [Remember: eigenfunction (x) can be complex when eigenvalue λ is complex.]
In an experiment to determine the spring constant of an elastic cord of length 0.60m, a student hangs the cord form a rod as represented above and then attaches a variety of weights to the cord. For each weight, the student allows the weights to hang in equilibrium and then measures the entire length of the cord. The data are recorded in the table below.Use the data to plot a graph of weight versus length on the axes below. Sketch...
33 Heat capacity at a constant volume of a system with average energy <e 37e dT /N,V Cv Prove that Cv(E-CEY)) kT2
33 Heat capacity at a constant volume of a system with average energy
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