It is possible to combine both Friedmann-Lemaitre equations to create another useful equation for cosmology (the...
It is possible to combine both Friedmann-Lemaitre equations to create another useful equation for cosmology (the so-called fluid equation). Derive that equation by first taking the time derivative of the first Friedmann-Lemaitre equation and then by using the second one to redefine the terms. Assume ? = 0.
Homework Answers
Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density. Some cosmologists call the second of these two equations the Friedmann acceleration equation and reserve the term Friedmann equation for only the first equation.
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It is possible to combine both Friedmann-Lemaitre equations to
create another useful equation for cosmology (the...
2. The acceleration equation We have derived in lectures the Friedmann and fluid equations, that describe...
2. The acceleration equation We have derived in lectures the Friedmann and fluid equations, that describe the evolution of simple cosmologies. The Friedmann equation is usually given (adopting c 1) as where a(t) is the scale factor of the universe, p is the mass (energy) density, and k is constant. (a) (5 points) The simplest (although spectacularly uninteresting) cosmology has zero mass-energy, i.e. ρ 0. Solve the Friedmann equation under this condition to obtain an expression for at(t). Describe the...
5. (Telegraph equations) Using basic electrical theory, it can be shown that the current I(r, t)...
5. (Telegraph equations) Using basic electrical theory, it can be shown that the current I(r, t) and voltage V(z, t) in an electrical transmission (eg., a power line or telephone line) at position in the lne andimt obey the equations Math 121 Applied DiffEqns II Homework 1 Due: Febuary 7, 2019 where C is the capacitance per u length, Gis the leakage per unit length, R is the resistance per unit length, and L is the inductance per unit length...
Linearity is an extremely useful property of equations in physics. If an equation is linear, it...
Linearity is an extremely useful property of equations in physics. If an equation is linear, it allows you to easily add together simple contributions from individual elements to get a solution for a more complex situation. One example of a simple linear equation in classical mechanics is F ma. If you find the individual accelerations of a particular body due to various individual forces you can then add those accelerations to find the acceleration due to all of the forces...
2. The equations of motion for a system of reduced mass moving subject to a force...
2. The equations of motion for a system of reduced mass moving subject to a force derivable from a spherically symmetric potential U(r) are AF –102) = (2+0 + rē) = 0 . (3) Using the second of these equations, show that the angular momentum L r 8 is a constant of the motion (b) Then use the first of these equations to derive the equation for radial motion in the form dU L i=- What is the significance of...
of Equations There are generally two approaches to solving systems of equations in physics, the substitution...
of Equations There are generally two approaches to solving systems of equations in physics, the substitution method and the addition method. Let us consider the system of two equations below: 4x + 2y 14 2x -y-1 We can solve these equations using both methods. First, the substitution method. The process is as follows: In one of the equations, solve for one of the variables . te this expression for that variable into the other equation. This will leave an expression...
From Acheson: Elementary Fluid Dynamics Equation 7.3 and 6.12 (Slow Flow Equations) 7.2. A rigid sphere...
From Acheson: Elementary Fluid Dynamics
Equation 7.3 and 6.12 (Slow Flow Equations)
7.2. A rigid sphere of radius a is immersed in an infinite expanse of viscous fluid. The sphere rotates with constant angular velocity Ω. The Reynolds number R-Ωα2/v is small, so that the slow flow equations apply (see eqns (7.3) and (6.12)). Using spherical polar coordinates (r, θ, φ) with θ-0 as the rotation axis, show that a purely rotary flow u us(r, e, s possible provided that...
PROBLEM 4.1 The parts of this problem are independent of each other (a) The derivative property...
PROBLEM 4.1 The parts of this problem are independent of each other (a) The derivative property of Fourier transforms states that if X(jw) is the Fourier transform of r(t), then jwX(ju) is the Fourier transform of (t). This is readily proved by writing down the inverse Fourier transform formula and taking the derivative with respect to t of both sides. Let's try proving this with another approach. Remember from your Freshman calculus class that a derivative could be defined as...
How do I get the equation on the second page from the two equations on the...
How do I get the equation on the second page from the two
equations on the first page?
Download Page of 5 ZOOM Calculating the path Once a projectile is launched its motion is completely determined by its initial velocity (speed and direction) and the acceleration of gravity. Thus, once the bean bag leaves the player hand the success of the throw is determined no matter how much they may try and will the bag through the hole. The motion...
(1 point) In this problem we find the eigenfunctions and eigenvalues of the differential equation B+...
(1 point) In this problem we find the eigenfunctions and eigenvalues of the differential equation B+ iy=0 with boundary conditions (0) + (0) = 0 W2) = 0 For the general solution of the differential equation in the following cases use A and B for your constants, for example y = A cos(x) + B sin(x)For the variable i type the word lambda, otherwise treat it as you would any other variable. Case 1: 1 = 0 (1a.) Ignoring the...
please explain and solve only the pre lab If you combine the equations from the previous...
please explain and solve only the pre lab
If you combine the equations from the previous page you should find B. Using the Lens/Mirror Equation So the same equation works for mirrors and lenses because the similar triangles are You should be following along. Start with the above relation and, in the box below, show essentially the same in each case, and the ratios of sides are the same. Now you get to try using this equation. the algebraic steps...
2. The acceleration equation We have derived in lectures the Friedmann and fluid equations, that describe the evolution of simple cosmologies. The Friedmann equation is usually given (adopting c 1) as where a(t) is the scale factor of the universe, p is the mass (energy) density, and k is constant. (a) (5 points) The simplest (although spectacularly uninteresting) cosmology has zero mass-energy, i.e. ρ 0. Solve the Friedmann equation under this condition to obtain an expression for at(t). Describe the...
5. (Telegraph equations) Using basic electrical theory, it can be shown that the current I(r, t) and voltage V(z, t) in an electrical transmission (eg., a power line or telephone line) at position in the lne andimt obey the equations Math 121 Applied DiffEqns II Homework 1 Due: Febuary 7, 2019 where C is the capacitance per u length, Gis the leakage per unit length, R is the resistance per unit length, and L is the inductance per unit length...
Linearity is an extremely useful property of equations in physics. If an equation is linear, it allows you to easily add together simple contributions from individual elements to get a solution for a more complex situation. One example of a simple linear equation in classical mechanics is F ma. If you find the individual accelerations of a particular body due to various individual forces you can then add those accelerations to find the acceleration due to all of the forces...
2. The equations of motion for a system of reduced mass moving subject to a force derivable from a spherically symmetric potential U(r) are AF –102) = (2+0 + rē) = 0 . (3) Using the second of these equations, show that the angular momentum L r 8 is a constant of the motion (b) Then use the first of these equations to derive the equation for radial motion in the form dU L i=- What is the significance of...
of Equations There are generally two approaches to solving systems of equations in physics, the substitution method and the addition method. Let us consider the system of two equations below: 4x + 2y 14 2x -y-1 We can solve these equations using both methods. First, the substitution method. The process is as follows: In one of the equations, solve for one of the variables . te this expression for that variable into the other equation. This will leave an expression...
From Acheson: Elementary Fluid Dynamics
Equation 7.3 and 6.12 (Slow Flow Equations)
7.2. A rigid sphere of radius a is immersed in an infinite expanse of viscous fluid. The sphere rotates with constant angular velocity Ω. The Reynolds number R-Ωα2/v is small, so that the slow flow equations apply (see eqns (7.3) and (6.12)). Using spherical polar coordinates (r, θ, φ) with θ-0 as the rotation axis, show that a purely rotary flow u us(r, e, s possible provided that...
PROBLEM 4.1 The parts of this problem are independent of each other (a) The derivative property of Fourier transforms states that if X(jw) is the Fourier transform of r(t), then jwX(ju) is the Fourier transform of (t). This is readily proved by writing down the inverse Fourier transform formula and taking the derivative with respect to t of both sides. Let's try proving this with another approach. Remember from your Freshman calculus class that a derivative could be defined as...
How do I get the equation on the second page from the two
equations on the first page?
Download Page of 5 ZOOM Calculating the path Once a projectile is launched its motion is completely determined by its initial velocity (speed and direction) and the acceleration of gravity. Thus, once the bean bag leaves the player hand the success of the throw is determined no matter how much they may try and will the bag through the hole. The motion...
(1 point) In this problem we find the eigenfunctions and eigenvalues of the differential equation B+ iy=0 with boundary conditions (0) + (0) = 0 W2) = 0 For the general solution of the differential equation in the following cases use A and B for your constants, for example y = A cos(x) + B sin(x)For the variable i type the word lambda, otherwise treat it as you would any other variable. Case 1: 1 = 0 (1a.) Ignoring the...
please explain and solve only the pre lab
If you combine the equations from the previous page you should find B. Using the Lens/Mirror Equation So the same equation works for mirrors and lenses because the similar triangles are You should be following along. Start with the above relation and, in the box below, show essentially the same in each case, and the ratios of sides are the same. Now you get to try using this equation. the algebraic steps...
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