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Homework Statement [/B] Given the dyad formed by two arbitrary position vector fields, u and v, use indicial notation in Cartesian coordinates to prove: $$\nabla^2 ({\vec u \vec v}) = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v} + 2\nabla {\vec u} \cdot {(\nabla \vec v)}^T$$ Homework...
2. ### Stagnation Point flow

Homework Statement The motion described by the Eulerian velocity, v, is given as v = c(x1e1 - x2e2) is called stagnation point flow. How long will it take a particle traveling on a streamline to reach the stagnation point? Homework Equations The Attempt at a Solution I really...
3. ### Fluids Homework Problem

Homework Statement Let R, L, C, K be suitable dimensional constants. In a certain motion of a fluid, the mass density function ρ is given in the Eulerian description by ρ = R*(x1^2+L^2). Is the velocity field v1 = C/(R*(x1^2+L^2)) e1 a possible velocity field? Is v2 = K*exp(-x1/L)...
4. ### Question on Equivalent/Hydraulic Diameter

Homework Statement Hello, I have a question on how to calculate the equivalent/hydraulic diameter for a problem that I'm doing. The problem states that I should use the cross-sectional flow area shown with an equivalent diameter based upon a closed rectangle (entire perimeter)...
5. ### Heat Transfer in 1D

Homework Statement A bare slab fuel element is 0.2 in thick. It has a kf = 10 Btu/(hr*ft*F) and a q''' = 5e7 Btu/(hr*ft^3). At x=0 (left hand face), h_lhs = 400 Btu/(hr*ft^2*F) with a fluid temp of 700F. At x=0.2 in (right hand face) the heat transfer coefficient is h_rhs = 300 Btu/(hr*ft*F)...
6. ### Second Moment of Area

Homework Statement I'm having a problem calculating the I value for this beam. I'm not sure on how to account for the missing circle in the front square and how to account for the beam changing from a 1" x 1" square to a 5" x 0.5" beam Homework Equations Ixx = bh^12/12 The...
7. ### Shaft Problem

Homework Statement I'm supposed to design and analyze a shaft including all features to retain the shaft in the bearings and attach the coupling and sheave and the shaft shouldn't fail in static loading or through fatigue. Homework Equations3 -The flexible coupling attaches your shaft to the...
8. ### Pressurizer Problem

Homework Statement A pressurizer is designed to accommodate an outsurge of 30,800 lbm and an insurge of 20900 lbm. The mass of the water necessary to cover the electric heaters is 4020 lbm. The system's operating pressure is 2250 psia. The hot leg is 600F and and the cold leg is 550F (For the...
9. ### Fourier Series

Homework Statement f(t) is given as: from 0 to 0.2s, f(t) = 5 from 0.2s to 0.6s, f(t) = 0 from 0.6s to 0.8s, f(t) = 5, etc Homework Equations for an odd function a0 = 2/p * integral(from -p/2 to p/2) of f(t) dt bn = 4/p * integral(from 0 to p/2) of f(t)*sin(2*pi*n*t/p) dt The Attempt at...
10. ### Convolution Integral

Homework Statement x'' + 2\zeta \omega_{n} x' + \omega_{n}^2 x = u_{s}(t) zeta is underdamped and u_{s}(t) is the unit step function and \omega_n is the natural frequency and there are zero initial conditions. Find the total response via the convolution integral. Homework...
11. ### Linear Algebra - Eigenvectors

Homework Statement A = [2,1,2;2,1,2;2,1,2] Find the Eigenvectors of A Homework Equations The Attempt at a Solution First I found the eigenvalues of A det(A - \lambda I) = 0 \lambda = 0,5 __________________________________________ A-5I...
12. ### Material Indices

I'm currently doing a project in my materials class and I wanted to check if these two material indices were right. The objectives are: 1. The material must not fracture 2. The material must be strong (no deformations) Material Index 1 ------------------------ Considering a yield-before-break...
13. ### Convergence of a series

Homework Statement Find the radius of convergence and the interval of convergence for \sum_{n=0}^\infty \frac{x^n}{n3^n} Homework Equations The Attempt at a Solution Ok, so I first applied the ratio test. \lim_{n\rightarrow\infty} | \frac{x^{n+1}}{(n+1)3^(n+1)} / \frac{x^n}{n3^n} |...
14. ### Trig Integration

Homework Statement \int sin^3(2x) dx Homework Equations sin^2(x) + cos^2(x) = 1 The Attempt at a Solution First I try to get the integral in the form of \intsin^3(u) and I do this by u-substitution. \int sin^3(2x) dx u = 2x du = 2dx dx = du/2 So the new integral looks as such...
15. ### Area of a parallelogram with vectors

Homework Statement Determine the area of the parallelogram spanned by the vectors < 0, 9, 6 > and < −10, −6, −4 > Homework Equations Area = A X B The cross product of < 0, 9, 6 > and < −10, −6, −4 > = 0i - 60j + 90k The Attempt at a Solution I know the area is the cross...
16. ### Electromagnetic Intensity

Homework Statement A helium-neon laser of the type often found in physics laboratories has a beam power of 3.50 mW at a wavelength of 633 nm. The beam is focused by a lens to a circular spot whose effective diameter may be taken to be equal to 2.00 wavelengths. Calculate the intensity of the...
17. ### Taylor Series question

Homework Statement Determine the Taylor Series for f(x) = ln(1-3x) about x = 0 Homework Equations ln(1+x) = \sum\fract(-1)^n^+^1 x^n /{n} The Attempt at a Solution ln(1-3x) = ln(1+(-3x)) ln(1+(-3x)) = \sum\fract(-1)^n^+^2 x^3^n /{n} Is that right?
18. ### Power Series

Homework Statement Determine the series of the given function: f(x) = 10 / (1-5*x) Homework Equations Power series of 1/(1-x) = Σ from n=0 to n=infinity of (x^n) The Attempt at a Solution f(x) = 10/(1-5x) = 10*(1/1-5x) = 10 * Σ(5x)^n = 10 * Σ(5^n)*(x^n) = Σ (50^n)*(x^n) <--- Not...
19. ### Torque on a charged particle moving in a circle in a uniform B field

Homework Statement A particle of charge q = e moves in a circle of radius = 0.44 m with speed v = 2.350×107 m/s. Treating the circular path as a current loop with constant current equal to its average current, find the maximum torque exerted on the loop by a uniform magnetic field of magnitude...
20. ### First Order Linear Differential Equation

Homework Statement 10 y′+ 10 y = 7 exp(6*x) y ( 0 ) = 7 Homework Equations The Attempt at a Solution First I divided everything by 10 because the coefficient of the y' should be 1 Then I found the integrating factor, which was exp(integral of 1*dx) = exp(x) = Integrating factor...
21. ### Semicircular Dielectric

Homework Statement http://nplq1.phyast.pitt.edu/res/westwolf/bookauthor/Capacitors/circle.jpg We use two circular metal plates of radius 0.55 m and thickness 8.7 mm to build a capacitor. We leave a gap of 2.5 mm between the plates and fill half of the space between the plates with a medium...
22. ### Differential Equation

Homework Statement Solve the inital value first order linear differential equation y'=y+x y(0) = 2 Homework Equations The Attempt at a Solution y'-y=x That's as far as I got. I'm not sure how to approach this. I've looked through my notes and book, and I don't have any examples that...
23. ### Constant Electric Field and Potential

Homework Statement The figure below shows two points in an electric field. Point 1 is at (X1,Y1) = (3,4), and point 2 is at (X2,Y2) = (12,9). (The coordinates are given in meters.) The electric field is constant with a magnitude of 59.3 V/m, and is directed parallel to the +X-axis. The...
24. ### Programs Help deciding engineering major

Hi, I have to decide my engineering major before Feb 25th 2009. I think I narrowed it down to Chemical Engineering or Mechanical Engineering. I want to work with nuclear power(I plan to get a nuclear certificate with either major), with the military or in a plant or business setting. Any...
25. ### U substitution

Homework Statement Integrate: x^2 dx / (1-x)^1/2 Homework Equations U substitution The Attempt at a Solution First I defined u = (1-x)^1/2 du = -dx/2(1-x)^1/2 dx = -2(1-x)^1/2 du then for x u^2 = 1-x x = 1-u^2 integral of: (1-u^2)^2/u = (u^4-2u^2+1) / u = u^3-2u+1/u integrating...
26. ### L'Hôpital's rule

Homework Statement Find the limit as x->0+ of (ln(x))^x *The answer is 1* Homework Equations l'Hôpital's rule The Attempt at a Solution lim (ln(x))^x = 0^0 I took the ln of that quantity to bring down the x lim = x*ln(ln(x)) lim = ln(ln(x)) / (1/x) Then I used l'Hôpital's rule...
27. ### What is the gravitational force homework

Homework Statement A solid sphere of uniform density has a mass of 3.0×10^4 kg and a radius of 1.0 m. What is the gravitational force due to the sphere on a particle of mass 1.0 kg located at a distance of 0.50 m from the center of the sphere? Homework Equations F = M1*M2*G / r^2 The...
28. ### Conservation of Angular Momentum of Star

Homework Statement The mass of a star is 1.51·1031 kg and its angular velocity is 1.50E-7 rad/s. Find its new angular velocity if the diameter suddenly shrinks to 0.49 times its present size. Assume a uniform mass distribution before and after. Icm for a solid sphere of uniform density is 2/5...
29. ### Where is the center of mass of the slab

Homework Statement The figure shows the dimensions of a composite slab; a fraction of the slab is made of aluminum (density = 2.70 g/cm3) and other part is made of iron (density = 7.85 g/cm3). They are not equal as indicated in the figure, but have x1 = 7.5 cm and x2 = 14.5 cm. As measured...
30. ### Rope Swing angle

Homework Statement The rope of a swing is 2.90 m long. Calculate the angle from the vertical at which a 82.0 kg man must begin to swing in order to have the same KE at the bottom as a 1450 kg car moving at 1.11 m/s (2.48 mph). Homework Equations k=(m*v^2) / 2 U = mgh The Attempt...