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1. Expectation operator - linearity

1. Homework Statement Show that the expectation operator E() is a linear operator, or, implying: E(a\bar{x}+b\bar{y})=aE(\bar{x})+bE(\bar{y}) 2. Homework Equations E(\bar{x})=\int_{-\infty}^{+\infty}xf_{\bar{x}}(x)dx With f_{\bar{x}} the probability density function of random variable...
2. Plate floating on oil - linear momentum equation

1. Homework Statement 1: Determine the wall shear stress that acts at the lower side of the plate. 2: Determine the force Fx that is needed to give the plate a speed of u = 1m/s. 3: Determine the speed V, that is leaves an air jet that blows against the plate and which creates the same force...
3. Definition: flow velocity

Good day, In my book, the following definition for flow velocity is given: So summarized, the flow velocity at a point in space is the velocity of an infinitesimally small fluid element as it sweeps through that point. But now my question; how is the velocity of an infinitesimally small...
4. Linearizing an explicit differentiation scheme

1. Homework Statement Consider the following implicit scheme: y_{n+1}=y_{n}+\frac{\Delta t}{2}\left [f(y_{n+1})+f(y_{n})] By linearization one can obtain an explicit scheme which is an approximation to this - with approximation error O(\Delta t^{3}) 2. Homework Equations The solution...
5. Second derivative of an integral

Good day, I don't understand the following: \frac{d^{2}}{dt^{2}}\int_{0}^{t}(t-\epsilon )\phi (\epsilon)d\epsilon=\phi''(t) All I know is: \frac{d^{2}}{dt^{2}}\int_{0}^{t}(t-\epsilon )\phi (\epsilon)d\epsilon=\frac{d^{2}}{dt^{2}}\int_{0}^{t}t \cdot \phi...
6. Inverse Laplace Transform

1. Homework Statement Take the Inverse Laplace Transform of: Y(s)=\frac{1}{\tau s+1}\cdot \frac{1}{s} 2. The attempt at a solution I know: L^{-1}(\frac{1}{\tau s+1})=\frac{1}{\tau}e^{\frac{-t}{\tau}} and: L^{-1}({\frac{1}{s}})=1 But how to continue?
7. How do I solve (a+b)^(-c)?

Good day, How do I work out (a+b)^(-c)? Thanks.
8. Unique solution of 1st order autonomous, homogeneous DE

Hello, 1st order autonomous, homogeneous differential equation have the general form: \dot{x}(t)=ax(t) It can be shown that the unique solution is always: x(t)=e^{at}x(t_{0}) I don't get this, could anyone help me? Thanks!
9. Drawing a graph with broken y-axis

1. Homework Statement With what program (freeware) can I draw a graph with broken y-axis. An example of what I want: I know that this is possible by using Excel, but just in a very cumbersome way, right? Thanks in advance.
10. Finding the confidence interval

1. Homework Statement What formula do I need to find the confidence interval, when I have got: - Number of samples - Level of Confidence - The assumed (1st guess) accuracy 2. Homework Equations I found the following equation online: µ = z * [p * (1 - p) / n] ^ (-1/2) 3. The...
11. Centroid and centre of pressure

1. Homework Statement Explain why the distance between the centroid and the centre of pressure for a plane submerged fluid decreases as the depth of fluid increases. I know that the centre of pressure for a plane submerged fluid is located below the centroid because pressure increases with...
12. Exponential form

1. Homework Statement Prove that: sinh(3x)=3sinh(x)+4sinh^{3}(x) 2. The attempt at a solution I know that: sinh(3x)=0.5(e^{3x}-e^{-3x}) and: 3sinh(x)=1.5(e^{x}-e^{-x}) But I have no idea how to rewrite 4sinh^{3}(x) in exponential form...
13. Fluid Mechanics - similarity

1. Homework Statement A body when tested in air of density 1.18 kg/m3 and dynamic viscosity 1.52x10^-5 Pas at a velocity of 35 m/s was found to produce a resistance of 250 N. A similarly shaped body 10 times longer than the original was tested in water of dynamic viscosity 1x10^-3 Pas...
14. Mass flow rate

1. Homework Statement A fluid of density 867 kg/m3 flows through a circular duct of diameter 40.60 mm with a mass flow rate of 14.63 kg/s. The fluid velocity in m/s is? 2. Homework Equations mass flow rate [kg/s] = density [kg/m3] x velocity [m/s] x area [m] 3. The Attempt at a Solution...
15. Rewriting an equation

Good day. I want to rewrite the following equation: 0.5p_{ss}=p_{ss}(1-e^{\frac{Lp}{Ix}t}) What I do is: ln(0.5)=ln(1)-ln(e^{\frac{Lp}{Ix}t}) ln(0.5)=-{\frac{Lp}{Ix}t} Though in the notes received it says: ln(0.5)={\frac{Lp}{Ix}t} (without the minus sign) Am I doing something wrong?
16. Control Engineering

1. Homework Statement If the steady-state error of the proportional-plus-derivative-plus acceleration system is to be less than 10% determine a suitable value for Kp. 2. Homework Equations The input of the system is: v The output of the system is: y The transfer function of the system...
17. Why is this true about the angle theta

1. Homework Statement From my notes: Why is: V_{e} \times sin \Theta \cong V_{e}\Theta ? The only thing I know is that the assumption: \Theta \rightarrow 0 has been made, but that doesn't make the above equation clear to me... Thanks in advance!
18. Indicating the onset of instability on a graph

Good day, 1. Homework Statement Indicate on the graph the onset of instability. The graph: I really have no idea how I can indicate the onset of instability on such a graph. Thanks in advance!
19. Partial derivatives

1. Homework Statement If z = 1 / (x^2+y^2-1) show that x(dz/dx)+y(dz/dy)=2z(1+z) 2. The attempt at a solution z = (x^2+y^2-1)^-1 dz/dx = -2x(x^2+y^2-1)^-2 = -2x * z^2 dz/dy = -2y(x^2+y^2-1)^-2 = -2y * z^2 (-2x^2 * z^2) - (2y^2 * z^2) = 2z(1+z) I can express x and y in something like z and...
20. Separation of Variables

Good day, I have to seperate the variables of the formula (dy/dx) + 1 = - (y/x) so I can determine the solution of the differential equation. I get: (dy/dx) + 1 = - (y/x) (dy/dx) = - (y/x) - 1 (dy) = (- (y/x) - 1)dx Though I cannot get rid of the y at the side of dx...
21. Young's modulus times second moment of area

Good day, Im am wondering what you get when you determine the following of a bar during a bending experiment: (E x I) / y E = the Young's Modulus [kgf/mm^2] I = Second moment of area [mm^4] y = half of the bar height [mm] Is there a name for this term? And what does this term...
22. Hyperbolic functions

1. Homework Statement If sinhx=tany show x=ln(tany±secy) 2. Homework Equations sinhx=0.5(e^x-e^(-x)) secy=1/cosy cosy=0.5(e^y+e^(-y)) tany=(e^(jx)-e^(-jx))/(e^(jx)+e^(-jx)) tany=siny/cosy 3. The Attempt at a Solution 0.5e^x -0.5e^-x=tany 0.5e^(2x) -0.5=tany e^x e^(2x) -2tany...
23. Hyperbolic function proof

1. Homework Statement Prove that: (1+tanhx)/(1-tanhx)=e^(2x) 2. Homework Equations 3. The Attempt at a Solution I tried substituting tanhx for (e^x-e^(-x))/(e^x+e^(-x)) and for (e^(2x)-1)/(e^(2x)+1)) But I really have no clue how to continue...
24. Laplace transform using the basic integral

The question: Find the Laplace transform of f(t)=te^2^t I have got: Though, this will not become f(s) = 1/(s+2)^2 Anyone got an idea about what I am doing wrong?
25. Laplace transform Heaviside step function

1. Homework Statement What is the Laplace transform of f(t) = 0 for 0 < t < 2 and f(t) = (4-t) for 2 < t < 3 and f(t) = 1 for 3 < t < 4 and f(t) = (5-t) for 4 < t < 5 and f(t) = 0 for t > 5? 2. Homework Equations 3. The Attempt at a Solution f(t) = H(t-2)(4-t) - H(t-3)(4-t) + H(t-3)...