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...
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...
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...
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...
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...
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?
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!
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.
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...
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...
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...
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...
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...
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?
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...
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!
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!
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...
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...
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...
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...
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?
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)...