we know that the center of instantaneous 0 velocity lies in the interception of 2 perpendicular lines to 2 points, which in this case lies above B. The velocity of any point of the rod can be described relative to the center of instantaneuous 0 velocity ##(Q)## as: $$\vec v_{P/Q}=\vec \omega...
a)
Our force can be represented as: $$\vec F= -k(r-H) \hat r$$ then the equations of motion are: $$\hat r: \ddot r -r {\dot{\theta}}^2=-\frac{k}{m_1}(r-H)$$ $$\hat{\theta}: r \ddot{\theta} + 2 \dot r \dot{\theta}=0$$
Plus we know that angular momentum is constant then $$|\vec L|=m r^2...
1) the motion equations for ##m_2## are: $$T-m_2 g=0 \rightarrow T=m_2 g$$
##m_1##: $$T=m_1\frac{v^2}{r_0} \rightarrow \vec {v_0}=\sqrt{\frac{r_0 g m_2}{m_1}}\hat{\theta}$$
2) This is where I am stuck, first I wrote ##m_2## motion equation just like before, but in polar coordinates...
So my problem isn't actually finding the components, but knowing if the initial approach I took is correct. So what I did was:
At first I found that at the same instant, ##x_{B/A}=10500 m## so then I wrote the equation of motion for plane B respect to A:
so $$\vec a_{B/O}- \vec a_{A/O}=\vec...
I think my approach is quite wrong, still I gave it a shot:
First I know that ##v_A=13.3 m/s=r\omega=60\omega \rightarrow \omega=0.2 \frac{rad}{s}##
Then $$\vec a_A=-r\omega^2 e_r=-2.4 e_r$$
But ##e_r=\cos{\theta}i+\sin{\theta}j## and substituing the latter in the acceleration equation I have...
So what I did was at first consider the case the kid is below the branch, so that x=0,t=0, then I thought that the lenght L of the rope should be ##L=2h## because we know the radius from the branch to the kid is just ##x^2+y^2=r^2## and when x=0, y=h. So then I wrote the motion equations for the...
So I know that ##a_t = \frac{dv}{dt}=-ks## and ##\frac{dv}{dt}=v\frac{dv}{ds}## then: $$v dv=-ks ds \rightarrow (v(s))^2=-ks^2+c$$ and using my initial conditions it follows that: $$(3.6)^2=c \approx 13$$ and $$(1.8)^2=13-5.4k \rightarrow k=1.8 \rightarrow (v(s))^2=13-1.8s$$
What bothers me is...
I tried to workout the problem but I find motion in different coordinates systems a bit weird at the moment, so only thing I could do is realise that the x component of ##\vec r(t)## is: $$vt +x_0$$ but for simplicity we will use the initial condition ##x_0=0## so that ##t_0## is the moment the...
Okay, so the answer is quite easy if you draw a diagram and notice that cosine law solves everything rapidly. But at first, I tried doing some vector algebra and apply properties to see if I could get to something. This is what I could develop.
Consider ##|\vec u|##=12, then $$\langle \vec...
Homework Statement
Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$
Homework Equations
$$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$
The Attempt at a Solution
First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...
Hello! I have been searching the web and textbooks for a certain theorem which generalizes the value of the integral around a infinitesimal contour in the real axis, or also called indented contour over a nth order pole.
It is easy to prove that if the pole is of simple order, the value of the...
Homework Statement
The following is a problem from "Applied Complex Variables for Scientists and Engineers"
It states:
The following integral occurs in the quantum theory of collisions:
$$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$
where p is real. Show that
$$I=\begin{cases}0 &...
Homework Statement
The currents ##I_{{a}}## and ##I_{{b}}## of the circuit have values 4A and -2A in that respective order.
A) Find ##I_{{g}}##
B) Find the power dissipated by each resistance
C) Find ##V_{{g}}## (voltage drop across the current source)
D) Show that the power delivered by the...
Homework Statement
It’s a rather confusing circuit, I’m having problems trying to understand the way current circulates through the circuit.
Homework Equations
Kirchoff’s DC circuits laws only, no resistive simplification neither nodal analysis
The Attempt at a Solution
I’ll attach my intent...
Hello! Currently I own Differential Equations by H.B Phillips, a really old book, but difficult and does it´s purpose. I have only 1 problem, certain exercises require certain geometrical functional study I suppose, for example:
"find the equation of the curves so that the part of every tangent...
I am having a hard time trying to understand this transformation from lorentz:
https://imgur.com/a/WYWMO
(You should ignore the spanish part and just focus on the math). I can’t understand well why they turn into what you can see in the second picture, when taking really small values of x...