All Questions
Tagged with calculus classical-mechanics
60
questions
0
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1
answer
167
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Time to travel a set distance given variable acceleration
Trying to solve a problem for the acceleration of an automated shuttle car at my work, been a while since I studied this stuff so thought I'd reach out for help.
I have a shuttle car that is tasked ...
-1
votes
1
answer
62
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Derivative of distance [duplicate]
I know that $speed = |\frac{\vec{dr}}{dt}|$
and first derivative of distance with respect time will be $\frac{d\vec{|r|}}{dt}|$
These 2 expressions don't seem to represent the same thing. But when I ...
1
vote
0
answers
37
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Is $n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} =\frac{60}{2\pi}\sqrt{g\frac{\int y_idx}{\int y_i^2dx}}\quad ?$
I have a question about this formula used to calculate the first critical speed of a drive shaft.
$$
n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} \tag {1} \quad .$$
It is the ...
- 11
0
votes
3
answers
426
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Goldstein: derivation of work-energy theorem
I am reading "Classical Mechanics-Third Edition; Herbert Goldstein, Charles P. Poole, John L. Safko" and in the first chapter I came across the work-energy theorem (paraphrased) as follows:
...
- 232
6
votes
7
answers
228
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Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
2
votes
2
answers
298
views
Solving for the radius of a sphere as a function of time
I have tried to realistically model the famous game Agar.io, which can described as the following: A sphere of initial mass $m_0$ expels part of its mass at a given rate ($\frac{dm_l}{dt}$) for thrust ...
- 300
0
votes
1
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43
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Doubt in derivation of bending of beam, It's about derivatives and intergration
Radius of curvature of the beam in above picture is given as:
$$ \frac{1}{R} = \frac{d^2 y}{dx^2}$$
Please help me two points used as steps of a derivation in my book:
How was the radius of ...
- 71
0
votes
0
answers
41
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Minimum seperation of moving objects doubt
Let there be $2$ objects $P_1$(initial velocity $u$ $ms^{-1}$ & acceleration $a$ $ms^{-2}$) & $P_2$ (initial velocity $U$ $ms^{-1}$ & acceleration $A$ $ms^{-2}$) initially separated by ...
1
vote
2
answers
129
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Time derivative of unit velocity vector?
Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
1
vote
2
answers
223
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Confused about the solution to the pendulum differential equation
So I’ve learned how to derive the exact solution to the pendulum differential equation (in respect to its period), $\ddot{\theta} + \frac{g}{l}\sin\theta=0$, where $g$ is gravitational acceleration ...
- 113
1
vote
1
answer
114
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Brachistochrone Problem without Trigonometric Substitution
I'm trying to numerically reproduce the cycloid solution for the brachistochrone problem. In doing so, I eventually ended up with the following integral:
$$ x = \int{\sqrt{\frac{y}{2a-y}} dy} $$
...
- 165
11
votes
3
answers
2k
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Why do we ignore the second-order terms in the following expansion?
Consider the expansion done for the kinetic energy of a system executing small oscillations as done in Goldstein:
A similar series expansion can be obtained for the kinetic energy. Since the ...
2
votes
2
answers
674
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Equilibrium and the derivative of potential energy
In his Classical Mechanics popular lectures ( Lecture 3, at the beginning) , Susskind illustrates the idea of a stationary quantity using an example involving the notion of equilibrium.
Link : https:/...
- 373
0
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1
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55
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Which is the differential $\text{d} p_i$ of a generalized momentum?
I want to get a partition function, but I introduce a generalized momentum, my doubt is about, when I define a differential respect to $p$, it means $\text{d} p$, which is the correct form to get it?
...
- 13
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1
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25
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Issue with a derivation in Marion's Dynamics [closed]
I was solving problem 2-14 in Marion's "Classical dynamics of particles and systems" edition 5. In this problem we calculate the range of a trajectory to be $d=\frac{2{v_0}^2\cos{\alpha}\sin{...
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