All Questions
Tagged with calculus newtonian-mechanics
122
questions
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Integration of equation of motion in polar coordinates
We have the equation of motion in polar coordinates:
$$\frac{d^{2}\vec r}{dt^2} = (\frac{d^2 |\vec r|}{dt^2} - |\vec r|\cdot (\frac{d\theta}{dt})^2)\hat r + (|\vec r|\cdot \frac{d^2\theta}{dt^2}+2\...
1
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1
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142
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Can you 'derive' mathematical approximations made from Taylor approximations from limiting cases in real life?
Here, the natural length of the string is $l_o$, and pulling the string up by $x$ increases its length by $ \sqrt{ l_{o}^{2} +x^2}$; thus, the increase in length can be approximated as
$$ \delta l = \...
-2
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1
answer
154
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What is wrong with my approach? What error did I make mathematically? [closed]
A $4$ kg object is moving in one dimension along the x-axis. The linear momentum of the object increases with the position of the object according to the following equation:
$p(x)=6+3x$
At $t = 0$ s ...
1
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1
answer
131
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What is the difference between zero and an infinitesimal number?
In a standard Atwood machine physics problem, the string going over the pulley is considered massless. So does that imply mass = 0 or mass = dm? General question: what is the difference between 0 and ...
3
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3
answers
878
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Newton's Second Law in vertical launch of a rocket
Consider a rocket being launched vertically.
Let $T(t)$ denote the thrust from the engine and $M(t)$ be the total mass of the rocket at time $t$.
At $t=0$, $T(0)=M(0)g$ (so that the normal force due ...
1
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2
answers
2k
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Acceleration as a function of position and time
I know if you have an acceleration as a function of $t$, $a(t)$, to find the velocity you simply integrate $a(t)$ with respect to $t$. Moreover, if the acceleration was a function of position, $a(x)$, ...
11
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6
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2k
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Does the logarithm of a non-dimensionless quantity make any sense?
A train consists of an engine and $n$ trucks. It is travelling along a straight horizontal section of track. The mass of the engine and of each truck is $M$. The resistance to motion of the engine and ...
4
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1
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How does the small angle approximation work for cosine?
In newtonian mechanics equation of motion of a simple pendulum:
$$\ddot{\theta}=\frac{g}{l}\sin\theta$$
And then I approximated for small angles $\sin\theta\simeq\theta$ that yields the equation of ...
5
votes
2
answers
301
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Clarify definite integration of differentials in physics problems
I realized there is an issue with integration in physics problems that I had always taken for granted.
As an example, the relation between work and potential energy is
$dW=-dU_p$
when integrating ...
0
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1
answer
130
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Working with infinitesimal quantities and the motivation behind it
So in my freshman physics class, in classical mechanics the homework was (it's solved already, this isn't a homework thread) the following:
"A thin, spinning ring is placed on a table, that divides ...
1
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1
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169
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Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]
We apply a force to a particle with a mass $m$ and inicial velocity $v_0$:
$$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$}
\\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\...
2
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3
answers
1k
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Calculating the moment of inertia of a uniform sphere [closed]
Currently trying to calculate the moment of inertia of a uniform sphere, radius R, I know the answer is $\frac{2}{5}MR^2$ but I keep getting $\frac{1}{5}MR^2$
Setup:
Assume mass per unit volume $\...
0
votes
0
answers
372
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Why don't we define time derivative of acceleration? [duplicate]
When we started the study of kinematics we defined position and its change with respect to time. After that we defined time derivative of velocity which gave us acceleration.
These 3 concepts really ...
-2
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1
answer
222
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If kinetic energy is mass times the integral of velocity, isn't it just a product of mass times distance? [closed]
I'm still learning Calculus at the moment and I'm currently on integration. The moment I realized the "$1/2$" and square value in $v^2$ are just products of integration, can't one just use ...
5
votes
1
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228
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What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?
In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as:
$a(t) = a_0$
$v(t) = a_0t + v_0$
$x(t) = \frac{1}{2}...
-1
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1
answer
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Calculating the distance between two masses with respect to gravitational force [duplicate]
Call them $m_1,m_2$. They are compressed to their center of masses, if you wish. If the initial distance at $t=0$ is $d$, is there a formula or an efficient way to calculate the distance between them ...
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1
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How to use Newton's second law to derive conservation of momentum and how to use derive conservation of momentum to derive the second law?
I know if taking the integral of $F=ma$, then I can get $p=mv$.
I'm weak in calculus, so I wondered how to do this exactly.
Is there anything wrong in my logic below?
\begin{align}\int F\left(t\...
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3
answers
306
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Issue with deriving the work-energy theorem
I'm a little confused regarding the way Total work = Change in kinetic energy is derived using calculus. My issue can be seen at 3:26 of this video: https://youtu.be/2dqO4sy4Njg?t=3m20s
Why can the ...
-1
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1
answer
62
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The proof of the fact that the summation of infinitesimal forces distributed on an object equal to the sum of concentrated external forces on it
When concentrated external forces is applied to an object, the integration of infinitesimal forces (df) distributed on the whole object due to these concentrated forces is equal to the sum of these ...
0
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1
answer
950
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Proving the centre of mass formula with integral [closed]
I came across a question:
Find $f(r)$ and prove the centre of mass formula:
$$\vec{r_{cm}} = \frac{1}{V} \int f(r) \vec{dS} $$
Where $V$ is the total volume and our surface integral is ...
0
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1
answer
177
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Acceleration as the second derivative of $e^{-\frac{1}{t^2}}$ [duplicate]
If we have, say, a material point with a zero velocity at the time $t=0$, and this point starts moving at a time $t>0$ , then we look at the force impressed on the point by inspecting the second ...
0
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3
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561
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What is meant by $dy/y$?
Consider the language in the following example:
What is meant by $dg$ and $dR$, and also by $dg/g$? Why does $dR/R=-2/100$ (negative for shrinks)? Is $4\%$ unity change? I mean $dg/g=4\%$ or $dg=...
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1
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360
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Dropping a block on a vertical spring - derivation with pure kinematics/dynamics (no work-energy) [closed]
Consider a block of mass $m$ falling on a vertical spring initially contacting the spring at equilibrium point with velocity $v_0$. The spring has constant $k$. I was trying to see if there was a way ...
0
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1
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2k
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calculating the length of a hanging spring
If we assume the slinky to have a uniform mass (mass per unit length around the circumference of slinky to be constant, or simply slinky is made of same material and has uniform thickness) and that ...
3
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4
answers
1k
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Can a proof of the work-energy theorem be made, that doesn't use Leibniz notation to cancel differentials?
I've been doing some reading, and even though many people say different things, i think i'm pretty confident in saying that we can't treat differentials as fractions. In some scenarios it works out (...
0
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1
answer
147
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Newtonian mechanics and calculus, looking for a good intro book to both together [duplicate]
I am looking for a book on Newtonian mechanics which is very careful to explain why, where and how you need to use calculus to develop physics. Or even, a book which introduce basic notion and ...
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1
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183
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Linear Momentum/Impulse for a Rigid Body
Problem
Consider a system of $N$ point-masses $m_i$, each travelling at a velocity $\mathbf{v}_i$. Then, the total linear momentum / impulse $\mathbf{p}$ of the system can be calculated as (see Smith,...
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2
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336
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Consider a semi pulling a tractor trailer–if the truck turns, will the trailer straighten out completely?
My friends and I are having a very heated debate about this question: Under perfect conditions (i.e. only considering friction from the road and no other forces), if a semi is pulling a tractor ...
0
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1
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1k
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Infinitesimal work
I am a newbie in Physics (Senior on highschool) and our teacher wrote in a proof
$$\dfrac{dK}{dt}=\dfrac{dW}{dt},$$
where $K$ is the Kinetic energy of a body and $W$ is the Work.
So now that I am ...
1
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3
answers
121
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Bounds of Integration (with respect to something that is not time)
I have been reading Richard Feynman's lectures and came across an interesting proof regarding the Earth's gravitational force. At one point in the proof, Feynman uses the following the integral:
$\...