All Questions
Tagged with calculus newtonian-mechanics
120
questions
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Position Dependence in Equation of Motion
Our lecturer gives study material which contained that Newton's second law could be written as:
$$ \begin{aligned} F &= m \ddot{x} \\ &= m \frac{d \dot{x}}{dt} \\ &= m \frac{dx}{dx} \frac{...
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1
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141
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Differential Equation & MacLaurin Series for Newton’s Second Law
I am currently working with a differential equation, where I think I need to take the derivative of $ma$ (corrected as per comment). I am trying to write $F = ma$ as a MacLaurin series and eventually ...
- 151
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2
answers
562
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What does $d$ stand for in this formula?
Context: I am building a tennis ball machine and am having trouble interpreting the following formula for the flight path of the ball. I know all of the other values in the formula but the source I am ...
1
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1
answer
80
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Problem finding Centre of Mass [closed]
My Question: For finding the Center of Mass ($COM$) of a hollow cone, why do we use its area to define its elemental mass ($dm$) and not its volume, which we use to find the $COM$ of a solid cone.
The ...
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1
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1
answer
77
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Equation for stationary string
I have some doubts on the following derivation of the EOM of a stationary string.
Let $F_x, F_y$ be horizontal and vertical tension of the string
$\mu$ be the mass per unit length of the string [kg/m]
...
- 180
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4
answers
1k
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Acceleration due to gravity during its journey up and down
When we throw an object up into the air, ignoring air resistance, etc, we define acceleration to be -9.8 m/s^2. When it goes down after its journey up, like a parabola, do we define the acceleration ...
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2
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5
answers
333
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Significance of $\frac{dv}{dx}=0$
Suppose an object is moving with varying acceleration in time.
What does it mean when it hits a point where $\frac{dv}{dx}=0$?
Does it mean the object has hit maximum velocity?
Assume the object ...
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1
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2
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83
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For regular moving objects around us, how many times can I differentiate their position with respect to time until I reach a constant? [duplicate]
When I practise problems, I come across ideal situations like constant velocities, constant accelerations, etc. But in real situations, objects usually don't magically gain momentum or acquire ...
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151
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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\...
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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 = \...
- 7,980
-2
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1
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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 ...
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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 ...
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3
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3
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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 ...
- 1,047
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2
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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)$, ...
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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 ...
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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$$
And then I approximated for small angles $\sin\theta\simeq\theta$ that yields the equation of ...
- 1,989
5
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2
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301
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Clarify definite integration of differentials in physics problems
I realized there is an issue with integration in physics problems that I had always taken for granted.
As an example, the relation between work and potential energy is
$dW=-dU_p$
when integrating ...
- 405
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votes
1
answer
130
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Working with infinitesimal quantities and the motivation behind it
So in my freshman physics class, in classical mechanics the homework was (it's solved already, this isn't a homework thread) the following:
"A thin, spinning ring is placed on a table, that divides ...
- 163
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1
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169
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Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]
We apply a force to a particle with a mass $m$ and inicial velocity $v_0$:
$$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$}
\\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\...
- 39
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3
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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 $\...
- 268
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0
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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 ...
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1
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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 ...
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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}...
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1
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4k
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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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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\...
- 1
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3
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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 ...
- 147
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1
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949
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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 ...
- 113
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1
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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 ...
- 139
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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=...
- 93
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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 ...
- 1,177
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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 ...
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4
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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 (...
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1
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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 ...
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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 ...
- 103
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3
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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:
$\...
- 954
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"Rigorous" derivation of kinetic energy
I've always wondered where the formula of (non-relativistic) kinetic energy we learn at high school comes from. This is the "derivation" I came up with:
$\Delta W:=\int_{r_0}^{r_1}drF=m\int_{r_0}^{r_1}...
- 1,012
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1
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154
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Is friction necessary for a Tractrix Curve?
Is friction necessary for a
Tractrix Curve?
If friction is necessary, what curve will the particle trace if friction is not present?
If friction is not necessary, what curve will the particle trace ...
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2
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560
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Writing Riemann sums for physics problems
If I want to find the mass of a rod of length l and density $\rho = kx$ where $x$ is the distance from one end.
If I want to find the gravitational potential due to a hollow sphere at a distance x ...
- 1,096
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5
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Question about the use of integration in physics
I've always thought of integration as a way to solve differential questions. I'd solve physics problems involving calculus by finding the change in the function $df(x) $when I increment the ...
- 1,096
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2
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183
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How can we treat dV like this?
Now, to calculate the gravitational potential due to a ring(or any object for that matter) at a distance $r$ we consider a tiny mass $dm$ on the ring, and calculate the potential $dV$ due to this ...
- 1,096
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116
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When can I assume a force to be constant?
If I have a force $F(x)$, can I assume it to be constant in any infinitesimal interval such as $Rd\theta$,$ dy \over cos\phi$, $dz$ etc. or can I assume it to only be constant in the interval $[x,x+dx)...
- 1,096
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2
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144
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Why can I assume the force to be constant in this particular interval?
If I have force, or any function $f(z)$, I was told that I can assume it to be constant only in the interval $dz$.
However, in this case, I had to calculate the work done by the spring force as a ...
- 1,096
3
votes
2
answers
200
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When exactly does error tend to zero in calculus?
I've come across many instances where sometimes the error tends to zero but other times it does not. Let me give you a few examples.
1.
When I calculate the area of a sphere summing up discs of ...
- 1,096
2
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3
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355
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Question about the application of calculus in physics
The way I've been taught to apply calculus to physics problems is to consider a small element at a general position and write an equation for that element and then to integrate it.
For e.g
To find ...
- 1,096
4
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2
answers
967
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Maximizing Time of Flight in Projectile Motion [closed]
Is (or How is) it possible to maximize the time of flight of projectile subject to the following conditions?
Given :
Fixed horizontal range
Interval in which velocity lies
For example, let the ...
4
votes
2
answers
4k
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friction of rope wrapped around a cylinder - the Capstan Equation
I have the following problem:
A rope is wound round a fixed cylinder of radius $r$ so as to make n complete turns. The coefficient of friction between the rope and cylinder is $\mu$. Show that if ...
- 3,997
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2
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Proof that SHM is sinusodial?
If we have an object attached to a spring, and the net force on that object is $-kx,$ how do we prove that its motion (if you move the object to $x\ne 0$) is sinusoidal? I know that you must ...
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