All Questions
Tagged with calculus newtonian-mechanics
122
questions
4
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3
answers
2k
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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}...
- 1,022
1
vote
1
answer
154
views
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 ...
- 42
0
votes
2
answers
561
views
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
-1
votes
5
answers
1k
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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
0
votes
2
answers
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
2
votes
1
answer
116
views
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
1
vote
2
answers
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
votes
3
answers
355
views
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
votes
2
answers
967
views
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
1
vote
2
answers
1k
views
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 ...
- 215
0
votes
1
answer
75
views
Question from Kline's Calculus: A physical and intuitive approach [closed]
A train runs at a velocity of 66 ft/ sec along a straight track. When the brakes are applied, the deceleration is $4/3$ ft/$sec^2$. For how long and how far should the brakes be applied so that the ...
- 379
-1
votes
1
answer
207
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Is there any reason why acceleration should not be the first derivative of the absolute value of velocity? [closed]
I ask mainly because I am not familiar enough with newtonian mechanics and higher-level physics in general to know the repercussions of such a change, but on the simpler plane of existence, I have ...
- 1
0
votes
2
answers
294
views
Line integral confusion
Hi , so I was solving this example . I have no problem in calculation . But at the end of it , when they asked about the closed line integral , I wondered how did the line integral on both paths be ...
user65035
0
votes
2
answers
111
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Is the motion of a particle non-analytic?
I really can't understand what happens during the time $t(0)$ to $t(0+dt)$ in the following crackpot arguement:
A particle is at rest (in an ideal frictionless world) until $t(0)$. So every order ...
user77648
2
votes
1
answer
81
views
Satellite and gravitational acceleration
According to $0.5gt^2$ object will fall 5m in first second.
Earth curve is 5m for 8km
So if we can project object at 8000 m/s speed object will never fall into ground.
Above scenario is correct for ...
- 29
0
votes
1
answer
530
views
The dot product integral in the proof of the Parallel axis theorem
The first picture is a question about proving the Parallel axis theorem and the second is the solution. I have no problem with the solution except for the part which says that
$$
\int 2\vec h \cdot \...
- 147
4
votes
2
answers
3k
views
Why dont you take derivative of force in definition of power ? P=F.v
The derivative of work is $\bf F\cdot v .$
$$P(t)= \frac{\mathrm dW}{\mathrm dt}= \mathbf{F\cdot v}=-\frac{\mathrm dU}{\mathrm dt}\;.$$
But why not $$\frac{\mathrm{d}\mathbf{F}}{\mathrm{d}t}\cdot \...
- 105
1
vote
1
answer
124
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How to model terminal velocity as a function of gravitational acceleration?
Taking the most simplistic form of terminal velocity, $v=\sqrt{\frac{mg}{c}}$
I want to try and derive an equation that models the velocity as g changes in height.. Because obviously the terminal ...
- 546
0
votes
2
answers
5k
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Can you take the integral of $ d^2x\over dt^2$? [closed]
I am messing around with physics problems, and as silly as this maybe how do you take the integral of
$$\int_0^\infty xd^2x$$
For example taking Newton's Second law $F=ma$
$$
F=m{d^2x\over dt^2}
$$...
- 546
-2
votes
2
answers
103
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Find $\oint xdx+2ydy+3zdz$ on the line segment formed by points $A(1,0,-1)$,$B(2,5,-1)$ [closed]
Find $\oint xdx+2ydy+3zdz$ on the line segment formed by points $A(1,0,-1)$,$B(2,5,-1)$
My approach:
First I find the line segment formed by $A,B$ which is $\vec l(t)=\vec{OA}+t\vec{AB}=(1,0,-1)+t(1,...
- 107
0
votes
2
answers
223
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Differentiating displacement with respect to speed in order to obtain time
I have this problem where I am trying to calculate $d(t)$ and $v(t)$ of a mass m on a spring, dropped from a displacement $A$, without using anything else than Hooke's law and energy calculations. ...
- 75
3
votes
2
answers
2k
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Falling rain drop problem [closed]
EDIT: I've read that a ball moving in a rectilinear motion with a non-constant radio, $r$ satisfies that
$$\frac{dV_c}{dy} = \pi r^2,$$
where $V_c$ is the volume swept by the ball and $y$ is the ...
- 160
0
votes
1
answer
81
views
Confusion regarding area from graph
This might be a trivial question but is illustrated below.
Why is the area 'below' the graph always taken for a velocity-time graph when finding the displacement? I mean why is the area with the time ...
- 51
24
votes
7
answers
12k
views
Zero velocity, zero acceleration?
In one dimension, the acceleration of a particle can be written as:
$$a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx}$$
Does this equation imply that if:
$$v = 0$$
Then,
$$\...
- 383
0
votes
2
answers
304
views
Is the Fundamental Theorem of Calculus really applicable to the definition of work?
When the force $F$ on an object is not constant, then the work it performs is defined as $$W = \int_{x_0}^{x} F(X)dX.$$
Now, the Fundamental Theorem of Calculus states that
$$\text{If}\,\,\, f(x) =...
user36790
1
vote
3
answers
170
views
Integral ambiguity
I'm a bit confused with some notation I encounter in physics calculus. Consider this:
Taken from here.
Integration operates on functions, correct? What does it mean to integrate $\frac{d{\bf p}}{dt} ...
2
votes
1
answer
1k
views
Taylor series expansion of $\ln$ and $\cosh$ in distance fallen in time $t$ equation
I want to find the Taylor expansion of $y=\frac {V_t^2}{g} \ln(\cosh(\frac{gt}{V_t}))$
I have tried using the fact $\cosh x= \frac {e^x}{2}$ for large t, which works, I just need help on small values ...
- 345
0
votes
1
answer
667
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Investigation of a pendulum's period, problem creating equation to sum the dynamic velocity
I am investigating the period of a pendulum swing. This is a simple harmonic pendulum and I am already aware of the common, but slightly inaccurate,
$2\pi \sqrt{\frac{L}{G}}$ formula.
My problem is ...
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