All Questions
Tagged with calculus newtonian-mechanics
130 questions
6
votes
4
answers
587
views
Limits of the integral for the calculation of work
To calculate the total displacement for a time dependent velocity, one can start from an infinitesimal displacement and integrate as follows
$$dx=vdt$$
$$ \int_{x_i}^{x_f}dx= \int_{t_i}^{t_f}vdt $$
$$ ...
- 2,410
0
votes
1
answer
41
views
Connection between gravitational potential function and Gaussian distribution function in 3 dimensions
I was reading Kai Lai Chung's book Green, Brown, and Probability. Consider the Gaussian distribution function in 3 dimensions:-
Now, this is a function of y, mean is x, and variance is t, which is ...
- 181
2
votes
1
answer
96
views
Why take the derivative of variables such as area, mass, and radius?
I'm taking a module on stars and the solar system; I've attached notes from our first lecture- hydrostatic equilibrium. I'm confused about the notation $\mathrm{d}$ for $\mathrm{d}A, \, \mathrm{d}r$, ...
0
votes
7
answers
152
views
Does centripetal acceleration affects the magnitude of velocity?
A vector has no component in the perpendicular direction, obviously.
But let us consider a situation where a ball is moving in a uniform circular motion tied to a string, then $a = \dfrac {v_1^2} r$.
...
-2
votes
1
answer
84
views
Where did $1/2$ of this come from? [duplicate]
Work done by an external force $F$ upon a particle displacing from point 1 to point 2 is defined as
$$
W_{12} = \int_1^2 F \cdot dr
\, .$$
Kinetic energy and work-energy theorem: According to Newton's ...
0
votes
0
answers
63
views
Force in thermodynamic configuration space
Consider a thermodynamic system whose internal energy $U$ may not be conserved in general. It's a direct consequence of the First Principle that the variations in internal energy do not depend on the ...
- 1,880
1
vote
1
answer
56
views
Reduced mass in a Harmonic Oscillator [closed]
I recently came across the harmonic oscillator and the concept of reduced mass, i.e
$$
\mu = \frac{m_1m_2}{m_1 + m_2}
$$
To begin, I understand the derivation from the point of view of sitting on one ...
- 13
0
votes
5
answers
186
views
Why the normal vector addition does not seem to work in centripetal acceleration? [duplicate]
It is known that centripetal acceleration acts at an angle of 90 degrees to the tangential velocity. This acceleration vector then causes an increment $\mathbf{a} \Delta t$ to be added to the original ...
- 105
2
votes
1
answer
154
views
Adding equations of motion
Consider an object of (constant) mass $m$ subject to forces $F_i$ where $i=1,\ldots,n$. Now assume $s_i, v_i, a_i$ are the corresponding equations of motion (position $s$, velocity $v$, acceleration $...
- 35
0
votes
2
answers
81
views
What is $F$ and what is $P$ in the sentence "$F × dP$ is a total differential"?
The physicist Emilio Segrè, as a student, attended lessons of Calculus given by Francesco Severi and of Analytical Mechanics given by Tullio Levi-Civita. Segrè wrote in his autobiography1
For many ...
1
vote
1
answer
76
views
How do force and mass work with all derivatives of position?
I think if $F(t) = kt^0$ then $$x(t) = x_0 + v_0t + \frac{k}{m}\frac{t^2}{2!},$$ and if $F(t) = kt^1$ then $$x(t) = x_0 + v_0t + \frac{k}{m} \frac{t^2}{2!} + \frac{k}{m} \frac{t^3}{3!},$$ and so on, ...
- 31
1
vote
3
answers
93
views
What is the actual meaning of $dx$ in $W=-F.dx $, in work in thermodynamics?
what I want to ask is that the $dx$ in that formula is the displacement of piston or the displacement of the center of mass of the gas. also is there any situation where this clarity is useful.
- 41
-1
votes
1
answer
170
views
How to Find Trajectory of Particle?
Let’s say I have a particle, and I know all the forces acting on it at every position. (Let’s say the particle is in an electric/gravitational field to simplify the mathematics involved.) Now, is ...
3
votes
4
answers
199
views
Is is true to say $F(x) = ma(x)$?
Considering the equation $F(t) = ma(t)$, I'm trying to figure out if the following is also always true:
$$F(x(t)) = m\cdot a(x(t))$$
I.e.: $F$ as a function of $X$ (the position, which itself is a ...
- 605
0
votes
0
answers
45
views
Why is time taken to go around the Sun to cover a small fixed angle proportional to the square of the distance?
I am reading Feynman's lost lecture. At this point, he asks us to consider points J, K, L and M which subtend equal angles at the sun S. And then he claims that triangles JKS and KLS are similar ...
- 430
0
votes
0
answers
49
views
Energy Dissipated by Damper Infinitesimal Derivation
If we consider a damper (dashpot) element that exerts a force opposite the direction of motion proportional to the velocity, i.e. $$ \vec{F} = -c \vec{v}$$
Therefore, we can consider an infinitesimal ...
- 115
0
votes
1
answer
106
views
Understanding the double integral solution to Newtons second law?
I was following this lecture on Newtons Laws.
https://youtu.be/2tHpgQmnH3A?si=Wbp36oBS_4b1HhIi
At 31:56 in the video, the board has a very general solution to Newton's second law.
However the second ...
0
votes
3
answers
211
views
What does it mean in terms of energy if power is increasing with time? [closed]
Recently I have been studying work and energy and in this chapter I encountered with the term power. In terms of work power is written as $dW/dt$. So I have a doubt that suppose power is increasing. ...
- 37
-2
votes
1
answer
101
views
What is $V$ in $a$=$V$$dv$/$dx$? [duplicate]
$a$=instantaneous acceleration
$V$=instantaneous velocity
$x$=position
$dx$=small Chang in position
$a$=$dv$/$dt$
multiplying numerator and denominator by $dx$,we get
$a$=$dv$.$dx$/$dx$.$dt$
now we ...
- 161
0
votes
2
answers
99
views
Equilibrium of a body with potential energy as a function of position
We know that if the potential energy of a body, say $U(x)$ of a body is known as a function of its x-coordinate, for equilibrium, $$\frac{dU(x)}{dx} = 0$$ Also, several sources suggest that for the ...
- 301
0
votes
1
answer
68
views
Can someone help me with differential equation please? [duplicate]
here is the topic of the problem:
You are given $2$ baseballs (consider them as perfect solid spheres) have equal properties with mass $m = 0,142kg$, radius $r_0 = 0.037m$ in the space and thay are $...
- 1
0
votes
0
answers
31
views
The choice of the direction of the displacement vector when calculating potential energy of a system
Here, when referring to potential energy, I will take gravitational potential energy as an example. Consider the following diagram where two point masses $m_1$ and $m_2$ at a distance $r$ from each ...
-1
votes
2
answers
299
views
(Physics 2, Waves) Why does $\tan(\theta) = dy/dx$? [closed]
In the following example:
At the very last step, how does the author get that $\tan(\theta) = dy/dx$? To which $dy$ and $dx$ is this referring to? It can't be the same $dx$ that is labelled in the ...
0
votes
2
answers
53
views
Work-Energy Theorem for a path that is not smooth
In the analysis of Newtonian Mechanics for a single particle, we come across the definition of work and also the Work-Kinetic Energy theorem:
For a single particle, the work done on a particle by a ...
- 3,340
0
votes
3
answers
178
views
How do I write the gradient in angular coordinates ($\theta_1$, $\theta_2$, $\theta_3$)?
I have to find $\tau$ by finding the gradient of $U(\theta_1, \theta_2, \theta_3)$, where my coordinates are $(\theta_1, \theta_2, \theta_3)$. I assume the gradient is not the simple Cartesian ...
2
votes
1
answer
1k
views
Why is the gravitational potential of a uniform disc not symmetric about its center?
Consider a uniform, infinitely thin disc of surface mass density $\sigma$ and radius $R$ placed in the $xy$-plane with its center as the origin.
The gravitational potential at a point on the axis of ...
- 113
1
vote
1
answer
70
views
Conservation Principle
We are introduced to Principle of Conservation of Linear Momentum via the Newton's Second Law
$$\vec{F_{net}}=\frac{d\vec{p}}{dt}$$
It states when net external force equals zero then $\vec{p}=$...
- 13
4
votes
4
answers
2k
views
Help me understand the derivation of the kinetic energy formula please
In my physics textbook, kinetic energy is defined as $W$Net $=$ $\int m\frac {dv}{dt}$ $dx$ This makes sense to me just fine. The book goes on to rearrange the integral to say the following:
$W$Net $=...
0
votes
2
answers
358
views
Circular motion equivalent in three dimensions [closed]
Are there equations or even a concept of circular motion/tangential acceleration/centripetal acceleration in three dimensions? Maybe something called "spherical acceleration"? or am I just ...
- 75
1
vote
3
answers
391
views
Is the acceleration vector half of the gradient of velocity squared?
Consider the differentiation of speed squared with respect to time:
$$\frac{d(v^2)}{dt}=\frac{d(\mathbf v\cdot\mathbf v)}{dt}$$
$$=2\mathbf v\cdot\frac{d\mathbf v}{dt}$$
$$=2\mathbf v\cdot\mathbf a$$
$...
0
votes
6
answers
117
views
Deriving Work-Kinetic Energy Theorem
I am currently reading Physics for Scientists and Engineers (Ninth Edition) by Serway and Jewett and in Chapter 7.5, a derivation of the work-kinetic energy theorem was shown.
To give context, ...
- 43
0
votes
1
answer
91
views
2D rotation dynamics/control systems as a complex number
I have a dynamic system (it's a rocket in a 2D plane), that I'd like to model the orientation of using complex numbers to remove the need for trig functions in my ode.
I'm having trouble defining the ...
- 1
15
votes
3
answers
4k
views
Why does solving the differential equation for circular motion lead to an illogical result?
In uniform circular motion, acceleration is expressed by the equation
$$a = \frac{v^2}{r}. $$
But this is a differential equation and solving it gets the result $$v = -\frac{r}{c+t}.$$
This doesn’t ...
- 153
0
votes
2
answers
320
views
Finding angular frequency via integration of Newton's Second Law for a physical pendulum
For context: I am a student enrolled in AP Physics C with prior knowledge from AP Calculus AB and AP Physics 1.
We just collected data for a lab to determine an experimental value for g. The setup ...
1
vote
1
answer
596
views
Acceleration as a function of displacement
I am given a question such that a 0.280kg object has a displacement (in meters) of $x=5t^3-8t^2-30t$. I need to find the average net power input from the interval of $t=2.0s$ to $t=4.0s$.
I know the ...
- 115
0
votes
0
answers
12
views
Issue with work vs force for calculating spring constant [duplicate]
Lets say I have a spring with spring constant k. I put a 10kg weight on the spring and it compresses the spring one meter before stopping. We know that at this point the downwards force is equal to ...
3
votes
5
answers
2k
views
Guidelines to calculate moment of inertia
The moment of inertia is defined as
$$I = \int r^2 dm$$
but I am not sure how to proceed with solving the above integral. All examples I have seen seem to be done with different strategies. They ...
- 2,180
1
vote
2
answers
213
views
How do we show that the work done by a variable force (in one dimension) is the area under the $F$ vs. $x$ curve?
In my physics textbook, to show that work is the area under the $F$ vs. $x$ curve, the author first writes the relation $dw = F dx$. This part makes sense to me. From there, the author writes, $$W = \...
- 11
1
vote
1
answer
298
views
Tidal forces mathematics
Let's calculate the difference in force, $\Delta F$, experienced by
the rocks. Because $\Delta r$ is very small compared to $r$,
$$\Delta F = F_{\text{out}} - F_{\text{in}} \approx\frac{dF}{dr}\Delta ...
0
votes
0
answers
145
views
Work-Energy Principle Derivation
I am currently in Mechanics I and both my professor and my book have derived the work principle in this way and I even asked about its derivation during class, but it has me puzzled.
I don't ...
0
votes
2
answers
89
views
Kinematics confusion regarding sign of integration
I was solving some problems regarding non-inertial frames, and Newtonian mechanics in general, when I faced a major doubt regarding one of the seemingly simple topics, and I'd appreciate it if someone ...
- 1,853
0
votes
1
answer
82
views
Doubt in finding center of mass of extended bodies [closed]
While finding center of mass of extended bodies, we generally write the coordinate of the starting point of the infinitesimal mass element. Why not the ending or starting point? How does the error ...
0
votes
1
answer
39
views
Velocity while falling from table
Suppose there is a chain of mass $m$ in a semicircular tube of radius $r$. If it's pushed, then find the velocity of the chain while leaving the tube.
This is a simple problem which we can do using ...
- 1,197
2
votes
4
answers
686
views
Work done by a vector field (Force field) on a particle travelling along a curve
Assume a particle travelling along a curve, the work done by any Force field on the particle while moving along a curve is given by the line integral of $\vec{\bf{F}} \cdot \vec{\bf{dr}}$, but shouldn'...
- 367
1
vote
1
answer
136
views
Finding velocity $v$ and position $r$, given a time $t$ under the acceleration of a gravitational force [closed]
I was messing with the maths, when I tried to find the velocity as a function of time, $v(t)$, and the position, also, as a function of time, $r(t)$ under the gravity force.
$$ m \ddot{r} = -G \frac{...
- 1,573
2
votes
1
answer
68
views
Forces along and perpendicular to a curve
A uniform rope of length $l$ is suspended from two hinges, making an angle of $\theta$ with the horizontal at the hinges. Find the depth $d$ of the lowest point of the rope.
Similar questions include ...
- 23
0
votes
1
answer
168
views
Maximum height of a projectile when $g$ is not constant [closed]
How can I calculate the maximum height of a projectile that is launched from the surface of the earth with a given initial velocity? (ignoring air resistance in the atmosphere)
I understand how to ...
27
votes
14
answers
4k
views
Explaining how we cannot account for changing acceleration questions without calculus
For context, I am a high school physics teacher.
I am teaching students about the basics of electromagnetic force between two point charges. The equation we use is $F=\frac{kq_1q_2}{r^2}$.
This gives ...
1
vote
2
answers
73
views
How do you differentiate this differential equation? [closed]
I have to differentiate this equation (Gravitational force between N-Bodies)
$\begin{align}
\frac{d^2}{dt^2}\vec{r_i}(t)=G
\sum_{k=1}^{n}
\frac
{m_k(\vec{r}_k(t)-\vec{r}_i(t))}
{\lvert\...
0
votes
1
answer
26
views
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{...
- 93