Questions relating to Newton's Laws of Motion

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Necc. and suff. conditions for a canonical transformation.

Let $\mathbf{P} = C^{−1}\mathbf{p} + B\mathbf{q}, \mathbf{Q} = C\mathbf{q}$, where $C$ is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on $C$ for the transformation ...
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1-D Motion , direction of air resistance in N2L

I cant seem to understand why mcx(dot) is not positive on the RHS of ma=F because it is acting in the same direction as the weight, but the answer doesnt agree, please help, I can get the DE but with ...
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How would the following graphs differ in shape?

This is a mechanics question but is pretty much mathematical so I figured I should post it here. If I had a particle dropped from rest and it had resistance $mkv$ where mass is $m$, $v$ is ...
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Simple Harmonic Motion - Particle Projection

Given $x=A\cos(\omega t) + B\sin(\omega t)$, how do you find the values of constants $A$ and $B$? I am aware that that depends on initial conditions, but I am unsure of the how. The initial conditions ...
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Confused on the argument of this function?

So say I wish to go from $$12\sin (t)+4\cos(t)$$ to the form $$A\cos (t+k)$$ by using the double angle formula I can get that $$\cos(k)=4$$ and $$\sin(k)=-12$$ and so we can find ...
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How would you answer this mechanics question?

This is not a homework question, it is from a past paper which I am using to practice. The question is shown in the image below: I really don't know much about mechanics, so I don't even know where ...
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Determine the Stress Vector on a plane given the normal?

Let $$\sigma=\left(\begin{array}{lcr} 1 & 1 & 0\\ 1&1&1\\0&1&1 \end{array}\right)$$ be the stress tensor. Find the stress vector acting on a plane through the point whose ...
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Proving that a certain function is an integral of motion for a Hamiltonian

Let $H=q_1p_1-q_2p_2-aq_1^2+bq_2^2$ (with $a,b$ constant) be a Hamiltionian. Show that $G=\dfrac{p_1-aq_1}{q_2}$ is a first integral (integral of motion) of this system. According to the ...
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Finding the trajectory of the centre of mass

Consider N particles $P_1, · · · , P_N$ with masses $m_1, · · · , m_N$, respectively, and with initial conditions $(x_1(0), x_1'(0)), · · · ,(x_N (0), x_N' (0))$. Show that the trajectory ...
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Deriving an equation for acceleration in circular motion

I have a question: A particle starts to move from rest in a circle of radius 3m, so after $t$ seconds its speed is $5t+1$m/s. Find its acceleration after 1 second. I have tried differentiating ...
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Mechanics speed of boat across river [closed]

If a boat travels with a speed of 6.0 miles per hour on a lake and it points straight across a river that Flow at a speed of 2.5 miles per hou, then what is its velocity (magnitude and direction)? Do ...
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Distance travelled on a curvelinear path and the coordinate of points

A race car travels in a curvilinear path at points A, B, and C. The following data is given: At point A time 0 seconds Speed is 195.1696800 $\large{\frac{m}{s}}$ tangential acceleration is 0.22 ...
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Center of mass calculation

Calculate the center of mass for : The area bounded by parabola $y = x^2/b$ and the line $y = b$. I got the following integral I just need verification that my work is correct. First I got ...
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1answer
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Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
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Trouble with a Statement in Arnold's “Mathematical Methods of Classical Mechanics”

On Pg 6 of Arnold's Mathematical Methods of Classical Mechanics (2nd Edition), there is a line which reads One can speak of two events occuring simultaneously in different places, but the ...
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Forms and conservative forces

According to Tongs notes on Classical Mechanics; a force is called conservative when $F=-\nabla V$ And iff $\nabla \times F = 0$. This is in $R^3$. Also the potential $V=\int_{x_o}^{x^1} F(x)$ $dx$ ...
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Books on Classical Mechanics

Can I get a book for a one semester course in these topics? My requisites are I dont want a very detailed book on these topics.It should just deal with the concepts clearly and provide some solved ...
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Which ellipses settle to 1-point contacts within a snow-globe circle?

Suppose you have a solid ellipse with axes $a$ and $b$, $(x/a)^2 + (y/b)^2 = 1$, confined inside a unit-radius circle. You shake the circle like a snow globe, and the ellipse settles to the bottom ...
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Position vector is given by r(t)=3sin(4t)i+3cos(4t)j+5tk Determine the velocity and acceleration of the particle at any time t>0.

I understand that the velocity and acceleration are found by the first and second derivatives of the position vector respectively. Also that the magnitude of the velocity is speed, given by ||v(t)||. ...
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How to solve an overconstrained system of equations?

What is the easiest codeable way to solve an overconstrained static model? How does Force Effect https://forceeffect.autodesk.com do it? Given a 10m long bar angled as the hypotenuse of a 3, 4, 5 ...
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Determine the resultant force of the inclined plane

So I was given this problem for my statics subject. I'm just having problems getting the x and y components of the force $120$ lb. I know that $60$-lb Force $$F_x = (60\mbox{ lb})\cos20^\circ$$ ...
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How to derive velocity out of acceleration in a circular motion?

A car starts moving in a circle with a radius of 200 meters. It has a constant tangential acceleration of $1{\text{m}\over {\text{sec}}^{2}}$. a. What is the angular acceleration? b. What is the ...
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Axioms of Newtonian Mechanics

Axiomatically speaking, could Newton's laws be derived (as theorems) from the conservation of momentum and energy -- along with a few suitable definitions of things like an inertia frame and force? ...
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Questions about the mechanics of a spinning CD.

A CD is spinning counterclockwise with a radial velocity of $\omega=30\text{rad}/\sec$. The preface of what I did manage to solve and further details: I was asked what is the period time (12 seconds), ...
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How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
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Finding the range of a cannonball- proof verification.

I asked such a question before but I do learn best by mistakes and corrections.(I didn't fully understand it yet.) I could really use your verification: A cannonball is being fired with a velocity of ...
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1answer
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How to get a range of a body being shot\thrown? Theoretical question.

I already asked a similar question here, but the answers were technical, leading me to no genuine comprehension of what I am doing.(I am not complaining or anything, I really thanked you then.) ...
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Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
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1answer
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Under what condition does $A^T(B \times C) + (B\times C)^T A = 2A^T(B \times C)$, A,B,C vectors

In my classical mechanics text book there is a formula that states $(\dot r_c + \omega_i \times d_i)^T (\dot r_c + \omega_i \times d_i)$ give rise to $\dot r_c^T \dot r_c + 2\dot r_c^T(\omega_i ...
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Lagrangian of bead on a rotating hoop

I'm trying to find the Lagrangian for a bead on a rotating circular loop (constant angular velocity $\omega$, radius $a$) in two different ways and I'm unsure why these are giving different answers. ...
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How to find the range of a cannon ball?

A cannon is positioned with a direction of 60 degrees between the ground and itself. (Sorry, again, for my poor English. I hope you understood that sentence.). The shooting velocity is ...
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Where have I gone wrong? A basic question in physics (Mechanics)

A stone is dropped vertically in a velocity of $15$ minutes per second, from a point $40$ meters above the ground (excuse my poor English.). a. How long will it take to the stone to hit the ground? ...
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Expression for $dW$ for a 3D position dependent force $\vec{F}(\vec{r})$.

I was looking at the derivation of the infinitesimal element of work done for a 3d position dependent force and I couldn't get over the switching of $\text{d}\vec{v}$ and $\text{d}\vec{r}$ in the ...
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1answer
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Planetary Motion: A comet describe a parabola about the sun [closed]

A comet describe a parabola about the sun, show that the sum of the squares of the velocities at the extremities of a focal chord is constant. I have no idea how to solve. Please help. I only ...
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planetary motion: Particle describes an ellipse as a central orbit about a focus

A particle describes an ellipse as a central orbit about a focus. Show that the velocity at the end of the minor axis is the geometric mean between the greatest and least velocities. My attempt: ...
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Equilibrium problem. (Mechanics).

Find the maximum weight that water bucket can take if each of the cables can carry a maximum of 10lb. I have worked as follows; $\sum F_y = 0$ $10sin(60)+10sin(180-tan^{-1}(4/3)) = W_{max} = ...
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Tracking an Object

I have the following situation, two objects A and B, at a distance of x from each other. Both objects have their own 2d heading Ah and Bh and their own speeds As and Bs. I'm trying to determine the ...
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Mechanics question Help

Two particles $A$ and $B$ both with masses $0.2kg$ move in the same direction with speeds $5ms$ and $3ms$ respectively. Both receive an impulse of $0.3ns$, show that the speed of $A$ after the ...
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Gyroscopic rotation.

I have never encountered a gyroscopic movement question so i am going to require some assistance. At the end of a rod of length $l$ is a solid disk with radius $R$, spinning with angular velocity ...
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Green's function in the context of classical mechanics

I am following this paper entitled "The classical mechanics of non-conservative systems". I would like to discuss equation (2) since I cannot get what the autor says. This is the problem: let's ...
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Derivative of an Infinitesimal?

I am currently studying calculus of variations (for my classical mechanics course). I have, on multiple occasions, seen the derivative of an infinitesimal quantity defined like below $$\frac{d}{dt} ...
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Inequality with Logarithms!

I need some help solving this inequality for a question involving the number of bounces, $n$, of ball such that the max. height of the ball is less than 5cm. This is the equation I have gathered from ...
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How to rearrange this equation and find the constant?

Okay so I've been working a mechanics problem and it has boiled down to this. I want to find $v(t)$ and I currently have that. $$t+c_1=\frac{1}{2\sqrt{gk}}\ln{\frac{\sqrt{g/k}-v}{\sqrt{g/k}+v}}$$ ...
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Mechanics limiting speed with variable radius.

Okay so I'm trying to solve this problem and have ran into some difficulties. Using impulse change of momentum principles I managed to figure out that the equation of motion for the hailstone is ...
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Question about dropping a ball and coefficient of restitution.

If I drop a ball from a height $h$ and the ball rebounds from the floor it will bounce back up to a height of $e^2h$ where $e$ is the coefficient of restitution between the floor and the ball. Why is ...
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Tension in the stretched string in equilibrium state

A smooth cylinder with circular cross-section of radius a is held with its axis horizontal. A light elastic band of unstretched length 2¼a and modulus of elasticity ¸ is wrapped round the ...
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Solving particle in vertical motion with air resistance using conservation of energy

A particle of unit mass is projected vertically upwards with speed u. At any height x, while the particle is moving upwards, it is found to experience a total force F, due to gravity and air ...
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Equation of motion and differential equations problem.

Hello I just worked through an old question I found online and was wanting some feedback on my answer (mainly if it was correct) or other improvements. Question Answer (a) $\frac{dF}{dt}=-k$ ...
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Classical mechanics Hamiltonian vector field.

On page 188 of Abraham and Marsden Foundations of classical mechanics, how "by construction" does \begin{equation} i_{X_{H}}dq^i=\frac{\partial H}{\partial p_i}\ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ ...
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Meaning of Torricelli's Equation ($v^2=u^2+2as$)

The equation of motion $v^2=u^2+2as$ is usually presented as the particular formulation of the SUVAT system which doesn't involve t. It is derived from the others using some (perhaps well-motivated) ...