Questions relating to Newton's Laws of Motion

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2answers
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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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3answers
55 views

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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1answer
16 views

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
46 views

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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1answer
87 views

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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0answers
38 views

Energy conservation $\iff \frac{dE}{dt} = 0\ $? [migrated]

If I'm asked to prove that a system is/ isn't conservative and compare it to whether or not the Hamiltonian is conserved, does that mean I need to compute the time derivative of energy $(T+U)$? Doing ...
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0answers
19 views

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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1answer
17 views

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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0answers
21 views

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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1answer
14 views

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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0answers
29 views

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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1answer
39 views

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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1answer
44 views

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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1answer
33 views

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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2answers
20 views

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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2answers
20 views

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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0answers
13 views

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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1answer
9 views

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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1answer
13 views

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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0answers
35 views

derivation of a central force field equation?

If we assume a particle of mass m is moving under the influence of a central force given by $F=f(r)\hat r$ a distance of $r$ away from an origin $O$. Derive the equation ...
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1answer
27 views

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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0answers
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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) ...
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1answer
30 views

Show the Euler Lagrange equations can also be written as

Show that the Euler Lagrange Equations can also be written as $$\frac{\partial \dot{T}}{\partial \dot{q}_j} - 2\frac{\partial T}{\partial q_j}=Q_j$$
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1answer
15 views

Find out the angular speed in terms of time.

Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: $u=\frac el\cos\theta+\frac 1l$, where $u$ is the reciprocal of the radial distance ...
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2answers
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Confusion regarding ($d = rt$) vs ($x_1 = x_0 + v_o t + 0.5 at^2$) usage.

I'm following an online physics course and I can't understand why for the question below the equation distance = speed $\times$ time can't be used while the equation $x_1 = x_0 + v_0t + 1/2at^2$ can. ...
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1answer
44 views

Cross product for vector angular position?

The angular velocity of a particle $\omega = r \times v$ is a pseudovector because it is formed by the cross product of two vectors (position and linear velocity). Likewise the angular acceleration ...
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1answer
25 views

How does this differential equation solve to give this?

Hello I would like to see how this differential equation solves to give the result on the picture. $c$ is a constant and I believe $\frac{dm}{dt}=-k$ Obviously they are dividing by $m$ and then ...
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1answer
17 views

Where is the error in this answer Newton's motion

I am asked to find the distance between the Earth and the Moon given only the constant $G$, the mass of the Earth and that it takes the Moon 28 days to orbit the Earth once as assumed. I said we have ...
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0answers
38 views

classic derivation of the proportionality between angular momentum and magnetic moment problem

The question is given in paranthesis. It is a classic derivation of the proportionality between angular momentum and magnetic moment $m=-IA=-I\pi r^2$ We start with the charge on a ring of Say the ...
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0answers
45 views

complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
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1answer
58 views

Vectors problem, Please Help!

A buoy is floating in water and is tied to a post. The water is creating a force of $3$ N on a bearing of $125^\circ$ and the wind is creating a force of $2$ N on a bearing of $230^\circ$ What force ...
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0answers
58 views

Attempt to solve the brachistochrone problem

I am attempting to solve the brachistochrone problem for fun. I don't have too much experience with differential equations and wanted to see if I am on the right path. My attempt Assumptions ...
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0answers
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Oscillation of a disc about a rod perpendicular to the disc but not through the centre.

A thin uniform circular disc of radius a and centre A, with density p, has a circular hole cut in it of radius b and centre B, where $AB = c < a−b$. The disc is free to oscillate in a vertical ...
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2answers
35 views

Calculating a 3 way circle collision

I need to calculate the resultant velocities of 3 circles/masses/particles if they was to collide at the exact same time. I understand that this is theoretically impossible (or incredibly unlikely) to ...
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2answers
20 views

Is the work integral decomposable?

Work is defined as $W = \int_{\gamma} \vec F \cdot d\vec l$ which I think means $W = \int (F_x, F_y, F_z) \cdot (dx, dy, dz)$. So by the linearity of the integral, could we always decompose work into ...
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1answer
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Curiosity with the Cartesian Notation of the Vector Cross Product

In my opinion Hibbeler's book on statics (Engineering Mechanics Statics, 12th ed) is one of the most approachable on the subject. On pg.123 he defines the Vector Cross Product in its Cartesian ...
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0answers
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Mechanics problem need to find expression for distance traveled.

Okay so I'm stuck on this problem. I have a ball that is projected at a cliff edge with a velocity $v$ and at an angle $\theta$ to the horizontal. The cliff edge is a height $h$ above the ground. The ...
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0answers
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Mechanics - Kinematics in two dimensions, how to find the time given a quadratic?

I have this information: $$u = 2i + 3j$$ $$r0(\text{initial position}) = 40i + 20j$$ $$r = 52i + 128j$$ $$a = -0.06i -0.04j$$ I need to find the $t(\text{time})$ at this point. I can use the equation ...
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0answers
31 views

How to show the planet always stays in the same plane?

I know angular momentum $q \times p$ is conserved, where $p=L_{\dot q}$ is linear momentum. How to apply this to a planet orbiting the star, described by the position vector $q$ relative to the star. ...
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0answers
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Mass Density Function Verification in Continuum Mechanics

I'm having problems knowing where to go with this question: "Suppose that a certain material initially fills an open region O ⊆ R3, with Ω := [−1, 1]^3 ⊂ O, so in particular the cube centered at zero ...
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0answers
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Integration over a second order tensor

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) ...
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0answers
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Find the value of P when the sliding begins

I was able to do the first part. For the second part, I wrote down $$F_b=\frac{5}{6}N_b$$Then taking moments about $C$ I got $$Pcos\theta +N_bsin\theta=F_bcos\theta$$ Then resolving horizontally and ...
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1answer
20 views

Show that $P=\frac{3W}{5+3\sin\theta -\cos\theta}$

I am stuck on the first part. Here is what I have done so far. First I resolved $P$ into $P\sin\theta$ and $P\cos$$\theta$. Then resolving in the horizontal plane, I obtained $F_a + ...
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1answer
23 views

Find the distance travelled by $P$ before it changes direction. (Mechanics)

A particle $P$ starts at the point $O$ and travels in a straight line. At time $t$ seconds after leaving $O$ the velocity of $P$ is v $m/s$, where $v = 0.75t^2 − 0.0625t^3$. Find (i) the positive ...
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0answers
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Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
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1answer
41 views

Expression for Hamiltonian vector field!

How does one prove that the Hamiltonian vector field has the following expression, what is the reasoning? \begin{equation} X_H=\sum ^n_{i=1}\frac{\partial H}{\partial q_i}\frac{\partial }{\partial ...
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0answers
26 views

Angular momentum in Cylindrical Coordinates

How to calculate the angular momentum of a particle in a cylindrical coordinates system $$x_1 = r \cos{\theta}$$ $$x_2 = r \sin{\theta}$$ $$x_3 = z$$ Thanks.
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0answers
108 views

Classical perturbation theory + KAM theory

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
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0answers
45 views

Stopping distance of a car on inclined surface

I am making a simulator of car stopping distances in various conditions for a computing project. My math knowledge of this subject is quite limited and i was hoping for a equation or formula to use to ...
2
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1answer
95 views

Applying linear algebra to solve a problem in mechanical equilibrium

I came across the following problem in "Introduction to Applied Mechanics" by Gilbert Strang, and am a little confused about the solution to this problem. The following figure shows the problem. ...