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13

As you probably know, Newton thought that energy is linearly proportional to motion. The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "change of motion is proportional to the force impressed" It was Gottfried Leibniz, as early as 1686, who first realized that KE (m = unitary mass) is proportional ...


7

You say: Imagine a book that we lift it with a force that is exactly equal to the force of gravity so the forces cancel out and the book moves with a constant velocity. so I'm guessing your reasoning is that the net force on the book is zero so the amount of work done on the book is zero. And you are absolutely correct - no work is done on the book and ...


6

You should think of the formula the other way around, i.e. $$ \mathrm{d}W = F\mathrm{d}x$$ which means that the infinitesimal work done along an infinitesimal path is just the force $F$ times the length $\mathrm{d}x$ of the path along which the force was exerted. If we are now given a real path $\gamma : [a,b] \to \mathbb{R}^3$, the total work done along ...


4

I believe the difference comes from the fact that forces can do different amounts of work in different reference frames. In particular, the normal force by the ramp does no work in the "lab" frame, but does do work in the moving frame (since there is a component of velocity that is now parallel to the normal force). I don't think you accounted for this work ...


4

The energy is conserved but it becomes "lumpy"- more in some places (directions), less in others. Total over all directions is the same.


2

Yes - the coiled spring has a certain amount of (potential) energy. When it gives up the energy to the ball, you could say the ball does negative work on the spring, so it loses (potential) energy.


2

You've already got some answers, but nobody mentioned Noether's Theorem yet. Noether's theorem maps a conserved quantity to each continuous symmetry. The relevant continuous symmetry needed to prove the conservation of energy is the one that leaves the laws of nature invariant, meaning the laws of physics don't change with time. Each continuous symmetry ...


2

This formula is used for conservative forces like gravitational force. This formula is used when the force is not constant i.e. variable force. This formula conveys that conservative force is equal to negative potential gradient. This formula establishes relation between a vector quantity and a scalar quantity(PE).


2

In inelastic collisions, kinetic energy is not conserved, so I'm going to assume you mean a totally elastic collision since you say energy is conserved. O.K, so when the ball hits the wall, the speed of the wall before and after is 0, so that means the kinetic energy of the ball is conserved and thus the magnitude of the velocity is the same before and after ...


2

There is a small error in your math by the way - the factor of $4$ should be a factor of $1$ - check the denominator of the quadratic equation. Why is this so unreasonable? At a glance, it looks like the momentum of the sail is a bit larger than what it started out with. If I make the reasonable assumptions that: The sail is non-relativistic ...


1

Imagine a book that we lift it with a force that is exactly equal to the force of gravity so the forces cancel out Ok, so sum of the forces is 0 and the acceleration is zero. and the book moves with a constant velocity. Spooky. Was the book moving initially? ...after the book has been lifted, and it has come to rest once again. According to ...


1

The quantity $mv^2$ is not constant. You can argue this by an example. Consider a ball in a uniform gravitational field. If you drop the ball, the quantity $mv^2$ changes as it falls, so it's not constant. Or, more simply, $mv^2$ is twice the kinetic energy. Think of any system where the kinetic energy changes, and you've shown that the quantity can't be ...


1

It is basically the same as two ordinary objects colliding and sticking together. The combined object's rest mass is the sum of the total energy of the original objects in the center-of-mass frame, which is their rest mass/energy plus their kinetic energy in that frame. Some of that would be carried away as gravitational radiation, but typically only a small ...


1

Let's say the book starts and stops from rest, as I believe you are assuming. The motion within this interval is unimportant, as you'll see. The increase in gravitational potential energy of the Earth-book system came from your body. You did positive work on the system since your hand force and displacement are in the same direction, resulting in an ...


1

Write down the potential and kinetic energy as a function of position. When the spring is in the middle of the motion, all is kinetic. When it is at the extreme of the range, all is potential. Somewhere between these two extremes, the potential and kinetic energies will be the same; their sum should always be constant (when there is no loss). Recall that ...


1

Suppose you have a constant angle $\theta$ slope in the original frame (we suppose that the transitions from horizontal movements to the slope is quasi-instantaneous). Call $x$ and $x'$ the horizontal displacements in the original and moving frame. Call $T$ the total time for going to $z=h$ to $z=0$. Then you have $x'= x- v_0 T$ With $ x= h \cot \theta$, ...


1

You can zoom in and look at just the wire; if you do that E and J are parallel(given some resistivity, we can call it z-hat) and B is parallel to the surface of the wire(we can call that theta-hat), so by definition you do get a poynting vector into your wire(z cross theta = -r). This is a transfer of energy from the flowing charges to the heat in the wire. ...


1

Some conservation laws are related to conservation of angular momentum. There is a famous example (from Feynman if I recall correctly), where you assume an infinite flat space and conservation of angular momentum about any point, and then you get conservation of linear momentum for free. Intuitively, to get say the $x$ component of linear momentum is ...



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