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9

To apply Noether's theorem, which is what you are alluding to here, one needs to look at continuous symmetries of a Lagrangian description of a system's dynamics. The damped oscillation equation you have written, although it is invariant with respect to a time translation as you rightly say, is not a Lagrangian description. If you write the Lagrangian for ...


4

Conservation of energy is related to time translation invariance for systems that can be described by a Lagrangian. Dissipative systems in general are not describable by Lagrangian mechanics (without altering the formalism that is) and so Noether's theorem cannot be applied to check whether or not energy is conserved. EDIT: the dissipative system the OP ...


2

When the wheel comes in contact with the belt friction will act as there will be relative motion between the belt and the point of contact. Now friction will tend to act on the belt opposite to the velocity of the belt until slipping ceases. To find the work done by the external agency lets consider the energy changes : 1) The K.E of the wheel increases. 2) ...


1

The force of friction acts both towards the centre of the circle and opposite the velocity vector of the car. Strictly speaking, the diagram you have does not show all forces acting on the car but it is enough for purposes of explaining the circular motion. As the text also explains, circular motion always requires a force pointed radially inwards because ...


1

I would look at this in a slightly different way. Rearranging it: $$ m \ddot{x} = -(a|\dot{x}|+k) x = -k_{eff} x$$ If you look at it that way, it is really a variable, non-linear stiffness $k_{eff}$ that depends on the velocity, rather than a damping that depends on the position. In this respect (assuming $a > 0$), the stiffness coefficient has a lower ...


1

As already explained in other answers and comments in General Relativity (GR) energy is not conserved. Some people and physicists say it is, it simply gets lost by the matter-energy and gained by the gravitational field, or viceversa; this is more pleasing to our sense of conservation of something, but it has problems in that the gravitational energy, is not ...


1

Half of your questions are concerning Newton's law: Why is it non-uniform? Because the object would have different densities in different parts, so weight would be greater in some parts? Yes. Think of a car. It is in contact with the ground in four places and pushes down causing four normal forces. If the car is heavily loaded with bagage in the ...


1

The differential equation you quote is fairly standard in university physics/engineering course but definitely requires some calculus to solve. As a first step, if you know how to differentiate products and chains, you can substitute the given solution into the differential equation and verify that it is indeed a solution. It contains two arbitrary constants ...


1

It depends also on the slope of the surface, for example, during WW2: "Among the features of the Soviet tank [T34] considered most significant were the sloping armour, which gave much improved shot deflection ...", sloping amour helps deflect incoming projectiles (look here).


1

The primary mechanism by which explosions cause damage to materials is through the momentum transferred by the shock wave. I don't see how a smoother surface would mitigate this. However, there are plenty of other ways to strengthen materials, an ancient example being work hardening.


1

The only way that friction can appear is for there to be tension, since it is tension that will give rise to the normal force needed for friction. Now if we note that the friction must result in a difference in tension between the left and right strings (if the masses are different) then there will be a continuous change in tension. The normal force at ...


1

If there is no friction the energy you put in initially will be conserved and the flywheel will rotate forever, then you don't need to put in any extra power to keep it running.



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