From what more general expression is the kinetic energy formula derived?
The formula can be deduced from the definition of work as the difference of kinetic energies A = Ek2-Ek1.
And formulas: work A = F * S (power * way).
As F = m * a then A = m * a * S
Moreover, from the kinematics acceleration: a = (V2-V1) / t
S = (V2 + V1) * t / 2-path with uniformly accelerated movement.
We substitute these quantities in the formula of work: A = m * ((V2-V1) / t) * ((V2 + V1) * t / 2)
we reduce the expression by t and the brackets with the sum and the difference of speeds, we transform into the difference of the squares of speeds:
We expand the brackets: A = m * V2 ^ 2/2 - m * V1 ^ 2/2.
Thus, the difference in the last formula corresponds to the very first formula.
We obtain formulas for kinetic energy at each point:
Ek2 = m * V2 ^ 2/2
Ek1 = m * V1 ^ 2/2
First, the potential energy formula is derived, and the kinetic energy formula is already derived from it. The formula of potential energy was received by Isaac Newton in his famous book "Mathematical Principles of Natural Philosophy". He reasoned roughly as follows.
Let some object lie on my palm. I will raise the palm with the object very slowly and evenly so that the reaction force of the palm N is balanced by the gravity of the object P, and the kinetic energy would be practically zero due to the very low speed. Where does the work A = INT (P dh) = mgh, which I do on the subject, go? It is transformed into the latent potential energy of the object, which can turn into a clear kinetic energy if the object is allowed to fall freely.
Now look at the mistake Newton made. If several forces F1, F2, F3 and so on act at once on an object, then to calculate the total energy produced by all forces together, you need to substitute the resulting force, and not one of the particular forces, under the integral sign. And Newton framed private power, the power of weight. Since in the case considered by him, the resulting force is zero (the weight force is balanced by the force of the palm reaction), a correct calculation will show zero work. And if the work is zero, then the energy of the object does not change. And if it was equal to zero at the starting point of the rise, it will remain equal to zero regardless of the height of the rise. In other words, potential energy does not exist in nature. But in practice, we are well aware that the lifting of any heavy object is accompanied by the expenditure of energy. So the conclusion about zero work is wrong? No, he is correct. It’s just that the work will not be carried out on the item being lifted, but on something else. And the mgh formula does not describe the potential energy of an object, but the energy of something else.
Now we turn to kinetic energy. In kinematics (the science of uniform and non-uniform motion) there is such a formula V1 V1 - V0 V0 = 2aS for accelerated motion, where V0 is the initial speed, V1 is the final speed, a is the acceleration, S is the length of the path traveled. If at the initial moment of time the speed of the object V0 was equal to zero, then expressing the product of acceleration by the length and substituting it in the potential energy formula, we obtain mVV / 2, that is, the kinetic energy formula. And now we will reason. If the mgh complex does not describe the potential energy of the object, but something else, then the formula mVV / 2 obtained from it will also describe not the kinetic energy of the object, but the energy of something else. And what exactly - I’ll try to explain it now.
When we raise any object, we do not overcome the resistance of the object, but of the gravitational field. Therefore, we will work on the gravitational field and increase its energy by the value of E = mgh. And when we throw an object, through its accelerated movement we deform the structure of the physical vacuum surrounding us, we work on it and increase its energy by E = mVV / 2. Thus, instead of potential energy, there is the energy of the gravitational field, and instead of kinetic energy, there is the energy of a physical vacuum.
9. Conservative and non-conservative forces. The connection between power and
potential energy. Gradient of potential energy. Condition equal
The scalar energy approach in mechanics is especially fruitful in the case of the so-called conservativeinteractions, in which the work of stationary forces does not depend on the shape of the trajectory, but is determined only by the initial and final positions of the body.
the forces of gravitational interaction, the forces of elasticity, but not the forces of friction and resistance, are conservative. For conservative forces, one can introduce such an energy characteristic aspotential energy, which is an unambiguous function of coordinates (position) and which, together with kinetic energy - a function of velocities, forms the total mechanical energy of the body (systems).
Unlike kinetic energy Eto = m 2 2, which is a unique, uniformly expressed function of velocities and, in the sense, a scalar dynamic measure of motion, potential energy EP - is a scalar measure of conservative interactions and does not have a uniform expression through the coordinates (position) of the body.
Conservative forces - forces whose work does not depend on the shape of the trajectory along which the body moves and is determined at the start and end points of the trajectory, the work of these forces in a closed loop = 0
Dissipative forces - forces whose work depends on the shape of the trajectory along which the body moves.
The interaction in the result of the cat between the bodies results in a sweat of force, which is carried out by means of a force sweat field.
The relationship between power and potential energy. Gradient of potential energy.
On the body, the position of which in the sweat field is determined by the radius vector r: F = xi + yj + zk
Gradient - an operator showing what actions need to be carried out with a scalar function. Is a vector directed towards the most rapid increase in the scalar function. Then the connection between F and En is formed as follows: force = gradEn taken with the opposite sign => F is directed towards the opposite grad.
Forces that depend only on coordinates (Forces that do not depend on time, are called stationary), can be set using force fields - areas of space, at each point of which a certain force acts on the body. Examples of force fields are the gravitational field and, in particular, the field of gravity, electrostatic field, etc.
Forces (and fields), workBUT12which on the path between any two points 1 and 2 does not depend on the shape of the trajectory between themare called potential, and if they are stationary, they are called toonservative. Potential are all homogeneous fields (at each point of such fields the force is unchanged), as well as fields central forces (they depend only on the distance between the interacting points and are directed along the straight line connecting them).
We obtain the formula for the relationship between the strength of such fields and potential energy. From the relationship of work with potential energy A12 = Fdr = Ep1 - En2 , or, for elementary work: А = Fdr = - dЕP. Bearing in mind that Fdr = Fsds, where ds = dr is the elementary path / displacement /, and Fr = Fcos - projection of the vector F to move drwrite: Frds = - dЕPwhere - dЕP - there is a decrease in potential energy in the direction of displacement dr. From here fr= - ЕPr, the partial derivative r is taken in some given direction.
In vector form, the resulting differential relationship of force with potential energy can be written as follows:
F = -(iЕPx + jЕPU + kЕPz) = - grad ЕP = - EPwhere is the symbolic vector operator (the vector sum of the first partial derivatives with respect to spatial coordinates) is called the Nabl operator or gradient scalar function (in this case, potential energy).
So power F = - grad EP = - EP in the potential field there is an anti-gradient / gradient with a minus sign / potential energy, or, otherwise - the spatial derivative, the speed of decrease of potential energy in space in a certain direction.
The meaning of the gradient can be clarified by introducing the concept of epotential surface - in all points of which potential energy EPhas the same meaning, i.e.. EP=const.
From the formula F = - EP it follows that the projection of the vector F to the direction of the tangent to the equipotential surface at any point equal to zero. This means that the vector F normal to equipotential surface EP = const.
If, further, take the dr drPthen dЕP 0, i.e., a vector F directed down EP. The gradient from EP there is a vector directed normal to the equipotential surface in the direction of the fastest increase in the scalar function / here - potential energy /.
By the example of a gravitational field, the force of which is directly proportional to the mass of the body, i.e., F = m1m2r 2, we can assume that each of the interacting bodies is in the force field of the other: F = mМr 2 = gm, where g = Fm = Мr 2 is the gravitational field strength / specific force - calculated per unit mass / created by a body of mass M.
From the relationship of force with potential energy follows:
or gdr = 1 - 2 where = EP/ m is the potential of the gravitational field, which is the specific / per unit mass / potential energy.
Or g = - grad = - is the formula for the relationship between tension and the potential of the gravitational field, tension is the antigradient of the potential.
Let the particle move in a one-dimensional potential field whose profile, i.e., the dependence EP (x) is presented in the figure in the form of the so-called potential curve.
From the law of conservation of mechanical energy: E = Eto + EP = m 2 2 + EP/ x / = const it follows that in the region where EP > E particle can not get. Thus, if the total energy E of a particle is equal to E1 /cm. fig. /, then the particle can move in the region between the x coordinates1 them2 (oscillates in this region, called the potential well), or in the region , to the right of the x coordinate3. But the particle cannot go from region I to region II or vice versa, a potential barrier of height E prevents thisb E1separating these areas.
Particle with energy E2greater height of the potential barrier (E2 Eb), can move in the entire area to the right of xabout. Its kinetic energy will increase (in the region of xabout to x ), then fall (in the region from x to x ) and then increase again in the region x x .
At the point x there is a stable equilibrium, here EP = En min and Fx = -gradx EP = - ЕPх = 0. When a body is displaced from it by dx 0, dЕP 0 and the force acts on the body
Fx = - ЕPx 0, which is of a character that returns the body to the equilibrium position.
At the point x there is an unstable equilibrium,
here EP = En max and F = - grad EP = - ЕPх = 0. When a body is displaced from it by dx 0, dЕP 0, and the force F acts on the bodyx = - ЕPх 0, which has a character that deviates the body from the equilibrium position.