Understanding and Derivation of the Kinetic Energy Theorem Formula

Kinetic energy theorem

The kinetic energy is instantaneous, which means that the work done by the force on the object in one process is equal to the change of the kinetic energy in the process. Kinetic energy is a state quantity with no negative values.

The external force (the sum of the external forces received by the object, according to the direction and the magnitude of the force can be calculated by the orthogonal method [1] can calculate the direction and size of the final resultant force of the object). The work done on the object is equal to the change of the kinetic energy of the object. The end kinetic energy reduces the initial kinetic energy.

The kinetic energy theorem generally only refers to the beginning and end state of the motion of the object, and the amount of change in the state of the beginning and the end is obtained by the transformation of the energy during the movement. But the total energy can follow the law of conservation of energy, and the transformation of energy includes changes in energy such as kinetic energy, potential energy, thermal energy, and light energy (not involved in high school).

Analysis of the understanding and derivation of the kinetic energy theorem formula :

(1) Determine the research object. The research object can be a single particle (single) or a system.

(2) Analyze the stress condition and motion of the research object, whether it is the problem of solving the relationship between force, displacement and speed.

(3) If yes, solve it according to the kinetic energy theorem ΔW=ΔE.

Handling multiple process issues:

The key to applying the kinetic energy theorem to deal with multi-process motion problems lies in the fact that there are several forces in the whole process, and the kinetic energy of the state at the beginning and end. The kinetic energy theorem is used to deal with the problem without considering the specific motion process. The total distance and number of motion problems, if using Newton's law and kinematics formula, must use the method of summation of series, but for solving some of these problems, such as solving the kinetic energy theorem, many complicated mathematical deductions can be omitted. To simplify the problem solving process.

Derivation steps:

For uniform acceleration linear motion:

According to Newton’s second law of motion,

1

Uniform acceleration linear motion law,

2

1×2,

External force Remember

which is

For non-uniform acceleration linear motion,

Infinitely subdivided into n segments, so each segment can be seen as a uniform acceleration linear motion (micro-element method)

For every sport,

W1=Ek1-Ek0

W2=Ek2-Ek1

......

Wn=Ekn-Ek(n-1) adds all the above formulas

Derived

The basic steps of applying kinetic energy to understand the problem

The basic steps of using kinetic energy to understand the problem can be summarized in six sentences and eighteen words, namely, the fixed object, the Ming process, the force analysis, the expression work, the column equation, and the solution.

The key to applying kinetic energy to understand the problem

It is not difficult to see from the kinetic energy theorem that the key to the application of kinetic energy is to be on the left side of the equation, that is, the algebraic sum of the work of each force. In addition to knowing that the object is subjected to several forces, it is more important to grasp the characteristics of each force. The method of solving the size of the work. The characteristics and solutions of several types of work are summarized as follows:

1) Gravity, electric field force and other conservative forces do not depend on the path of movement, only related to the two positions. For example, gravity work is only related to the height difference between the beginning and the end;

2) The direction is always perpendicular to the direction of the speed of the force does not work. Typically, the elastic force of the string or the light rod does not work when the string or the light rod is held in a circular motion in a vertical plane. The elastic force of the contact surface when the object moves on the contact surface and the Lorentz force of the charged particle in the magnetic field also do not work (this conclusion can be proved by the micro-element method);

3) The same size, the direction is always opposite to the speed direction, such as friction or air resistance, its work is equal to one F road, where s road represents the total distance traveled by the object, pay attention to the total distance instead of the total Displacement! (This conclusion can also be proved by the micro-element method)

4) The sum of the work done by the same rope or light rod on the object connected to it must be zero. This conclusion is very effective in solving the problem of conservation of mechanical energy of the system;

5) The sum of the work of a pair of static friction forces must be zero (meaning that static friction does not produce thermodynamic energy). The sum of the pair of sliding friction forces must be negative (meaning that the sliding friction work must produce thermodynamic energy).

Kinetics

As shown in Fig. 1, the dynamic friction factor between the object of mass m and the horizontal turntable is that the distance between the object and the rotating shaft of the turntable is R, and the object is rotated with the turntable from the standstill. When the speed increases to a certain value, the object is about to start to slide. In this process, the work done by the friction on the object is ()

Analysis: The frictional force of the object provides the centripetal force required for the circular motion of the object. Obviously, its size and direction change, so it belongs to the general variable force. The work of solving the work can only use the kinetic energy theorem. When the object is about to start sliding, it means that the frictional force of the object is reached, 2 max. From the centripetal force equation Then by the kinetic energy theorem (the object is subjected to three forces, of which only friction works) Two-way joint solution The answer is A.

Borrow the title:

Applying the kinetic energy theorem to solve the problem of changing the force to do work; 1 The variable force sought is not necessarily the total work, so the work of the force required is not necessarily equal to AE; 2 the total work done by the force on the object corresponds to the kinetic energy of the object Change, not the kinetic energy of the corresponding object; 3 If there are multiple forces to do work, it is necessary to clarify the positive and negative of each force. If the function of the variable force to be sought is negative work, it can be said that the work done by overcoming the force is W, then a W is applied in the expression, and the work of the variable force is W, and the letter w itself has a negative sign. 4 Although the kinetic energy theorem is derived from the action of constant force and linear motion, it is also applicable to the case of variable force action and curve motion.

Applying the kinetic energy theorem to solve the total distance of reciprocating motion (applying kinetic energy theorem to solve multi-process problems)

For the same magnitude, the direction is always opposite or the same as the speed direction, such as sliding friction, air resistance, etc. When the object is reciprocating, its work is equal to the product of the force and the path traveled by the object. According to this feature, the total distance traveled by the object can be solved.

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