Potential kinetic and total mechanical energy. §2.6 Kinetic energy
To increase the distance of a body from the center of the Earth (raise the body), work must be done on it. This work against gravity is stored in the form potential energy bodies.
In order to understand what it is potential energy body, we will find the work done by gravity when moving a body of mass m vertically down from a height above the Earth’s surface to a height .
If the difference is negligible compared to the distance to the center of the Earth, then the gravitational force during the movement of the body can be considered constant and equal to mg.
Since the displacement coincides in direction with the gravity vector, it turns out that the work of gravity is equal to
From the last formula it is clear that the work of gravity when transferring a material point of mass m into the gravitational field of the Earth is equal to the difference between two values of a certain quantity mgh. Since work is a measure of energy change, the right side of the formula contains the difference between the two energy values of this body. This means that the value mgh represents the energy due to the position of the body in the Earth's gravitational field.
The energy caused by the relative position of interacting bodies (or parts of one body) is called potential and denoted by Wp. Therefore, for a body located in the gravitational field of the Earth,
The work done by gravity is equal to the change body potential energy, taken with the opposite sign.
The work of gravity does not depend on the trajectory of the body and is always equal to the product of the gravity modulus and the difference in heights in the initial and final positions
Meaning potential energy of a body raised above the Earth depends on the choice of the zero level, that is, the height at which the potential energy is assumed to be zero. It is usually assumed that the potential energy of a body on the Earth's surface is zero.
With this choice of zero level body potential energy, located at a height h above the Earth’s surface, is equal to the product of the body’s mass by the gravitational acceleration modulus and its distance from the Earth’s surface:
From all of the above, we can conclude: the potential energy of a body depends on only two quantities, namely: from the mass of the body itself and the height to which this body is raised. The trajectory of a body does not affect the potential energy in any way.
A physical quantity equal to half the product of the rigidity of a body by the square of its deformation is called the potential energy of an elastically deformed body:
The potential energy of an elastically deformed body is equal to the work done by the elastic force when the body transitions to a state in which the deformation is zero.
There is also:
In the formula we used.
Work is performed in nature whenever any body in the direction of its movement or against it is acted upon by a force (or several forces) from another body (other bodies).
Job force is equal to the product of the moduli of force and displacement of the point of application of the force and the cosine of the angle between them.
A= F · S cos , Where AJ); F – force, ( N); S- movement, ( m).
Energy is neither created nor destroyed, but only transformed from one form to another: from kinetic to potential and vice versa. Given the value Ek and Ep, the law of conservation of mechanical
energy can be written as follows:
In state 2, the body has kinetic energy (since it has already developed speed), but the potential energy has decreased, since h 2 is less than h 1 . Part of the potential energy turned into kinetic energy.
State 3 is the state just before stopping. The body seemed to have just touched the ground, while the speed was maximum. The body has maximum kinetic energy. Potential energy is zero (the body is on Earth).
The total mechanical energies are equal if we neglect the force of air resistance.
Denoting "action". You can call an energetic person who moves, creates certain work, can create, act. Machines created by people, living things and nature also have energy. But this is in ordinary life. In addition, there is a strict one that has defined and designated many types of energy - electric, magnetic, atomic, etc. However, now we will talk about potential energy, which cannot be considered in isolation from kinetic energy.
Kinetic energy
This energy, according to the concepts of mechanics, is possessed by all bodies that interact with each other. And in in this case we're talking about about the movement of bodies.
Potential energy
A=Fs=Ft*h=mgh, or Ep=mgh, where:
Ep - potential energy of the body,
m - body weight,
h is the height of the body above the ground,
g is the acceleration of free fall.
Two types of potential energy
Potential energy has two types:
1. Energy in the relative position of bodies. A suspended stone has such energy. Interestingly, ordinary wood or coal also has potential energy. They contain unoxidized carbon that can oxidize. To put it simply, burnt wood can potentially heat up the water.
2. Energy elastic deformation. Examples here include an elastic band, a compressed spring, or a “bone-muscle-ligament” system.
Potential and kinetic energy are interrelated. They can transform into each other. For example, if a stone is upward, when it moves, it first has kinetic energy. When it reaches a certain point, it will freeze for a moment and gain potential energy, and then gravity will pull it down and kinetic energy will arise again.
Kinetic energy- energy mechanical system, depending on the speed of movement of its points. The kinetic energy of translational and rotational motion is often released. The SI unit of measurement is Joule. More strictly, kinetic energy is the difference between the total energy of a system and its rest energy; thus kinetic energy is part total energy, caused by movement.
Let us consider the case when a body of mass m there is a constant force (it can be the resultant of several forces) and force vectors and the movements are directed along one straight line in one direction. In this case, the work done by the force can be defined as A = F∙s. The modulus of force according to Newton's second law is equal to F = m∙a, and the displacement module s with uniformly accelerated rectilinear motion is associated with the modules of the initial υ 1 and final υ 2 speed and acceleration A expression
From here we get to work
A physical quantity equal to half the product of a body’s mass and the square of its speed is calledkinetic energy of the body .
Kinetic energy is represented by the letter E k .
Then equality (1) can be written as follows:
A = E k 2 – E k 1 . (3)
Kinetic energy theorem:
the work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body.
Since the change in kinetic energy is equal to the work of force (3), the kinetic energy of the body is expressed in the same units as the work, i.e. in joules.
If the initial speed of movement of a body of mass T is zero and the body increases its speed to the value υ , then the work done by the force is equal to the final value of the kinetic energy of the body:
(4)
Physical meaning kinetic energy:
The kinetic energy of a body moving with a speed v shows how much work must be done by a force acting on a body at rest in order to impart this speed to it.
Potential energy- the minimum work that must be done to move a body from a certain reference point to a given point in the field of conservative forces. Second definition: potential energy is a function of coordinates, which is a term in the Lagrangian of the system and describes the interaction of the elements of the system. Third definition: potential energy is the energy of interaction. Units [J]
The potential energy is assumed to be zero for a certain point in space, the choice of which is determined by the convenience of further calculations. The process of selecting a given point is called potential energy normalization. It is also clear that the correct definition of potential energy can only be given in the field of forces, the work of which depends only on the initial and final position of the body, but not on the trajectory of its movement. Such forces are called conservative.
The potential energy of a body raised above the Earth is the energy of interaction between the body and the Earth by gravitational forces. The potential energy of an elastically deformed body is the energy of interaction of individual parts of the body with each other by elastic forces.
Potential are calledstrength , the work of which depends only on the initial and final position of a moving material point or body and does not depend on the shape of the trajectory.
In a closed trajectory, the work done by the potential force is always zero. Potential forces include gravitational forces, elastic forces, electrostatic forces and some others.
Powers , the work of which depends on the shape of the trajectory, are callednon-potential . When a material point or body moves along a closed trajectory, the work done by the nonpotential force is not equal to zero.
Potential energy of interaction of a body with the Earth.
Let's find the work done by gravity F t when moving a body of mass T vertically down from a height h 1 above the Earth's surface to a height h 2 (Fig. 1).
If the difference h 1 –h 2 is negligible compared to the distance to the center of the Earth, then the force of gravity F T during the movement of the body can be considered constant and equal mg.
Since the displacement coincides in direction with the gravity vector, the work done by gravity is equal to
A = F∙s = m∙g∙(h l – h 2). (5)
Let us now consider the movement of a body along an inclined plane. When moving a body down an inclined plane (Fig. 2), the force of gravity F T = m∙g does work
A = m∙g∙s∙cos a = m∙g∙h, (6)
Where h– height of the inclined plane, s– displacement module equal to the length of the inclined plane.
Movement of a body from a point IN exactly WITH along any trajectory (Fig. 3) can be mentally imagined as consisting of movements along sections of inclined planes with different heights h", h" etc. Work A gravity all the way from IN V WITH equal to the sum of work on individual sections of the route:
Where h 1 and h 2 – heights from the Earth’s surface at which the points are located, respectively IN And WITH.
Equality (7) shows that the work of gravity does not depend on the trajectory of the body and is always equal to the product of the gravity modulus and the difference in heights in the initial and final positions.
When moving downward, the work of gravity is positive, when moving up it is negative. The work done by gravity on a closed trajectory is zero .
Equality (7) can be represented as follows:
A = – (m∙g∙h 2 – m∙g∙h l). (8)
A physical quantity equal to the product of the mass of a body by the acceleration modulus of free fall and the height to which the body is raised above the surface of the Earth is calledpotential energy interaction between the body and the Earth.
Work done by gravity when moving a body of mass T from a point located at a height h 2 , to a point located at a height h 1 from the Earth's surface, along any trajectory, is equal to the change in the potential energy of interaction between the body and the Earth, taken with the opposite sign.
A= – (ER 2 – ER 1). (9)
Potential energy is indicated by the letter ER.
The value of the potential energy of a body raised above the Earth depends on the choice of the zero level, i.e., the height at which the potential energy is assumed to be zero. It is usually assumed that the potential energy of a body on the Earth's surface is zero.
With this choice of the zero level, the potential energy ER body at height h above the Earth's surface is equal to the product of mass m bodies to the free fall acceleration module g and distance h it from the surface of the Earth:
Ep = m∙g∙h. (10)
Physical meaning potential energy of interaction of a body with the Earth:
the potential energy of a body on which gravity acts is equal to the work done by gravity when moving the body to the zero level.
Unlike the kinetic energy of translational motion, which can only have positive values, the potential energy of a body can be both positive and negative. Body mass m, located at a height h, Where h 0 ( h 0 – zero height), has negative potential energy:
Ep = –m∙gh
Potential energy of gravitational interaction
Potential energy of gravitational interaction of a system of two material points with masses T And M, located at a distance r one from the other is equal
(11)
Where G is the gravitational constant, and the zero of the potential energy reference ( Ep= 0) accepted at r = ∞. Potential energy of gravitational interaction of a body with mass T with the Earth, where h– height of the body above the Earth’s surface, M 3 – mass of the Earth, R 3 is the radius of the Earth, and the zero of the potential energy reading is chosen at h= 0.
(12)
Under the same condition of choosing zero reference, the potential energy of gravitational interaction of a body with mass T with Earth for low altitudes h(h« R 3) equal to
Ep = m∙g∙h,
where is the magnitude of the acceleration due to gravity near the Earth's surface.
Potential energy of an elastically deformed body
Let us calculate the work done by the elastic force when the deformation (elongation) of the spring changes from a certain initial value x 1 to final value x 2 (Fig. 4, b, c).
The elastic force changes as the spring deforms. To find the work of the elastic force, you can take the average value of the force modulus (since the elastic force depends linearly on x) and multiply by the displacement module:
Where 
From here
(14)
A physical quantity equal to half the product of the rigidity of a body by the square of its deformation is calledpotential energy elastically deformed body:
From formulas (14) and (15) it follows that the work of the elastic force is equal to the change in the potential energy of an elastically deformed body, taken with the opposite sign:
A = –(ER 2 – ER 1). (16)
If x 2 = 0 and x 1 = x, then, as can be seen from formulas (14) and (15),
ER = A.
Then physical meaning potential energy of a deformed body
the potential energy of an elastically deformed body is equal to the work done by the elastic force when the body transitions to a state in which the deformation is zero.
