Kinetic energy of a mechanical system is the energy of mechanical movement of this system.
Force F, acting on a body at rest and causing it to move, does work, and the energy of a moving body increases by the amount of work expended. So the work dA strength F on the path that the body has passed during the increase in speed from 0 to v, it goes to increase kinetic energy dT bodies, i.e.
Using Newton's second law F=md v/dt
and multiplying both sides of the equality by the displacement d r, we get
F d r=m(d v/dt)dr=dA
Thus, a body of mass T, moving at speed v, has kinetic energy
T = tv 2 /2. (12.1)
From formula (12.1) it is clear that kinetic energy depends only on the mass and speed of the body, i.e. kinetic energy of a system is a function of the state of its motion.
When deriving formula (12.1), it was assumed that the motion was considered in an inertial frame of reference, since otherwise it would be impossible to use Newton’s laws. In different inertial reference systems moving relative to each other, the speed of the body, and therefore its kinetic energy, will not be the same. Thus, kinetic energy depends on the choice of reference frame.
Potential energy - mechanical energy of a system of bodies, determined by their relative position and the nature of the interaction forces between them.
Let the interaction of bodies be carried out through force fields (for example, a field of elastic forces, a field of gravitational forces), characterized by the fact that the work done by the acting forces when moving a body from one position to another does not depend on the trajectory along which this movement occurred, and depends only on the start and end positions. Such fields are called potential, and the forces acting in them are conservative. If the work done by a force depends on the trajectory of the body moving from one point to another, then such a force is called dissipative; an example of this is the force of friction.
A body, being in a potential field of forces, has potential energy II. The work done by conservative forces during an elementary (infinitesimal) change in the configuration of the system is equal to the increase in potential energy taken with a minus sign, since the work is done due to the decrease in potential energy:
Work d A expressed as the dot product of force F to move d r and expression (12.2) can be written as
F d r=-dP. (12.3)
Therefore, if the function P( r), then from formula (12.3) one can find the force F by module and direction.
Potential energy can be determined based on (12.3) as
where C is the integration constant, i.e. the potential energy is determined up to some arbitrary constant. This, however, is not reflected in the physical laws, since they include either the difference in potential energies in two positions of the body, or the derivative of P with respect to coordinates. Therefore, the potential energy of a body in a certain position is considered equal to zero (the zero reference level is chosen), and the energy of the body in other positions is measured relative to the zero level. For conservative forces
or in vector form
F=-gradP, (12.4) where
(i, j, k- unit vectors of coordinate axes). The vector defined by expression (12.5) is called gradient of the scalar P.
For it, along with the designation grad P, the designation P is also used. (“nabla”) means a symbolic vector called operatorHamilton or by nabla operator:
The specific form of the function P depends on the nature of the force field. For example, the potential energy of a body of mass T, raised to a height h above the Earth's surface is equal to
P = mgh,(12.7)
where is the height h is measured from the zero level, for which P 0 = 0. Expression (12.7) follows directly from the fact that potential energy is equal to the work done by gravity when a body falls from a height h to the surface of the Earth.
Since the origin is chosen arbitrarily, the potential energy can have a negative value (kinetic energy is always positive. !} If we take the potential energy of a body lying on the surface of the Earth as zero, then the potential energy of a body located at the bottom of the shaft (depth h"), P = - mgh".
Let's find the potential energy of an elastically deformed body (spring). The elastic force is proportional to the deformation:
F X control = -kx,
Where F x control - projection of elastic force onto the axis X;k- elasticity coefficient(for a spring - rigidity), and the minus sign indicates that F x control directed in the direction opposite to the deformation X.
According to Newton’s third law, the deforming force is equal in magnitude to the elastic force and directed oppositely to it, i.e.
F x =-F x control =kx Elementary work dA, performed by force F x at an infinitesimal deformation dx, is equal to
dA = F x dx = kxdx,
a full job
goes to increase the potential energy of the spring. Thus, the potential energy of an elastically deformed body
P =kx 2 /2.
The potential energy of a system, like kinetic energy, is a function of the state of the system. It depends only on the configuration of the system and its position in relation to external bodies.
Total mechanical energy of the system- energy of mechanical movement and interaction:
i.e., equal to the sum of kinetic and potential energies.
The part of mechanics in which motion is studied without considering the reasons causing this or that character of motion is called kinematics.Mechanical movement called a change in the position of a body relative to other bodies
Reference system called the body of reference, the coordinate system associated with it and the clock.
Body of reference name the body relative to which the position of other bodies is considered.
Material point is a body whose dimensions can be neglected in this problem.
Trajectory called a mental line that a material point describes during its movement.
According to the shape of the trajectory, the movement is divided into:
A) rectilinear- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of a curve.
Path is the length of the trajectory that a material point describes over a given period of time. This is a scalar quantity.
Moving is a vector connecting the initial position of a material point with its final position (see figure).
It is very important to understand how a path differs from a movement. The most main difference is that movement is a vector with a beginning at the point of departure and an end at the point of destination (it does not matter at all what route this movement took). And the path is, on the contrary, a scalar quantity that reflects the length of the trajectory traveled.
Uniform linear movement is a movement in which a material point makes equal movements in any equal intervals of time
Speed of uniform linear motion is called the ratio of movement to the time during which this movement occurred:
For uneven motion they use the concept average speed. Average speed is often introduced as a scalar quantity. This is the speed of such uniform motion in which the body travels the same path in the same time as during uneven motion:
Instant speed call the speed of a body at a given point in the trajectory or at a given moment in time.
Uniformly accelerated linear motion- this is a rectilinear movement in which the instantaneous speed for any equal periods of time changes by the same amount
The dependence of the body coordinates on time in uniform rectilinear motion has the form: x = x 0 + V x t, where x 0 is the initial coordinate of the body, V x is the speed of movement.
Free fall called uniformly accelerated motion with constant acceleration g = 9.8 m/s 2, independent of the mass of the falling body. It occurs only under the influence of gravity.
Free fall speed is calculated using the formula:
Vertical movement is calculated using the formula:
One type of motion of a material point is motion in a circle. With such movement, the speed of the body is directed along a tangent drawn to the circle at the point where the body is located (linear speed). You can describe the position of a body on a circle using a radius drawn from the center of the circle to the body. The displacement of a body when moving in a circle is described by rotating the radius of the circle connecting the center of the circle with the body. The ratio of the angle of rotation of the radius to the period of time during which this rotation occurred characterizes the speed of movement of the body in a circle and is called angular velocity ω:
Angular velocity is related to linear velocity by the relation
where r is the radius of the circle.
The time it takes a body to complete a full revolution is called circulation period. The reciprocal of the period is the circulation frequency - ν
Since during uniform motion in a circle the velocity module does not change, but the direction of the velocity changes, with such motion there is acceleration. He is called centripetal acceleration, it is directed radially towards the center of the circle:
Basic concepts and laws of dynamics
The part of mechanics that studies the reasons that caused the acceleration of bodies is called dynamics
Newton's first law:
There are such reference systems relative to which a body maintains its speed constant or is at rest if other bodies do not act on it or the action of other bodies is compensated.
The property of a body to maintain a state of rest or uniform linear motion with balanced external forces acting on it is called inertia. The phenomenon of maintaining the speed of a body under balanced external forces is called inertia. Inertial reference systems are systems in which Newton's first law is satisfied.
Galileo's principle of relativity:
in all inertial reference systems at the same initial conditions all mechanical phenomena proceed in the same way, i.e. subject to the same laws
Weight is a measure of body inertia
Force is a quantitative measure of the interaction of bodies.
Newton's second law:
The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:
$F↖(→) = m⋅a↖(→)$
The addition of forces consists of finding the resultant of several forces, which produces the same effect as several simultaneously acting forces.
Newton's third law:
The forces with which two bodies act on each other are located on the same straight line, equal in magnitude and opposite in direction:
$F_1↖(→) = -F_2↖(→) $
Newton's III law emphasizes that the action of bodies on each other is in the nature of interaction. If body A acts on body B, then body B acts on body A (see figure).
Or in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull a sled if these bodies interact with equal forces? This is possible only through interaction with the third body - the Earth. The force with which the hooves press into the ground must be greater than the frictional force of the sled on the ground. Otherwise, the hooves will slip and the horse will not move.
If a body is subjected to deformation, forces arise that prevent this deformation. Such forces are called elastic forces.
Hooke's law written in the form
where k is the spring stiffness, x is the deformation of the body. The “−” sign indicates that the force and deformation are directed in different directions.
When bodies move relative to each other, forces arise that impede the movement. These forces are called friction forces. A distinction is made between static friction and sliding friction. Sliding friction force calculated by the formula
where N is the support reaction force, µ is the friction coefficient.
This force does not depend on the area of the rubbing bodies. The friction coefficient depends on the material from which the bodies are made and the quality of their surface treatment.
Static friction occurs if the bodies do not move relative to each other. The static friction force can vary from zero to a certain maximum value
By gravitational forces are the forces with which any two bodies are attracted to each other.
Law of universal gravitation:any two bodies are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Here R is the distance between the bodies. The law of universal gravitation in this form is valid either for material points or for spherical bodies.
Body weight called the force with which the body presses on a horizontal support or stretches the suspension.
Gravity- this is the force with which all bodies are attracted to the Earth:
With a stationary support, the weight of the body is equal in magnitude to the force of gravity:
If a body moves vertically with acceleration, its weight will change.
When a body moves with upward acceleration, its weight
It can be seen that the weight of the body is greater than the weight of the body at rest.
When a body moves with downward acceleration, its weight
In this case, the weight of the body is less than the weight of the body at rest.
Weightlessness is the movement of a body in which its acceleration is equal to the acceleration of gravity, i.e. a = g. This is possible if only one force acts on the body - gravity.
Artificial Earth satellite- this is a body that has a speed V1 sufficient to move in a circle around the Earth
There is only one force acting on the Earth's satellite - the force of gravity directed towards the center of the Earth
First escape velocity- this is the speed that must be imparted to the body so that it revolves around the planet in a circular orbit.
where R is the distance from the center of the planet to the satellite.
For the Earth, near its surface, the first escape velocity is equal to
1.3. Basic concepts and laws of statics and hydrostatics
A body (material point) is in a state of equilibrium if the vector sum of the forces acting on it is equal to zero. There are 3 types of equilibrium: stable, unstable and indifferent. If, when a body is removed from an equilibrium position, forces arise that tend to bring this body back, this stable balance. If forces arise that tend to move the body further from the equilibrium position, this unstable position; if no forces arise - indifferent(see Fig. 3).
When we are not talking about a material point, but about a body that can have an axis of rotation, then in order to achieve an equilibrium position, in addition to the equality of the sum of forces acting on the body to zero, it is necessary that the algebraic sum of the moments of all forces acting on the body be equal to zero.
Here d is the force arm. Shoulder of strength d is the distance from the axis of rotation to the line of action of the force.
Lever equilibrium condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
Pressure is a physical quantity equal to the ratio of the force acting on a platform perpendicular to this force to the area of the platform:
Valid for liquids and gases Pascal's law:
pressure spreads in all directions without changes.
If a liquid or gas is in a gravity field, then each layer above presses on the layers below, and as the liquid or gas is immersed inside, the pressure increases. For liquids
where ρ is the density of the liquid, h is the depth of penetration into the liquid.
A homogeneous liquid in communicating vessels is established at the same level. If liquid with different densities is poured into the elbows of communicating vessels, then the liquid with a higher density is installed at a lower height. In this case
The heights of liquid columns are inversely proportional to densities:
Hydraulic Press is a vessel filled with oil or other liquid, in which two holes are cut, closed by pistons. The pistons have different areas. If a certain force is applied to one piston, then the force applied to the second piston turns out to be different.
Thus, the hydraulic press serves to convert the magnitude of the force. Since the pressure under the pistons must be the same, then 
Then A1 = A2.
A body immersed in a liquid or gas is acted upon by an upward buoyant force from the side of this liquid or gas, which is called by the power of Archimedes
The magnitude of the buoyancy force is determined by Archimedes' law: a body immersed in a liquid or gas is acted upon by a buoyant force directed vertically upward and equal to the weight of the liquid or gas displaced by the body:
where ρ liquid is the density of the liquid in which the body is immersed; V submergence is the volume of the submerged part of the body.
Body floating condition- a body floats in a liquid or gas when the buoyant force acting on the body is equal to the force of gravity acting on the body.
1.4. Conservation laws
Body impulse is a physical quantity equal to the product of a body’s mass and its speed:Momentum is a vector quantity. [p] = kg m/s. Along with body impulse, they often use impulse of power. This is the product of force and the duration of its action
The change in the momentum of a body is equal to the momentum of the force acting on this body. For an isolated system of bodies (a system whose bodies interact only with each other) law of conservation of momentum: the sum of the impulses of the bodies of an isolated system before interaction is equal to the sum of the impulses of the same bodies after the interaction.
Mechanical work called a physical quantity that is equal to the product of the force acting on the body, the displacement of the body and the cosine of the angle between the direction of the force and the displacement:
Power is the work done per unit of time:
The ability of a body to do work is characterized by a quantity called energy. Mechanical energy is divided into kinetic and potential. If a body can do work due to its motion, it is said to have kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula
If a body can do work by changing its position relative to other bodies or by changing the position of parts of the body, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated using the formula
where h is the lift height
Compressed spring energy:
where k is the spring stiffness coefficient, x is the absolute deformation of the spring.
The sum of potential and kinetic energy is mechanical energy. For an isolated system of bodies in mechanics, conservation law mechanical energy : if there are no frictional forces between the bodies of an isolated system (or other forces leading to energy dissipation), then the sum of the mechanical energies of the bodies of this system does not change (the law of conservation of energy in mechanics). If there are friction forces between the bodies of an isolated system, then during interaction part of the mechanical energy of the bodies turns into internal energy.
1.5. Mechanical vibrations and waves
Oscillations movements that have varying degrees of repeatability over time are called. Oscillations are called periodic if the values of physical quantities that change during the oscillation process are repeated at regular intervals.Harmonic vibrations are called such oscillations in which the oscillating physical quantity x changes according to the law of sine or cosine, i.e.
The quantity A equal to the largest absolute value of the fluctuating physical quantity x is called amplitude of oscillations. The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T is the time it takes for an oscillating body to complete one complete oscillation. Frequency of periodic oscillations The number of complete oscillations completed per unit of time is called:
Frequency is measured in s -1. This unit is called hertz (Hz).
Mathematical pendulum is a material point of mass m suspended on a weightless inextensible thread and oscillating in a vertical plane.
If one end of the spring is fixed motionless, and a body of mass m is attached to its other end, then when the body is removed from the equilibrium position, the spring will stretch and oscillations of the body on the spring will occur in the horizontal or vertical plane. Such a pendulum is called a spring pendulum.
Period of oscillation of a mathematical pendulum determined by the formula 
where l is the length of the pendulum.
Period of oscillation of a load on a spring determined by the formula 
where k is the spring stiffness, m is the mass of the load.
Propagation of vibrations in elastic media.
A medium is called elastic if there are interaction forces between its particles. Waves are the process of propagation of vibrations in elastic media.
The wave is called transverse, if the particles of the medium oscillate in directions perpendicular to the direction of propagation of the wave. The wave is called longitudinal, if the vibrations of the particles of the medium occur in the direction of wave propagation.
Wavelength is the distance between two closest points oscillating in the same phase:
where v is the speed of wave propagation.
Sound waves are called waves in which oscillations occur with frequencies from 20 to 20,000 Hz.
The speed of sound is different in different environments. The speed of sound in air is 340 m/s.
Ultrasonic waves are called waves whose oscillation frequency exceeds 20,000 Hz. Ultrasonic waves are not perceived by the human ear.
Due to its location in the field of action of forces. Another definition: potential energy is a function of coordinates, which is a term in the Lagrangian of the system and describes the interaction of elements of the system. The term "potential energy" was coined in the 19th century by Scottish engineer and physicist William Rankine.
The SI unit of energy is the Joule.
Potential energy is assumed to be zero for a certain configuration of bodies in space, the choice of which is determined by the convenience of further calculations. The process of choosing this configuration is called potential energy normalization.
A correct definition of potential energy can only be given in a 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.
Also, potential energy is a characteristic of the interaction of several bodies or a body and a field.
Any physical system tends to a state with the lowest potential energy.
More strictly, kinetic energy is the difference between the total energy of a system and its rest energy; thus, kinetic energy is the part of the total energy due to motion.
Kinetic energy
Let's consider a system consisting of one particle and write the equation of motion:
There is a resultant of all forces acting on a body.
Let us scalarly multiply the equation by the displacement of the particle. Considering that , we get:
- moment of inertia of the body
- angular velocity of the body.
Law of energy conservation.
The law of conservation of energy is a fundamental law of nature, established empirically, which states that the energy of an isolated (closed) physical system is conserved over time. In other words, energy cannot come from nothing and cannot disappear into nothing, it can only move from one form to another.
From a fundamental point of view, according to Noether’s theorem, the law of conservation of energy is a consequence of the homogeneity of time and in this sense is universal, that is, inherent in systems of very different physical natures. In other words, for each specific closed system, regardless of its nature, it is possible to determine a certain quantity called energy, which will be conserved over time. Moreover, the fulfillment of this conservation law in each specific system is justified by the subordination of this system to its specific laws of dynamics, which generally differ for different systems. However, in different branches of physics, for historical reasons, the law of conservation of energy is formulated differently, and therefore they talk about conservation various types
energy. For example, in thermodynamics, the law of conservation of energy is expressed as the first law of thermodynamics.
Since the law of conservation of energy does not apply to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it is more correct to call it not a law, but the principle of conservation of energy.
From a mathematical point of view, the law of conservation of energy is equivalent to the statement that a system of differential equations describing the dynamics of a given physical system has a first integral of motion associated with- a measure of the movement of matter in all its forms. The main property of all types of energy is interconvertibility. The energy reserve that the body possesses is determined by the maximum work that the body can do after completely using up its energy. Energy is numerically equal to the maximum work a body can do and is measured in the same units as work. When energy transfers from one type to another, you need to calculate the energy of the body or system before and after the transition and take their difference. This difference is usually called work:
Thus, the physical quantity characterizing the ability of a body to do work is called energy.
The mechanical energy of a body can be caused either by the movement of the body at a certain speed, or by the presence of the body in a potential field of forces.
Kinetic energy.
The energy that a body possesses due to its motion is called kinetic. The work done on a body is equal to the increase in its kinetic energy.
Let us find this work for the case when the resultant of all forces applied to the body is equal to .
The work done by the body due to kinetic energy is equal to the loss of this energy.
Potential energy.
If at each point in space other bodies act on a body, then the body is said to be in a field of forces or a force field.
If the line of action of all these forces passes through one point - the force center of the field - and the magnitude of the force depends only on the distance to this center, then such forces are called central, and the field of such forces is called central (gravitational, electric field of a point charge).
A field of forces that are constant in time is called stationary.
A field in which the lines of action of forces are parallel straight lines located at the same distance from each other is homogeneous.
All forces in mechanics are divided into conservative and non-conservative (or dissipative).
Forces whose work does not depend on the shape of the trajectory, but is determined only by the initial and final position of the body in space, are called conservative.
The work done by conservative forces along a closed path is zero. All central forces are conservative. Powers elastic deformation are also conservative forces. If only conservative forces act in the field, the field is called potential (gravitational fields).
Forces whose work depends on the shape of the path are called non-conservative (friction forces).
Potential energy- this is the energy that bodies or parts of the body possess due to their relative position.
The concept of potential energy is introduced as follows. If a body is in a potential field of forces (for example, in the gravitational field of the Earth), each point in the field can be associated with a certain function (called potential energy) so that the work A 12, performed over the body by field forces when it moves from an arbitrary position 1 to another arbitrary position 2, was equal to the decrease in this function along the path 1®2:
,
where and are the values of the potential energy of the system in positions 1 and 2.
 |
In each specific problem, it is agreed that the potential energy of a certain position of the body is equal to zero, and the energy of other positions is taken in relation to the zero level. The specific form of the function depends on the nature of the force field and the choice of the zero level. Since the zero level is chosen arbitrarily, it can have negative values. For example, if we take the potential energy of a body located on the Earth’s surface as zero, then in the field of gravity near the Earth’s surface, the potential energy of a body of mass m raised to a height h above the surface is equal to (Fig. 5).
where is the movement of the body under the influence of gravity;
The potential energy of the same body lying at the bottom of a hole of depth H is equal to
In the example considered, we were talking about the potential energy of the Earth-body system.
Gravitational potential energy - energy of a system of bodies (particles) caused by their mutual gravitational attraction.
For two gravitating point bodies with masses m 1 and m 2, the gravitational potential energy is equal to:
,
where =6.67·10 -11 is the gravitational constant,
r is the distance between the centers of mass of bodies.
The expression for gravitational potential energy is obtained from Newton’s law of gravitation, provided that for bodies at infinity gravitational energy is equal to 0. The expression for the gravitational force has the form:
On the other hand, according to the definition of potential energy:
Then .
Potential energy can be possessed not only by a system of interacting bodies, but by an individual body. In this case, the potential energy depends on the relative position of the parts of the body.
Let us express the potential energy of an elastically deformed body.
Potential energy of elastic deformation, if we assume that the potential energy of an undeformed body is zero;
Where k- coefficient of elasticity, x- body deformation.
In the general case, a body can simultaneously possess both kinetic and potential energies. The sum of these energies is called total mechanical energy body: .
The total mechanical energy of a system is equal to the sum of its kinetic and potential energies. The total energy of a system is equal to the sum of all types of energy that the system possesses.
The law of conservation of energy is the result of a generalization of many experimental data. The idea of this law belongs to Lomonosov, who outlined the law of conservation of matter and motion, and the quantitative formulation was given by the German physician Mayer and naturalist Helmholtz.
Law of conservation of mechanical energy: in a field of only conservative forces, the total mechanical energy remains constant in an isolated system of bodies. The presence of dissipative forces (friction forces) leads to dissipation (dissipation) of energy, i.e. converting it into other types of energy and violating the law of conservation of mechanical energy.
Law of conservation and transformation of total energy: the total energy of an isolated system is a constant quantity.
Energy never disappears or appears again, but only transforms from one type to another in equivalent quantities. This is the physical essence of the law of conservation and transformation of energy: the indestructibility of matter and its movement.
Example of the law of conservation of energy:
During the fall, potential energy is converted into kinetic energy, and the total energy is equal to mgH, remains constant.
Energy is a scalar quantity. The SI unit of energy is the Joule.
Kinetic and potential energy
There are two types of energy - kinetic and potential.
DEFINITION
Kinetic energy- this is the energy that a body possesses due to its movement:
DEFINITION
Potential energy is energy that is determined by the relative position of bodies, as well as the nature of the interaction forces between these bodies.
Potential energy in the Earth's gravitational field is the energy due to the gravitational interaction of a body with the Earth. It is determined by the position of the body relative to the Earth and is equal to the work of moving the body from this provision to zero level:
Potential energy is the energy caused by the interaction of body parts with each other. It is equal to the work of external forces in tension (compression) of an undeformed spring by the amount:
A body can simultaneously possess both kinetic and potential energy.
The total mechanical energy of a body or system of bodies is equal to the sum of the kinetic and potential energies of the body (system of bodies):
Law of energy conservation
For a closed system of bodies, the law of conservation of energy is valid:
In the case when a body (or a system of bodies) is acted upon by external forces, for example, the law of conservation of mechanical energy is not satisfied. In this case, the change in the total mechanical energy of the body (system of bodies) is equal to the external forces:
The law of conservation of energy allows us to establish a quantitative relationship between various forms movement of matter. Just like , it is valid not only for, but also for all natural phenomena. The law of conservation of energy says that energy in nature cannot be destroyed just as it cannot be created from nothing.
In the most general view The law of conservation of energy can be formulated as follows:
- Energy in nature does not disappear and is not created again, but only transforms from one type to another.
Examples of problem solving
EXAMPLE 1
| Exercise | A bullet flying at a speed of 400 m/s hits an earthen shaft and travels 0.5 m to a stop. Determine the resistance of the shaft to the movement of the bullet if its mass is 24 g. |
| Solution | The resistance force of the shaft is external force, so the work done by this force is equal to the change in the kinetic energy of the bullet: Since the resistance force of the shaft is opposite to the direction of movement of the bullet, the work done by this force is: Change in bullet kinetic energy: Thus, we can write: where does the resistance force of the earthen rampart come from: Let's convert the units to the SI system: g kg. Let's calculate the resistance force: |
| Answer | The shaft resistance force is 3.8 kN. |
EXAMPLE 2
| Exercise | A load weighing 0.5 kg falls from a certain height onto a plate weighing 1 kg, mounted on a spring with a stiffness coefficient of 980 N/m. Determine the magnitude of the greatest compression of the spring if at the moment of impact the load had a speed of 5 m/s. The impact is inelastic. |
| Solution | Let us write down a load + plate for a closed system. Since the impact is inelastic, we have: where does the velocity of the plate with the load after impact come from:
According to the law of conservation of energy, the total mechanical energy of the load together with the plate after impact is equal to the potential energy of the compressed spring: |
