Mechanical work is an energy characteristic of the movement of physical bodies, which has a scalar form. It is equal to the modulus of the force acting on the body, multiplied by the modulus of the displacement caused by this force and by the cosine of the angle between them.
Formula 1 - Mechanical work.
F - Force acting on the body.
s - Body movement.
cosa - Cosine of the angle between force and displacement.
This formula has general form. If the angle between the applied force and the displacement is zero, then the cosine is equal to 1. Accordingly, the work will be equal only to the product of the force and the displacement. Simply put, if a body moves in the direction of application of force, then mechanical work is equal to the product of force and displacement.
Second special case, when the angle between the force acting on the body and its displacement is 90 degrees. In this case, the cosine of 90 degrees is equal to zero, so the work will be equal to zero. And indeed, what happens is that we apply force in one direction, and the body moves perpendicular to it. That is, the body clearly does not move under the influence of our force. Thus, the work done by our force to move the body is zero.
Figure 1 - Work of forces when moving a body.
If more than one force acts on a body, then the total force acting on the body is calculated. And then it is substituted into the formula as the only force. A body under the influence of force can move not only rectilinearly, but also along an arbitrary trajectory. In this case, the work is calculated for a small section of movement, which can be considered rectilinear, and then summed up along the entire path.
Work can be both positive and negative. That is, if the displacement and force coincide in direction, then the work is positive. And if a force is applied in one direction, and the body moves in another, then the work will be negative. An example of negative work is the work of a frictional force. Since the friction force is directed counter to the movement. Imagine a body moving along a plane. A force applied to a body pushes it in a certain direction. This force does positive work to move the body. But at the same time, the friction force does negative work. It slows down the movement of the body and is directed towards its movement.
Figure 2 - Force of motion and friction.
Mechanical work is measured in Joules. One Joule is the work done by a force of one Newton when moving a body one meter. In addition to the direction of movement of the body, the magnitude of the applied force can also change. For example, when a spring is compressed, the force applied to it will increase in proportion to the distance traveled. In this case, the work is calculated using the formula.
Formula 2 - Work of compression of a spring.
k is the spring stiffness.
x - moving coordinate.
1. Mechanical work \(A \) is a physical quantity equal to the product of the force vector acting on the body and the vector of its displacement:\(A=\vec(F)\vec(S) \) . Work is a scalar quantity, characterized by a numerical value and a unit.
A unit of work is taken to be 1 joule (1 J). This is the work done by a force of 1 N along a path of 1 m.
\[ [\,A\,]=[\,F\,][\,S\,]; [\,A\,]=1Н\cdot1m=1J\]
2. If the force acting on the body makes a certain angle \(\alpha \) with the displacement, then the projection of the force \(F \) onto the X axis is equal to \(F_x \) (Fig. 42).
Since \(F_x=F\cdot\cos\alpha \) , then \(A=FS\cos\alpha \) .
Thus, the work of a constant force is equal to the product of the magnitudes of the force and displacement vectors and the cosine of the angle between these vectors.
3. If force \(F\) = 0 or displacement \(S \) = 0, then mechanical work is zero \(A \) = 0. Work is zero if the force vector is perpendicular to the displacement vector, t .e. \(\cos90^\circ \) = 0. So, the work of the force imparting centripetal acceleration to the body when it uniform motion along a circle, since this force is perpendicular to the direction of movement of the body at any point of the trajectory.
4. The work done by a force can be either positive or negative. The work is positive \(A \) > 0, if the angle is 90° > \(\alpha \) ≥ 0°; if the angle 180° > \(\alpha \) ≥ 90°, then the work is negative \(A \) < 0.
If the angle \(\alpha \) = 0°, then \(\cos\alpha \) = 1, \(A=FS \) . If the angle \(\alpha \) = 180°, then \(\cos\alpha \) = -1, \(A=-FS \) .
5. In free fall from a height \(h\) a body with mass \(m\) moves from position 1 to position 2 (Fig. 43). In this case, the force of gravity does work equal to:
\[ A=F_тh=mg(h_1-h_2)=mgh \]
When a body moves vertically downward, the force and displacement are directed in one direction, and the force of gravity does positive work.
If a body rises upward, then the force of gravity is directed downward, and if it moves upward, then the force of gravity does negative work, i.e.
\[ A=-F_тh=-mg(h_1-h_2)=-mgh \]
6.
The work can be presented graphically. The figure shows a graph of the dependence of gravity on the height of a body relative to the surface of the Earth (Fig. 44). Graphically, the work of gravity is equal to the area of the figure (rectangle) bounded by the graph, coordinate axes and perpendicular to the abscissa axis
at point \(h\) .
The graph of the elastic force versus the elongation of the spring is a straight line passing through the origin of coordinates (Fig. 45). By analogy with the work of gravity, the work of the elastic force is equal to the area of the triangle bounded by the graph, the coordinate axes and the perpendicular to the abscissa at the point \(x\) .
\(A=Fx/2=kx\cdot x/2 \) .
7. The work done by gravity does not depend on the shape of the trajectory along which the body moves; it depends on the initial and final positions of the body. Let the body first move from point A to point B along the trajectory AB (Fig. 46). The work of gravity in this case
\[ A_(AB)=mgh \]
Let now the body move from point A to point B, first along inclined plane AC, then along the base of the inclined plane BC. The work done by gravity when moving along the aircraft is zero. The work of gravity when moving along the AC is equal to the product of the projection of gravity onto the inclined plane \(mg\sin\alpha \) and the length of the inclined plane, i.e. \(A_(AC)=mg\sin\alpha\cdot l \). Product \(l\cdot\sin\alpha=h \) . Then \(A_(AC)=mgh \) . The work of gravity when moving a body along two different trajectories does not depend on the shape of the trajectory, but depends on the initial and final positions of the body.
The work of the elastic force also does not depend on the shape of the trajectory.
Suppose that a body moves from point A to point B along the trajectory ACB, and then from point B to point A along the trajectory BA. When moving along the ACB trajectory, gravity does positive work; when moving along the BA trajectory, the work of gravity is negative, equal in magnitude to the work when moving along the ACB trajectory. Therefore, the work done by gravity along a closed path is zero. The same applies to the work of the elastic force.
Forces whose work does not depend on the shape of the trajectory and is equal to zero along a closed trajectory are called conservative. Conservative forces include gravity and elasticity.
8. Forces whose work depends on the shape of the path are called non-conservative. The friction force is non-conservative. If a body moves from point A to point B (Fig. 47) first along a straight line and then along a broken line ACB, then in the first case the work of the friction force \(A_(AB)=-Fl_(AB) \) , and in the second \(A_(ABC)=A_(AC)+A_(CB) , \(A_(ABC)=-Fl_(AC)-Fl_(CB) \) .
Therefore, work \(A_(AB) \) is not equal to work \(A_(ABC) \) .
9. Power is a physical quantity equal to the ratio of work to the period of time during which it is performed. Power characterizes the speed at which work is done.
Power is denoted by the letter \(N\) .
Unit of power: \([N]=[A]/[t] \) . \([N] \) = 1 J/1 s = 1 J/s. This unit is called the watt (W). One watt is the power at which 1 J of work is performed in 1 s.
10. The power developed by the engine is equal to: \(N = A/t \) , \(A=F\cdot S \) , whence \(N=FS/t \) . The ratio of movement to time is the speed of movement: \(S/t = v\) . Where \(N = Fv \) .
From the resulting formula it is clear that with a constant resistance force, the speed of movement is directly proportional to the engine power.
In various machines and mechanisms, mechanical energy is converted. Due to energy, work is done during its transformation. In this case, only part of the energy is spent on performing useful work. Some of the energy is spent doing work against friction forces. Thus, any machine is characterized by a value indicating what part of the energy transferred to it is used usefully. This quantity is called efficiency factor (efficiency).
The efficiency coefficient is a value equal to the ratio of useful work \((A_p) \) to all completed work \((A_s) \) : \(\eta=A_p/A_s \) . Efficiency is expressed as a percentage.
Part 1
1. The work is determined by the formula
1) \(A=Fv \)
2) \(A=N/t\)
3) \(A=mv\)
4) \(A=FS \)
2. The load is evenly lifted vertically upward by a rope tied to it. The work of gravity in this case
1) equal to zero
2) positive
3) negative
4) more work elastic forces
3. The box is pulled by a rope tied to it, making an angle of 60° with the horizontal, applying a force of 30 N. What is the work done by this force if the modulus of displacement is 10 m?
1) 300 J
2) 150 J
3) 3 J
4) 1.5 J
4. An artificial Earth satellite, whose mass is equal to \(m\) , moves uniformly in a circular orbit with radius \(R\) . The work done by gravity in a time equal to the period of revolution is equal to
1) \(mgR \)
2) \(\pi mgR \)
3) \(2\pi mgR \)
4) \(0 \)
5. A car weighing 1.2 tons travels 800 m along a horizontal road. How much work was done by the friction force if the friction coefficient is 0.1?
1) -960 kJ
2) -96 kJ
3) 960 kJ
4) 96 kJ
6. A spring with a stiffness of 200 N/m is stretched by 5 cm. How much work will the elastic force do when the spring returns to equilibrium?
1) 0.25 J
2) 5 J
3) 250 J
4) 500 J
7. Balls of the same mass roll down a slide along three different chutes, as shown in the figure. In which case will the work done by gravity be greatest?
1) 1
2) 2
3) 3
4) the work is the same in all cases
8. Work along a closed path is zero
A. Friction forces
B. Elastic forces
The correct answer is
1) both A and B
2) only A
3) only B
4) neither A nor B
9. The SI unit of power is
1) J
2) W
3) J s
4) Nm
10. What is it equal to useful work, if the work done is 1000 J and the engine efficiency is 40%?
1) 40000 J
2) 1000 J
3) 400 J
4) 25 J
11. Establish a correspondence between the work of force (in the left column of the table) and the sign of work (in the right column of the table). In your answer, write down the selected numbers under the corresponding letters.
WORK OF FORCE
A. Work of elastic force when stretching a spring
B. Work of friction force
B. Work of gravity when a body falls
WORK SIGN
1) positive
2) negative
3) equal to zero
12. From the statements below, choose two correct ones and write their numbers in the table.
1) The work of gravity does not depend on the shape of the trajectory.
2) Work is done during any movement of the body.
3) The work done by the sliding friction force is always negative.
4) Work of elastic force on closed loop not equal to zero.
5) The work of the friction force does not depend on the shape of the trajectory.
Part 2
13. A winch uniformly lifts a load weighing 300 kg to a height of 3 m in 10 s. What is the power of the winch?
Answers
The horse pulls the cart with some force, let's denote it F traction. Grandfather, sitting on the cart, presses on it with some force. Let's denote it F pressure The cart moves along the direction of the horse's traction force (to the right), but in the direction of the grandfather's pressure force (downward) the cart does not move. That's why in physics they say that F traction does work on the cart, and F the pressure does not do work on the cart.
So, work of force on the body or mechanical work– a physical quantity whose modulus is equal to the product of the force and the path traveled by the body along the direction of action of this force s:
In honor of the English scientist D. Joule, the unit mechanical work got the name 1 joule(according to the formula, 1 J = 1 N m).
If a certain force acts on the body in question, then some body acts on it. That's why the work of force on the body and the work of the body on the body are complete synonyms. However, the work of the first body on the second and the work of the second body on the first are partial synonyms, since the moduli of these works are always equal, and their signs are always opposite. That is why there is a “±” sign in the formula. Let's discuss the signs of work in more detail.
Numerical values of force and path are always non-negative quantities. In contrast, mechanical work can have both positive and negative signs. If the direction of the force coincides with the direction of motion of the body, then the work done by the force is considered positive. If the direction of force is opposite to the direction of motion of the body, the work done by a force is considered negative(we take “–” from the “±” formula). If the direction of motion of the body is perpendicular to the direction of the force, then such a force does not do any work, that is, A = 0.
Consider three illustrations of three aspects of mechanical work.
Doing work by force may look different from the perspective of different observers. Let's consider an example: a girl rides up in an elevator. Does it perform mechanical work? A girl can do work only on those bodies that are acted upon by force. There is only one such body - the elevator cabin, since the girl presses on its floor with her weight. Now we need to find out whether the cabin goes a certain way. Let's consider two options: with a stationary and moving observer.
Let the observer boy sit on the ground first. In relation to it, the elevator car moves upward and passes a certain distance. The girl’s weight is directed in the opposite direction - down, therefore, the girl performs negative mechanical work on the cabin: A dev< 0. Вообразим, что мальчик-наблюдатель пересел внутрь кабины движущегося лифта. Как и ранее, вес девочки действует на пол кабины. Но теперь по отношению к такому наблюдателю кабина лифта не движется. Поэтому с точки зрения наблюдателя в кабине лифта девочка не совершает механическую работу: A dev = 0.
What does it mean?
In physics, “mechanical work” is the work of some force (gravity, elasticity, friction, etc.) on a body, as a result of which the body moves.
Often the word “mechanical” is simply not written.
Sometimes you can come across the expression “the body has done work,” which in principle means “the force acting on the body has done work.”
I think - I'm working.
I'm going - I'm working too.
Where is the mechanical work here?
If a body moves under the influence of a force, then mechanical work is performed.
They say that the body does work.
Or more precisely, it will be like this: the work is done by the force acting on the body.
Work characterizes the result of a force.
The forces acting on a person perform mechanical work on him, and as a result of the action of these forces, the person moves.
Work is a physical quantity equal to the product of the force acting on a body and the path made by the body under the influence of a force in the direction of this force.
A - mechanical work,
F - strength,
S - distance traveled.
Work is done, if 2 conditions are met simultaneously: a force acts on the body and it
moves in the direction of the force.
No work is done(i.e. equal to 0), if:
1. The force acts, but the body does not move.
For example: we exert force on a stone, but cannot move it.
2. The body moves, and the force is zero, or all forces are compensated (i.e., the resultant of these forces is 0).
For example: when moving by inertia, no work is done.
3. The direction of the force and the direction of movement of the body are mutually perpendicular.
For example: when a train moves horizontally, gravity does no work.
Work can be positive and negative
1. If the direction of the force and the direction of motion of the body coincide, positive work is done.
For example: the force of gravity, acting on a drop of water falling down, does positive work.
2. If the direction of force and movement of the body is opposite, negative work is done.
For example: the force of gravity acting on a rising balloon, does negative work.
If several forces act on a body, then full time job of all forces is equal to the work done by the resulting force.
Units of work
In honor of the English scientist D. Joule, the unit of work was named 1 Joule.
In the International System of Units (SI):
[A] = J = N m
1J = 1N 1m
Mechanical work is equal to 1 J if, under the influence of a force of 1 N, a body moves 1 m in the direction of this force.
When flying from a person's thumb to his index finger
the mosquito does work - 0.000 000 000 000 000 000 000 000 001 J.
The human heart performs approximately 1 J of work per contraction, which corresponds to the work done when lifting a load weighing 10 kg to a height of 1 cm.
GET TO WORK, FRIENDS!
Before revealing the topic “How work is measured,” it is necessary to make a small digression. Everything in this world obeys the laws of physics. Each process or phenomenon can be explained on the basis of certain laws of physics. For each measured quantity there is a unit in which it is usually measured. Units of measurement are constant and have the same meaning throughout the world.
The reason for this is the following. In nineteen sixty, at the Eleventh General Conference on Weights and Measures, a system of measurements was adopted that is recognized throughout the world. This system was named Le Système International d’Unités, SI (SI System International). This system has become the basis for determining units of measurement accepted throughout the world and their relationships.
Physical terms and terminology
In physics, the unit of measurement of the work of force is called J (Joule), in honor of the English physicist James Joule, who made a great contribution to the development of the branch of thermodynamics in physics. One Joule is equal to the work done by a force of one N (Newton) when its application moves one M (meter) in the direction of the force. One N (Newton) is equal to a force of one kg (kilogram) mass with an acceleration of one m/s2 (meter per second) in the direction of the force.
For your information. In physics, everything is interconnected; performing any work involves performing additional actions. As an example we can take household fan. When the fan is plugged in, the fan blades begin to rotate. The rotating blades influence the air flow, giving it directional movement. This is the result of the work. But to perform the work, the influence of other external forces is necessary, without which the action is impossible. These include electric current, power, voltage and many other related values.
Electric current, at its core, is the ordered movement of electrons in a conductor per unit time. Electric current is based on positively or negatively charged particles. They are called electric charges. Denoted by the letters C, q, Kl (Coulomb), named after the French scientist and inventor Charles Coulomb. In the SI system, it is a unit of measurement for the number of charged electrons. 1 C is equal to the volume of charged particles flowing through cross section conductor per unit time. The unit of time is one second. The formula for electric charge is shown in the figure below.
The strength of electric current is indicated by the letter A (ampere). Ampere is a unit in physics that characterizes the measurement of the work of force that is expended to move charges along a conductor. At its core, electricity- this is the ordered movement of electrons in a conductor under the influence electromagnetic field. A conductor is a material or molten salt (electrolyte) that has little resistance to the passage of electrons. The strength of electric current is affected by two physical quantities: voltage and resistance. They will be discussed below. Current strength is always directly proportional to voltage and inversely proportional to resistance.
As mentioned above, electric current is the ordered movement of electrons in a conductor. But there is one caveat: they need a certain impact to move. This effect is created by creating a potential difference. Electric charge may be positive or negative. Positive charges always tend towards negative charges. This is necessary for the balance of the system. The difference between the number of positively and negatively charged particles is called electrical voltage.
Power is the amount of energy expended to do one J (Joule) of work in a period of time of one second. The unit of measurement in physics is designated as W (Watt), in the SI system W (Watt). Since electrical power is considered, here it is the value of the expended electrical energy for execution certain action in a period of time.