Formula for calculating mechanical work. Mechanical work. Power (Zotov A.E.)
The energy characteristics of motion are introduced on the basis of the concept of mechanical work or work of force.
Definition 1Work A performed by a constant force F → is a physical quantity equal to the product of the force and displacement modules multiplied by the cosine of the angle α , located between the force vectors F → and the displacement s →.
This definition is discussed in Figure 1. 18 . 1 .
The work formula is written as,
A = F s cos α .
Work is a scalar quantity. This makes it possible to be positive at (0° ≤ α< 90 °) , отрицательной при (90 ° < α ≤ 180 °) . Когда задается прямой угол α , тогда совершаемая сила равняется нулю. Единицы измерения работы по системе СИ - джоули (Д ж) .
A joule is equal to the work done by a force of 1 N to move 1 m in the direction of the force.
Picture 1 . 18 . 1 . Work of force F →: A = F s cos α = F s s
When projecting F s → force F → onto the direction of movement s → the force does not remain constant, and the calculation of work for small movements Δ s i is summed up and produced according to the formula:
A = ∑ ∆ A i = ∑ F s i ∆ s i .
This amount of work is calculated from the limit (Δ s i → 0) and then goes into the integral.
The graphical representation of the work is determined from the area of the curvilinear figure located under the graph F s (x) of Figure 1. 18 . 2.
Picture 1 . 18 . 2. Graphic definition of work Δ A i = F s i Δ s i .
An example of a force that depends on the coordinate is the elastic force of a spring, which obeys Hooke's law. To stretch a spring, it is necessary to apply a force F →, the modulus of which is proportional to the elongation of the spring. This can be seen in Figure 1. 18 . 3.
Picture 1 . 18 . 3. Stretched spring. The direction of the external force F → coincides with the direction of movement s →. F s = k x, where k denotes the spring stiffness.
F → y p = - F →
The dependence of the external force modulus on the x coordinates can be plotted using a straight line.
Picture 1 . 18 . 4 . Dependence of the external force modulus on the coordinate when the spring is stretched.
From the above figure, it is possible to find the work done on the external force of the right free end of the spring, using the area of the triangle. The formula will take the form
This formula is applicable to express the work done by an external force when compressing a spring. Both cases show that the elastic force F → y p is equal to the work of the external force F → , but with the opposite sign.
Definition 2
If several forces act on a body, then the formula for the total work will look like the sum of all the work done on it. When a body moves translationally, the points of application of forces move equally, that is, the total work of all forces will be equal to the work of the resultant of the applied forces.
Picture 1 . 18 . 5 . Model of mechanical work.
Power determination
Definition 3Power is called the work done by a force per unit time.
Recording the physical quantity of power, denoted N, takes the form of the ratio of work A to the time period t of the work performed, that is:
Definition 4
The SI system uses the watt (W t) as a unit of power, equal to the power of the force that does 1 J of work in 1 s.
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1. From the 7th grade physics course, you know that if a force acts on a body and it moves in the direction of the force, then the force does mechanical work A, equal to the product of the force modulus and the displacement modulus:
A=Fs.
Unit of work in SI - joule (1 J).
[A] = [F][s] = 1 H 1 m = 1 N m = 1 J.
A unit of work is taken to be the work done by a force 1 N on a way 1m.
It follows from the formula that mechanical work is not performed if the force is zero (the body is at rest or moving uniformly and linearly) or the displacement is zero.
Let us assume that the force vector acting on the body makes a certain angle a with the displacement vector (Fig. 65). Since the body does not move in the vertical direction, the projection of force Fy per axis Y does not do work, but the projection of force Fx per axis X does work that is equal to A = F x s x.
Because the Fx = F cos a, a s x= s, That
|
A = Fs cos a. |
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.
2. Let us analyze the resulting work formula.
If angle a = 0°, then cos 0° = 1 and A = Fs. The work done is positive and its value is maximum if the direction of the force coincides with the direction of displacement.
If angle a = 90°, then cos 90° = 0 and A= 0. The force does not do work if it is perpendicular to the direction of movement of the body. Thus, the work done by gravity is zero when a body moves along a horizontal plane. The work of the force imparting centripetal acceleration to the body during its uniform motion in a circle is equal to zero, since this force at any point of the trajectory is perpendicular to the direction of motion of the body.
If angle a = 180°, then cos 180° = –1 and A = –Fs. This case occurs when the force and displacement are directed in opposite directions. Accordingly, the work done is negative and its value is maximum. Negative work is performed, for example, by the sliding friction force, since it is directed in the direction opposite to the direction of movement of the body.
If the angle a between the force and displacement vectors is acute, then the work is positive; if angle a is obtuse, then the work is negative.
3. Let us obtain a formula for calculating the work of gravity. Let the body have mass m freely falls to the ground from a point A, located at a height h relative to the surface of the Earth, and after some time it ends up at a point B(Fig. 66, A). The work done by gravity is equal to
A = Fs = mgh.
In this case, the direction of motion of the body coincides with the direction of the force acting on it, therefore the work of gravity during free fall is positive.
If a body moves vertically upward from a point B exactly A(Fig. 66, b), then its movement is directed in the direction opposite to gravity, and the work of gravity is negative:
A= –mgh
4. The work done by a force can be calculated using a graph of force versus displacement.
Suppose a body moves under the influence of constant gravity. Gravity modulus graph F cord from the body movement module s is a straight line parallel to the abscissa axis (Fig. 67). Find the area of the selected rectangle. It is equal to the product of its two sides: S = F cord h = mgh. On the other hand, the work of gravity is equal to the same value A = mgh.
Thus, the work is numerically equal to the area of the rectangle bounded by the graph, the coordinate axes and the perpendicular raised to the abscissa axis at the point h.
Let us now consider the case when the force acting on the body is directly proportional to the displacement. Such a force, as is known, is the elastic force. Its module is equal F control = k D l, where D l- lengthening of the body.
Suppose a spring, the left end of which is fixed, is compressed (Fig. 68, A). At the same time, its right end shifted to D l 1. An elastic force has arisen in the spring F control 1, directed to the right.
If we now leave the spring to itself, its right end will move to the right (Fig. 68, b), the elongation of the spring will be equal to D l 2, and the elastic force F exercise 2.
Let's calculate the work done by the elastic force when moving the end of the spring from the point with coordinate D l 1 to the point with coordinate D l 2. We use a dependence graph for this F control (D l) (Fig. 69). The work done by the elastic force is numerically equal to the area of the trapezoid ABCD. The area of a trapezoid is equal to the product of half the sum of the bases and the height, i.e. S = AD. In the trapeze ABCD grounds AB = F control 2 = k D l 2 , CD= F control 1 = k D l 1 and the height AD= D l 1 – D l 2. Let's substitute these quantities into the formula for the area of a trapezoid:
S= (D l 1 – D l 2) =– .
Thus, we found that the work of the elastic force is equal to:
A =– .
5 * . Let us assume that a body of mass m moves from a point A exactly B(Fig. 70), moving first without friction along an inclined plane from a point A exactly C, and then without friction along the horizontal plane from the point C exactly B. Work of gravity on the site C.B. is zero because the force of gravity is perpendicular to the displacement. When moving along an inclined plane, the work done by gravity is:
A AC = F cord l sin a. Because l sin a = h, That A AC = Ft cord h = mgh.
Work done by gravity when a body moves along a trajectory ACB equal to A ACB = A AC + A CB = mgh + 0.
Thus, A ACB = mgh.
The result obtained shows that the work done by gravity does not depend on the shape of the trajectory. It depends only on the initial and final positions of the body.
Let us now assume that the body moves along a closed trajectory ABCA(see Fig. 70). When moving a body from a point A exactly B along the trajectory ACB the work done by gravity is A ACB = mgh. When moving a body from a point B exactly A gravity does negative work, which is equal to A BA = –mgh. Then the work of gravity on a closed trajectory A = A ACB + A BA = 0.
The work done by the elastic force on a closed trajectory is also zero. Indeed, suppose that an initially undeformed spring is stretched and its length increases by D l. The elastic force did the work A 1 = . When returning to equilibrium, the elastic force does work A 2 = . The total work done by the elastic force when the spring is stretched and returned to its undeformed state is zero.
6. The work done by gravity and elasticity on a closed trajectory is zero.
Forces whose work on any closed trajectory is zero (or does not depend on the shape of the trajectory) are called conservative.
Forces whose work depends on the shape of the trajectory are called non-conservative.
The friction force is non-conservative. For example, a body moves from a point 1 exactly 2 first in a straight line 12 (Fig. 71), and then along a broken line 132 . At each section of the trajectory the friction force is the same. In the first case, the work of the friction force
A 12 = –F tr l 1 ,
and in the second -
A 132 = A 13 + A 32, A 132 = –F tr l 2 – F tr l 3 .
From here A 12A 132.
7. From the 7th grade physics course you know that an important characteristic of devices that perform work is power.
Power is a physical quantity equal to the ratio of work to the period of time during which it is performed:
|
N = . |
Power characterizes the speed at which work is performed.
SI unit of power - watt (1 W).
[N] === 1 W.
A unit of power is taken to be the power at which work 1 J is completed for 1 s .
Self-test questions
1. What is work called? What is the unit of work?
2. In what case does a force do negative work? positive work?
3. What formula is used to calculate the work of gravity? elastic forces?
5. What forces are called conservative? non-conservative?
6 * . Prove that the work done by gravity and elasticity does not depend on the shape of the trajectory.
7. What is called power? What is the unit of power?
Task 18
1. A boy weighing 20 kg is carried evenly on a sled, applying a force of 20 N. The rope by which the sled is pulled makes an angle of 30° with the horizontal. What is the work done by the elastic force generated in the rope if the sled moves 100 m?
2. An athlete weighing 65 kg jumps into water from a platform located at a height of 3 m above the surface of the water. How much work is done by the force of gravity acting on the athlete as he moves to the surface of the water?
3. Under the action of an elastic force, the length of a deformed spring with a stiffness of 200 N/m decreased by 4 cm. What is the work done by the elastic force?
4 * . Prove that the work of a variable force is numerically equal to the area of the figure, limited by the graph of the force versus coordinate and the coordinate axes.
5. What is the traction force of a car engine if at a constant speed of 108 km/h it develops a power of 55 kW?
Do you know what work is? Without any doubt. Every person knows what work is, provided that he was born and lives on planet Earth. What is mechanical work?
This concept is also known to most people on the planet, although some individuals have a rather vague understanding of this process. But we are not talking about them now. Even fewer people have any idea what it is mechanical work from the point of view of physics. In physics, mechanical work is not human labor for food, it is a physical quantity that may be completely unrelated to either a person or any other living creature. How so? Let's figure it out now.
Mechanical work in physics
Let's give two examples. In the first example, the waters of the river, faced with an abyss, noisily fall down in the form of a waterfall. The second example is a man who holds a heavy object in his outstretched arms, for example, holding the broken roof over the porch of a country house from falling, while his wife and children are frantically looking for something to support it with. When is mechanical work performed?
Definition of mechanical work
Almost everyone, without hesitation, will answer: in the second. And they will be wrong. The opposite is true. In physics, mechanical work is described with the following definitions: Mechanical work is performed when a force acts on a body and it moves. Mechanical work is directly proportional to the force applied and the distance traveled.
Mechanical work formula
Mechanical work is determined by the formula:
where A is work,
F - strength,
s is the distance traveled.
So, despite all the heroism of the tired roof holder, the work he has done is zero, but the water, falling under the influence of gravity from a high cliff, does the most mechanical work. That is, if we push a heavy cabinet unsuccessfully, then the work we have done from the point of view of physics will be equal to zero, despite the fact that we apply a lot of force. But if we move the cabinet a certain distance, then we will do work equal to the product of the applied force and the distance over which we moved the body.
The unit of work is 1 J. This is the work done by a force of 1 Newton to move a body over a distance of 1 m. If the direction of the applied force coincides with the direction of movement of the body, then this force does positive work. An example is when we push a body and it moves. And in the case when a force is applied in the direction opposite to the movement of the body, for example, friction force, then this force does negative work. If the applied force does not affect the movement of the body in any way, then the force performed by this work is equal to zero.
Let the body, which is acted upon by a force, pass, moving along a certain trajectory, a path s. In this case, the force either changes the speed of the body, giving it acceleration, or compensates for the action of another force (or forces) opposing the movement. The action on the path s is characterized by a quantity called work.
Mechanical work is a scalar quantity equal to the product of the projection of the force on the direction of movement Fs and the path s traversed by the point of application of the force (Fig. 22):
A = Fs*s.(56)
Expression (56) is valid if the magnitude of the projection of the force Fs on the direction of movement (i.e., on the direction of velocity) remains unchanged all the time. In particular, this occurs when the body moves rectilinearly and a force of constant magnitude forms a constant angle α with the direction of movement. Since Fs = F * cos(α), expression (47) can be given the following form:
A = F * s * cos(α).
If is the displacement vector, then the work is calculated as the scalar product of two vectors and :
. (57)
Work is an algebraic quantity. If the force and direction of movement form an acute angle (cos(α) > 0), the work is positive. If the angle α is obtuse (cos(α)< 0), работа отрицательна. При α = π/2 работа равна нулю. Последнее обстоятельство особенно отчетливо показывает, что понятие работы в механике существенно отличается от обыденного представления о работе. В обыденном понимании всякое усилие, в частности и мускульное напряжение, всегда сопровождается совершением работы. Например, для того чтобы держать тяжелый груз, стоя неподвижно, а тем более для того, чтобы перенести этот груз по горизонтальному пути, носильщик затрачивает много усилий, т. е. «совершает работу». Однако это – «физиологическая» работа. Механическая работа в этих случаях равна нулю.
Work when moving under force
If the magnitude of the projection of force on the direction of movement does not remain constant during movement, then the work is expressed as an integral:
. (58)
An integral of this type in mathematics is called a curvilinear integral along the trajectory S. The argument here is a vector variable, which can change both in magnitude and direction. Under the integral sign is the scalar product of the force vector and the elementary displacement vector.
A unit of work is taken to be the work done by a force equal to one and acting in the direction of movement along a path equal to one. In SI The unit of work is the joule (J), which is equal to the work done by a force of 1 newton along a path of 1 meter:
1J = 1N * 1m.
In the CGS, the unit of work is the erg, equal to the work done by a force of 1 dyne along a path of 1 centimeter. 1J = 10 7 erg.
Sometimes the non-systemic unit kilogrammometer (kg*m) is used. This is the work done by a force of 1 kg along a path of 1 meter. 1 kg*m = 9.81 J.
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 the total work done by 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!
