Work of force, formula
A force applied to a body and moving it does work (Fig. 1).
Rice. 1. Force moves a body and does work
The work of a force is the scalar product of the force vector and the displacement vector.
The work done by a force can be calculated using a vector or scalar form of writing the following formula:
Vector record view
\[ \large \boxed{ A = \left( \vec{F} , \vec{S} \right) }\]
To solve problems, it is convenient to write the right side of this formula in scalar form:
\[ \large \boxed{ A = \left| \vec{F} \right| \cdot \left| \vec{S} \right| \cdot cos(\alpha) }\]
\( F \left( H \right) \) – force moving the body;
\( S \left( \text{м} \right) \) – movement of the body under the influence of force;
\( \alpha \) – the angle between the force vector and the body displacement vector;
Work is denoted by the symbol \(A\) and measured in Joules. Work is a scalar quantity.
In the case where the force is constant, the formula allows you to calculate the work done by the force during the total time of its action.
If the force changes with time, then at each specific moment in time we will receive instantaneous work. These instantaneous values will differ for different points in time.
Let's consider several cases following from the formula:
- When the angle between force and displacement is acute, the work done by the force is positive;
- And if the angle is obtuse, the work is negative, since the cosine of an obtuse angle is negative;
- If the angle is right, the work is zero. A force perpendicular to the displacement does no work!
Work done by gravity - potential energy difference
Let's now consider the following example. An apple weighing 0.2 kg fell onto a garden table from a branch located at a height of 3 meters from the surface of the earth. The tabletop is located at a height of 1 meter from the surface (Fig. 3). Let's find the work done by gravity in this process.
Rice. 3. The figure shows the initial 1st position of the body (apple) and its final 2nd position, marked heights for calculating the work of vertical movement of the body
Let's calculate the potential energy of the apple before it falls and the energy of the apple on the table top.
\( E_{p1} \left(\text{J} \right) \) – initial potential energy of the apple;
\( E_{p2} \left(\text{J} \right) \) – final potential energy of the apple;
Note: Work can be calculated through the difference in the potential energy of the body.
We will calculate potential energy using the formula:
\[ \large E_{p} = m \cdot g \cdot h\]
\( m \left( \text{kg}\right) \) – mass of the apple;
The quantity \( \displaystyle g \approx 10 \left(\frac{\text{m}}{c^{2}} \right) \) is the acceleration of free fall.
\( h \left( \text{m}\right) \) – the height at which the apple is located relative to the surface of the earth.
The initial height of the apple above the ground is 3 meters
\[ \large E_{p2} = 0.2 \cdot 10 \cdot 3 = 6 \left(\text{J} \right) \]
Potential energy of an apple on the table
\[ \large E_{p1} = 0.2 \cdot 10 \cdot 1 = 2 \left(\text{J} \right) \]
Now let's find the difference in the potential energy of the apple at the end of the fall and before it begins.
\[ \large \Delta E_{p} = E_{p2} — E_{p1} \]
\[ \large \Delta E_{p} = 2 – 6 = — 4 \left(\text{J} \right) \]
It is important to remember: When a body falls to the ground, its potential energy decreases. The force of gravity does positive work!
To make the work positive, we will additionally add a minus sign on the right side of the formula before \( \Delta E_{p}\).
\[ \large \boxed{ A = — \Delta E_{p} }\]
This means that the work required by gravity for an apple weighing 0.2 kg to fall from a height of 3 m to a height of 1 meter is equal to 4 Joules.
Notes:
- If a body falls to the ground, the work done by gravity is positive;
- When we lift a body above the ground, we do work against gravity. Our work is positive, and the work of gravity will be negative;
- Gravity is a conservative force. For conservative forces, we add a minus sign in front of the potential energy difference;
- The work done by gravity does not depend on the trajectory along which the body moved;
- The work for the force \(\displaystyle F_{\text{heavy}}\) depends only on the difference in heights at which the body was at the final and initial moments of time.
Figure 4 illustrates the fact that for the force \(\displaystyle F_{\text{heavy}}\) the work depends only on the difference in height and does not depend on the trajectory along which the body moved.
Rice. 4. The height difference between the initial and final position of the body in all cases in the figure is the same, therefore, the work of gravity for the presented cases will be the same
Power and power units
When purchasing a car (Fig. 226), a lawn mower, a microwave oven (Fig. 227), etc., a person is interested in their power. It is power that is the passport characteristic of machines and mechanisms. What is power? Why is it so important to know it?
Let's look at an example. A man uses a shovel to dig a hole for a cellar over several days. An excavator (Fig. 228) will dig the same hole in a few minutes. The work is done the same. The same mass of soil rises to the same height. But the speed of work performed by a person and an excavator is different. In a unit of time, an excavator performs many times more work than a person. To describe the speed of work done, power is introduced.
A physical quantity equal to the ratio of work to the period of time during which this work is performed is called power. Power is designated by the letter R.
The SI unit of power is the power at which the force acting on the body does A = 1 J of work in time t = 1 s. This unit of power is called the watt (W) in honor of the English inventor J. Watt. To measure large powers, multiple units are used: hectowatt (gW), kilowatt (kW), megawatt (MW). Note:
For low powers, submultiple units are used - milliwatt (mW), microwatt (µW):
In everyday life, the unit of power kilowatt is often thoughtlessly mistaken for a unit of work. But work from which it follows that the unit of work can only be a kilowatt-hour (kW • h), but not a kilowatt (kW). Let's express power through other units - force and speed. Power but work the way Then
Power is proportional to the force doing work and the speed of movement. Then, at constant power, the lower the speed, the greater the force. That is why the driver, when starting from a stop or climbing a hill (Fig. 229), when great force is required, drives at low speed. Thus, it increases the traction force of the car engine.
Main conclusions
- Power is a physical quantity that characterizes the speed of work.
- The SI unit of power is 1 watt.
- The same power can be obtained either with high speed and low force, or with low speed and high force.
For the curious
In the automotive industry, traditionally, an ancient unit of power is used - horsepower (hp). Using the figure, formulate your own definition of power of 1 horsepower.
Let's write down the connection 1 l. With. and watt: 1 l. With. = 736 W.
In these non-system units, the power of the first Belarusian tractor MTZ-2 (1953) was 37 hp. With. The Belarus-3023 tractor, mastered in 2010, has a 300 hp engine. With. Convert these power signs to SI units yourself and compare them.
An example of solving a problem:
During a physical education lesson, a boy with mass m = 40 kg climbed a rope to a height h = 5.0 m in a period of time t = 10 s. Determine the average power developed by the boy when lifting. Coefficient
Given:
Solution:
When climbing a rope, the muscular strength of the arms works to overcome the force of gravity.
Then
Answer: P = 0.20 kW.
Kinetic energy
Energy is one of the most important and complex concepts. And not only in physics, but also in other sciences. What is kinetic energy?
Let's look at two examples. The puck, hitting the goal net (Fig. 230), bends it. A hammer for driving piles (Fig. 231), falling on a pile, drives it into the ground to a certain depth. To bend the mesh more or drive the pile deeper, the washer and hammer must have greater speed. Both the washer and the hammer did the work. At the same time, the speed of their movement changed (decreased to zero). The work they performed was different, even if we assume that the speeds were the same. But the masses of the hammer and washer are not equal.
If a body is capable of doing work, then it has energy. In physics, the energy of a moving body is called kinetic (from the Greek kinetikos - setting in motion). Kinetic energy is denoted by the letter K (or ) and is measured in SI in the same units as work, i.e. in joules.
The large kinetic energy of moving bodies - a stone, a car, a train (Fig. 232), a meteorite, etc. - means, firstly, that when they were accelerated to a given speed by the accelerating force, a lot of work was done and, secondly, when they When stopping, the braking force will do the same amount of work.
From the examples it follows that kinetic energy depends on the mass of the body and the speed of its movement. What is this dependency?
Experiments show that kinetic energy is directly proportional to the mass of a body and the square of its speed of movement:
Increasing the speed of body movement, for example by 4 times, leads to an increase in kinetics - Please note! ical energy by 16 times. Drivers and pedestrians should always remember this.
Main conclusions:
- Kinetic energy expresses the ability of moving bodies to do work.
- Kinetic energy, like work, is measured in joules.
- The kinetic energy of a body depends on its mass and speed.
- You can change (increase or decrease) the kinetic energy of a body only by doing work (positive or negative).
An example of solving a problem:
The speed of movement of a loaded car with mass m = 4.0 t increased from to along the path s = 25 m. Determine the traction force of the car engine and the work done by this force. Ignore movement resistance.
Given:
Solution:
To increase the kinetic energy from to the traction force had to do work:
But work from here
Answer:
Potential energy
When any body (sleigh, car, etc.) accelerates, it acquires the ability to perform mechanical work—the moving body acquires kinetic energy. What if the body is motionless? Does it have the ability to do work?
Let's conduct two experiments. In the first, we will lift and fasten a weight on a thread above a box of sand (Fig. 235, a). In the second, between the stop and the ball we will place a spring pre-compressed and tied with a thread (Fig. 235, b). Both bodies (weight and spring) are motionless and do not possess kinetic energy. But both the weight and the spring have the ability to do work. To do this, in both cases it is enough to burn the thread. In physics they say that bodies (a raised weight interacting with the Earth and a compressed spring) have potential energy (from the Latin potentia - hidden ability). Potential energy in SI is measured in the same units as work - in joules.
It is important to understand that potential energy does not appear on its own. In these experiments, the weight was raised above the table, the spring was compressed by some force. This means that in order for the body to store potential energy, work must be done. The more the spring is compressed, the higher the body is raised, the greater the potential energy reserve. The bodies presented in Figure 236 already have potential energy. In a springboard it is caused by the deflection (deformation) of the board, in a mousetrap it is caused by the twisting of the spring, in a bow it is caused by a change in the position of the shaft and string. From these and other examples it follows that potential energy is the energy caused by the relative position of interacting bodies or parts of the body (weights and the Earth, arrows and bowstrings, spring links). Potential energy is denoted by the letter P (or).
It is thanks to the potential energy of a compressed (twisted) spring that mechanical watches, time relays for microwave ovens, washing machines, and some children's toys move. The potential energy of the water raised by the dam forces hydroelectric power stations to operate (Fig. 237).
Main conclusions:
- Fixed interacting bodies (a system of bodies) may have the ability to perform mechanical work, and therefore potential energy.
- The value of potential energy depends on the relative position of interacting bodies (body parts).
- Potential energy changes only when work is done.
Potential Energy Calculation
The kinetic energy of a body, depending on its mass and speed, is expressed by the formula This formula is valid both for the planet Earth, rushing at speed in its orbit around the Sun, and for an atom invisible to our eyes. Is there a single formula for calculating potential energy?
Let us consider two cases separately: the potential energy of attraction of a body above the Earth’s surface and the potential energy of a deformed body.
In the first case, the formula for calculating potential energy is easy to derive. If a body of mass m is raised relative to the surface of the Earth to a height h (Fig. 238), then when it falls, the force of gravity can do work:
This is the potential energy of a raised body:
The value of potential energy is relative. Thus, relative to the floor, the potential energy of a lamp (Fig. 239) with a mass m = 1.0 kg, the center of gravity of which is located at a height from the floor, is equal to:
Relative to the ceiling it is equal to:
Therefore, when giving the value of potential energy, it is necessary to indicate the level relative to which it is given - the zero level of potential energy (this could be, for example, the surface of the floor, ceiling, table, etc.).
The situation is much more complicated when calculating the potential energy of a deformed body. We can stretch or compress the spring, bend or twist it (Fig. 240). The spring will have potential energy in each of these cases. And the greater the elastic deformation, the greater the potential energy of the spring. In this example, potential energy will have to be calculated using various formulas. You will learn more about this in 9th grade.
Main conclusions:
- The potential energy of attraction of a body to the Earth depends on the mass of the body and the height of its rise above the zero energy level.
- The value of the body's potential energy depends on the choice of the zero energy level.
- The potential energy of a deformed body depends on the magnitude of the deformation.
An example of solving a problem:
A homogeneous paraffin cube with edge length a = 10 cm lies on the table at a height = 0.80 m from the floor. Determine the potential energy of the cube relative to the surfaces: a) floor; b) table. How much work must be done to lift the cube from the floor to the table? Coefficient
Given:
Solution:
The potential energy of the cube relative to the floor surface (Fig. 241) is determined by the position of its center (point O):
The mass of the cube is then:
Potential energy of the cube relative to the table surface:
The work done to raise the cube to a height is equal to the change in its potential energy. We get:
Answer:
Law of conservation of mechanical energy
Kinetic and potential energy are two types of mechanical energy. Are they related to each other? And if so, what is this connection?
Let's follow the movement of a metal ball thrown upward (Fig. 243). At the lowest point of the trajectory, the force of the hand on the ball imparts kinetic energy to it. The ball moves up. The speed of its movement, and therefore the kinetic energy, decreases. But does kinetic energy disappear without a trace? Rising higher, the ball acquires more and more potential energy (remember: ). At the top point, the speed and kinetic energy of the ball are zero, and the potential energy is maximum. This means that in the example considered, energy is converted from one type (kinetic) to another (potential). When the ball returns back, energy conversion will occur again: with a decrease in height (and potential energy), the speed of the ball (and kinetic energy) increases.
If air resistance is small (and can be neglected), a ball thrown upward returns back with almost the same speed and kinetic energy as at the moment of throwing.
What will be the value of the mechanical energy of the ball at intermediate points? For example, at a height (Fig. 243)? As the ball rises to a height, its kinetic energy decreases, but potential energy appears. What is their sum, i.e., total mechanical energy? This and similar experiments and calculations show that if there are no resistance forces, then the total mechanical energy of a body (system of bodies), equal to the sum of kinetic and potential energies, is conserved. This statement about the constancy of mechanical energy in physics is called the law of conservation of mechanical energy.
If the forces of friction or resistance to movement cannot be neglected, this law does not apply. Let us replace the metal ball in the experiment with a foam block of the same mass (Fig. 244). We will see that even with an initial speed greater than that of a metal ball, it will not rise to the same height and will return back at a noticeably lower speed. The kinetic energy of a puck moving along a horizontal ice surface decreases, but potential energy does not appear in return. Due to the kinetic energy of the puck, work is done against frictional forces.
In conclusion, we note that people have learned to use the phenomenon of transforming energy from one type into another for practical purposes. The energy of falling water powers water mills and hydroelectric power stations. The Republic of Belarus is successfully implementing a state program for the use of river energy. An important role in it is given to such rivers as the Neman and Western Dvina. The Grodno hydroelectric power station with a capacity of 17 MW operates in Neman. The installed capacity of the Vitebsk hydroelectric power station on the Western Dvina is 40 MW.
From ancient times, man began to use the kinetic energy of the wind with the help of a sail (Fig. 245), then began to use it in windmills. In recent years, the construction of wind power plants has begun in our country (Fig. 246). They are unique in that they do not have a harmful effect on the environment. Many countries successfully use the energy of the ebb and flow of the seas and oceans. Tidal power plants have been created there.
Main conclusions:
- Kinetic and potential energies are mutually convertible.
- In the absence of friction forces and resistance to movement, the total mechanical energy of the body (system of bodies) is conserved.
- The law of conservation of mechanical energy is not satisfied if the forces of friction (resistance) cannot be neglected.
- Order solutions to physics problems
An example of solving a problem:
A stone is thrown vertically upward with a speed. At what height from the point of throwing will the kinetic energy of the stone be 4 times less than its potential energy? Neglect the resistance to the movement of the stone. Coefficient
Given:
Solution:
Let us take the zero level of potential energy to be the O - O level passing through the point of throwing the stone (Fig. 247). Means,
Total mechanical energy of the stone at throwing point 1:
Total mechanical energy of the stone at point 2:
According to the condition, this means
Answer:
Power
In mechanics, power is often symbolized N or P and measured in Watts, after the Scottish inventor James Watt.
Note: The symbol \(\vec{N}\) is used to denote the ground reaction force - it is measured in Newtons and is a vector quantity. To avoid confusion, we will denote power instead of N by the symbol P. The symbol P is the first letter in the English word power.
Power is work done in one second (energy expended in 1 second).
We calculate the work using any of the formulas:
\[ \large A = \Delta E_{k} \]
\[ \large A = \Delta E_{p} \]
\[ \large A = F \cdot S \cdot cos(\alpha) \]
Dividing this work by the time during which it was performed, we get the power.
\[ \large \boxed{ P = \frac{A}{\Delta t} }\]
If the work was done in equal parts at equal time intervals, the power will be a constant value.
Power is variable when more work is done at some time intervals.
Another formula for calculating power
There is another way to calculate power, when a force moves a body and the speed of the body does not change:
\[ \large P = \left( \vec{F} , \vec{v} \right) \]
The formula can be written in scalar form:
\[ \large P = \left| \vec{F} \right| \cdot \left| \vec{v} \right| \cdot cos(\alpha) \]
\( F \left( H \right) \) – force moving the body;
\( \displaystyle v \left( \frac{\text{m}}{c} \right) \) – body speed;
\( \alpha \) – the angle between the force vector and the body velocity vector;
When the vectors \(\vec{F}\) and \(\vec{v}\) are parallel, the formula is simplified:
\[ \large \boxed{ P = F \cdot v }\]
Note: This formula for calculating power can be obtained from the expression for the work of force by dividing both sides of this expression by the time during which the work was performed (or more precisely, by finding the derivative of both sides of the equation).
Power power
The equation for power combines work done and time. Since we know that work is done by forces, and forces can move objects, we can derive another expression for instantaneous power:
- Work done by force when moving:
A = F x S x cos φ.
- If we put A in the universal formula for N , the power of the force is determined:
N = (F x S x cos φ)/t = F x V x cos φ, since V = S/t.
- If the force is parallel to the velocity of the particle, then the formula takes the form:
N = F x V.
Efficiency
Efficiency – efficiency factor. Usually denoted by the Greek symbol \(\eta\) "this". It has no units of measurement and is expressed either as a decimal fraction or as a percentage.
Notes:
- A percentage is a fraction whose denominator is 100.
- Efficiency is either a proper fraction or a fraction equal to unity.
Calculate the coefficient \(\eta\) for a device, mechanism or process.
\[ \large \boxed{ \eta = \frac{ A_{\text{useful}}}{ A_{\text{all}}} }\]
\(\eta\) – efficiency;
\( \large A_{\text{useful}} \left(\text{J} \right)\) – useful work;
\(\large A_{\text{all}} \left(\text{J} \right)\) – all the energy expended to perform the work;
Note: Efficiency is often less than unity, since there is always energy loss. The efficiency cannot be greater than one, since this contradicts the law of conservation of energy.
\[ \large \boxed{ \eta \leq 1 }\]
The quantity \(\eta\) is a fractional quantity. If the numerator and denominator of a fraction are divided by the same number, the resulting fraction will be equal to the original. Using this fact, you can calculate efficiency using power:
\[ \large \boxed{ \eta = \frac{ P_{\text{useful}}}{ P_{\text{all spent}}} }\]
Electric motor power concept
Power is perhaps the most important parameter when choosing an electric motor. Traditionally, it is indicated in kilowatts (kW), for imported models - in kilowatts and horsepower (hp, HP, Horse Power). For reference: 1 hp. approximately equal to 0.75 kW.
The engine nameplate indicates the rated net (output mechanical) power
. This is the power that the engine can deliver to a mechanical load with the stated parameters without overheating. In the formulas, the rated mechanical power is denoted by P2.
Electrical (consumption) power
motor P1 is always greater than the output P2, since there are losses in any energy conversion device. The main losses in an electric motor are mechanical, caused by friction. As you know from a physics course, losses in any device are determined through efficiency (ƞ), which is always less than 100%. In this case, the formula is valid:
Р2 = Р1 · ƞ
The efficiency in engines depends on the rated power - for low-power models it can be less than 0.75, for powerful ones it exceeds 0.95. The above formula is valid for active power consumption. But, since the electric motor is an active-reactive load, to calculate the total power consumption S
(taking into account the reactive component) it is necessary to take into account reactive losses. The reactive component is expressed in terms of power factor (cosϕ). Taking this into account, the formula for the rated engine power looks like this:
Р2 = Р1 · ƞ = S · ƞ · cosϕ
conclusions
- A force applied to a body and moving it does work;
- When the angle between the force and the displacement is acute, the work of the force is positive, and if the angle is obtuse, the work is negative; If the angle is right, the work is zero. A force perpendicular to the displacement does no work!
- Work can be calculated by measuring the kinetic energy of a body at the beginning and end of its motion;
- The work can be calculated through the difference in the potential energy of the body at the initial and final heights above the ground;
- When a body falls to the ground, its potential energy decreases. The force of gravity does positive work!
- We do work against gravity when we lift our body above the ground. Moreover, our work is positive, and the work of gravity is negative;
- Gravity is a conservative force. Therefore, the work of force \(\displaystyle F_{\text{heavy}}\) does not depend on the trajectory along which the body moved, but depends only on the difference in heights at which the body was at the final and initial moments of time;
- Power is work done in one second, or expended in 1 second. energy;
- The efficiency factor is denoted by the Greek symbol \(\eta\) “eta”, has no units of measurement, and is expressed either as a decimal fraction or as a percentage;
- Efficiency is either a proper fraction or a fraction equal to unity.
- You can calculate efficiency by substituting work or power into the formula
Power of rotating objects
Efficiency formula
Processes associated with the rotation of objects can be described by similar equations. The equivalent of the force for rotation is the torque M, the equivalent of the speed V is the angular velocity ω.
If we replace the corresponding values, we get the formula:
N = M x ω.
M = F xr, where r is the radius of rotation.
To calculate the power of a shaft rotating against a force, the formula is used:
N = 2π x M xn,
where n is the speed in rev/s (n = ω/2π).
This gives the same simplified expression:
N = M x ω.
Thus, the engine can achieve high power either at high speed or by having high torque. If the angular velocity ω is zero, then the power is also zero, regardless of the torque.
