Acceleration characterizes the rate of change in the speed of a moving body. If the speed of a body remains constant, then it does not accelerate.

Acceleration occurs only when the speed of a body changes. If the speed of a body increases or decreases by a certain constant amount, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m/s2) and is calculated from the values ​​of two speeds and time or from the value of the force applied to the body.

Steps

1. 1 a = Δv / Δt
2. 2 Definition of variables. You can calculate Δv And Δt in the following way: Δv = vк - vн And Δt = tк - tн, Where vк– final speed, vн- starting speed, tk– final time, tн– initial time.
3. 3
4. Write the formula: a = Δv / Δt = (vк - vн)/(tк - tн)
5. Write the variables: vк= 46.1 m/s, vн= 18.5 m/s, tk= 2.47 s, tн= 0 s.
6. Calculation: a
7. Write the formula: a = Δv / Δt = (vк - vн)/(tк - tн)
8. Write the variables: vк= 0 m/s, vн= 22.4 m/s, tk= 2.55 s, tн= 0 s.
9. Calculation: A

1. 1 Newton's second law.
2. Fres = m x a, Where Fres m- body mass, a– acceleration of the body.
3. 2 Find the mass of the body.
4. Remember that 1 N = 1 kg∙m/s2.
5. a = F/m = 10/2 = 5 m/s2

3 Testing your knowledge

1. 1 Direction of acceleration.
   
2. 2 Direction of force.
3. 3 Resultant force.
4. Solution: The conditions of this problem are designed to confuse you. In fact, everything is very simple. Draw a diagram of the direction of forces, so you will see that a force of 150 N is directed to the right, a force of 200 N is also directed to the right, but a force of 10 N is directed to the left. Thus, the resulting force is: 150 + 200 - 10 = 340 N. The acceleration is: a = F/m = 340/400 = 0.85 m/s2.

Determining the force or moment of force, if the mass or moment of inertia of the body is known, allows you to find out only the acceleration, that is, how quickly the speed changes

Shoulder of power– perpendicular lowered from the axis of rotation to the line of action of the force.

Bone links in the human body are levers. In this case, the result of a muscle’s action is determined not so much by the force it develops as by the moment of force. A feature of the structure of the human musculoskeletal system is the small values ​​of the shoulder forces of muscle traction. At the same time, external force, for example, gravity, has a large shoulder (Fig. 3.3). Therefore, to counteract large external torques, muscles must develop greater traction force.

Rice. 3.3. Features of human skeletal muscles

The moment of force is considered positive if the force causes the body to rotate counterclockwise, and negative when the body rotates clockwise. In Fig. 3.3. the gravity of the dumbbell creates a negative moment of force, as it tends to rotate the forearm at the elbow joint clockwise. The traction force of the forearm flexor muscles creates a positive torque as it tends to rotate the forearm at the elbow joint counterclockwise.

Momentum impulse(Sм) – a measure of the influence of the moment of force relative to a given axis over a period of time.

Kinetic moment (TO) & vector quantity, a measure of the rotational motion of a body, characterizing its ability to be transmitted to another body in the form of mechanical movement. The kinetic moment is determined by the formula: K=J .

Kinetic moment during rotational motion is an analogue of the body's momentum (momentum) during translational motion.

Example. When performing a jump into the water after taking off from the bridge, the kinetic moment of the human body ( TO) remains unchanged. Therefore, if you reduce the moment of inertia (J), that is, perform a tuck, the angular speed increases. Before entering the water, the athlete increases the moment of inertia (straightens up), thereby reducing the angular speed of rotation.

How to find acceleration through force and mass?

How much the speed has changed can be found by determining the impulse of force. Force impulse is a measure of the impact of force on a body over a given period of time (in translational motion): S = F*Dt = m*Dv. In the case of simultaneous action of several forces, the sum of their impulses is equal to the impulse of their resultant during the same time. It is the impulse of force that determines the change in speed. In rotational motion, the impulse of force corresponds to the impulse of the moment of force - a measure of the influence of force on a body relative to a given axis for a given period of time: Sz = Mz*Dt.

As a result of the impulse of force and the impulse of moment of force, changes in motion arise, depending on the inertial characteristics of the body and manifested in changes in speed (momentum and angular momentum - kinetic moment).

The amount of motion is a measure of the translational motion of a body, characterizing the ability of this movement to be transmitted to another body: K = m*v. The change in momentum is equal to the force impulse: DK = F*Dt = m*Dv = S.

Kinetic moment is a measure of the rotational motion of a body, characterizing the ability of this movement to be transmitted to another body: Kя = I*w = m*v*r. If a body is connected to an axis of rotation that does not pass through its CM, then the total angular momentum is composed of the angular momentum of the body relative to the axis passing through its CM parallel to the external axis (I0*w) and the angular momentum of some point that has the mass of the body and is distant from the axis rotation at the same distance as the CM: L = I0*w + m*r2*w.

There is a quantitative relationship between the angular momentum (kinetic torque) and the angular momentum of force: DL = Mz*Dt = I*Dw = Sz.

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Acceleration characterizes the rate of change in the speed of a moving body. If the speed of a body remains constant, then it does not accelerate. Acceleration occurs only when the speed of a body changes. If the speed of a body increases or decreases by a certain constant amount, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m/s2) and is calculated from the values ​​of two speeds and time or from the value of the force applied to the body.

Steps

1 Calculation of average acceleration at two speeds

1. 1 Formula for calculating average acceleration. The average acceleration of a body is calculated from its initial and final speeds (speed is the speed of movement in a certain direction) and the time it takes the body to reach its final speed. Formula for calculating acceleration: a = Δv / Δt, where a is acceleration, Δv is the change in speed, Δt is the time required to reach the final speed.
2. The units of acceleration are meters per second per second, i.e. m/s2.
3. Acceleration is a vector quantity, that is, it is given by both value and direction. Value is a numerical characteristic of acceleration, and direction is the direction of movement of the body. If the body slows down, then the acceleration will be negative.
4. 2 Definition of variables. You can calculate Δv And Δt in the following way: Δv = vк - vн And Δt = tк - tн, Where vк– final speed, vн- starting speed, tk– final time, tн– initial time.
5. Since acceleration has a direction, always subtract the initial velocity from the final velocity; otherwise the direction of the calculated acceleration will be incorrect.
6. If the initial time is not given in the problem, then it is assumed that tн = 0.
7. 3 Find the acceleration using the formula. First, write the formula and the variables given to you. Formula: a = Δv / Δt = (vк - vн)/(tк - tн). Subtract the initial speed from the final speed, and then divide the result by the time interval (time change). You will get the average acceleration over a given period of time.
8. If the final speed is less than the initial speed, then the acceleration has a negative value, that is, the body slows down.
9. Example 1: A car accelerates from 18.5 m/s to 46.1 m/s in 2.47 s. Find the average acceleration.
10. Write the formula: a = Δv / Δt = (vк - vн)/(tк - tн)
11. Write the variables: vк= 46.1 m/s, vн= 18.5 m/s, tk= 2.47 s, tн= 0 s.
12. Calculation: a= (46.1 - 18.5)/2.47 = 11.17 m/s2.
13. Example 2: A motorcycle starts braking at a speed of 22.4 m/s and stops after 2.55 s. Find the average acceleration.
14. Write the formula: a = Δv / Δt = (vк - vн)/(tк - tн)
15. Write the variables: vк= 0 m/s, vн= 22.4 m/s, tk= 2.55 s, tн= 0 s.
16. Calculation: A= (0 - 22.4)/2.55 = -8.78 m/s2.

2 Calculation of acceleration by force

1. 1 Newton's second law. According to Newton's second law, a body will accelerate if the forces acting on it do not balance each other. This acceleration depends on the net force acting on the body. Using Newton's second law, you can find the acceleration of a body if you know its mass and the force acting on that body.
2. Newton's second law is described by the formula: Fres = m x a, Where Fres– resultant force acting on the body, m- body mass, a– acceleration of the body.
3. When working with this formula, use metric units, which measure mass in kilograms (kg), force in newtons (N), and acceleration in meters per second per second (m/s2).
4. 2 Find the mass of the body. To do this, place the body on the scale and find its mass in grams. If you are considering a very large body, look up its mass in reference books or on the Internet. The mass of large bodies is measured in kilograms.
5. To calculate acceleration using the above formula, you need to convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.
6. 3 Find the net force acting on the body. The resulting force is not balanced by other forces. If two differently directed forces act on a body, and one of them is greater than the other, then the direction of the resulting force coincides with the direction of the larger force. Acceleration occurs when a force acts on a body that is not balanced by other forces and which leads to a change in the speed of the body in the direction of action of this force.
7. For example, you and your brother are in a tug of war. You are pulling the rope with a force of 5 N, and your brother is pulling the rope (in the opposite direction) with a force of 7 N. The resulting force is 2 N and is directed towards your brother.
8. Remember that 1 N = 1 kg∙m/s2.
9. 4 Rearrange the formula F = ma to calculate the acceleration. To do this, divide both sides of this formula by m (mass) and get: a = F/m. Thus, to find acceleration, divide the force by the mass of the accelerating body.
10. Force is directly proportional to acceleration, that is, the greater the force acting on a body, the faster it accelerates.
11. Mass is inversely proportional to acceleration, that is, the greater the mass of a body, the slower it accelerates.
12. 5 Calculate the acceleration using the resulting formula. Acceleration is equal to the quotient of the resulting force acting on the body divided by its mass. Substitute the values ​​given to you into this formula to calculate the acceleration of the body.
13. For example: a force equal to 10 N acts on a body weighing 2 kg. Find the acceleration of the body.
14. a = F/m = 10/2 = 5 m/s2

3 Testing your knowledge

1. 1 Direction of acceleration. The scientific concept of acceleration does not always coincide with the use of this quantity in everyday life. Remember that acceleration has a direction; acceleration is positive if it is directed upward or to the right; acceleration is negative if it is directed downward or to the left. Check your solution based on the following table:
   
2. 2 Direction of force. Remember that acceleration is always co-directional with the force acting on the body. Some problems provide data that is intended to mislead you.
3. Example: A toy boat with a mass of 10 kg is moving north with an acceleration of 2 m/s2. A wind blowing in a westerly direction exerts a force of 100 N on the boat. Find the acceleration of the boat in a northerly direction.
4. Solution: Since the force is perpendicular to the direction of movement, it does not affect the movement in that direction. Therefore, the acceleration of the boat in the north direction will not change and will be equal to 2 m/s2.
5. 3 Resultant force. If several forces act on a body at once, find the resulting force, and then proceed to calculate the acceleration. Consider the following problem (in two-dimensional space):
6. Vladimir pulls (on the right) a container of mass 400 kg with a force of 150 N. Dmitry pushes (on the left) a container with a force of 200 N. The wind blows from right to left and acts on the container with a force of 10 N. Find the acceleration of the container.
7. Solution: The conditions of this problem are designed to confuse you. In fact, everything is very simple.

   Newton's second law

   Draw a diagram of the direction of forces, so you will see that a force of 150 N is directed to the right, a force of 200 N is also directed to the right, but a force of 10 N is directed to the left. Thus, the resulting force is: 150 + 200 - 10 = 340 N. The acceleration is: a = F/m = 340/400 = 0.85 m/s2.

Sent by: Veselova Kristina. 2017-11-06 17:28:19

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Lesson 5. DEPENDENCE OF MASS ON SPEED. RELATIVISTIC DYNAMICS

Newton's laws of mechanics do not agree with the new space-time concepts at high speeds of motion. Only at low speeds of movement, when classical ideas about space and time are valid, Newton’s second law

does not change its shape when moving from one inertial frame of reference to another (the principle of relativity is fulfilled).

But at high speeds this law in its usual (classical) form is unfair.

According to Newton's second law (2.4), a constant force acting on a body for a long time can impart an arbitrarily high speed to the body. But in reality, the speed of light in a vacuum is limiting, and under no circumstances can a body move at a speed exceeding the speed of light in a vacuum. A very small change in the equation of motion of bodies is required for this equation to be correct at high speeds. Let us first move on to the form of writing the second law of dynamics that Newton himself used:

where is the momentum of the body. In this equation, body mass was considered independent of speed.

It is striking that even at high speeds, equation (2.5) does not change its form.

The changes concern only the masses. As the speed of a body increases, its mass does not remain constant, but increases.

The dependence of mass on speed can be found based on the assumption that the law of conservation of momentum is also valid under new concepts of space and time. The calculations are too complicated. We present only the final result.

If through m0 denote the mass of a body at rest, then the mass m the same body, but moving with speed, is determined by the formula

Figure 43 shows the dependence of body mass on its speed. The figure shows that the increase in mass is greater, the closer the speed of movement of the body is to the speed of light With.

At speeds of movement much lower than the speed of light, the expression differs extremely little from unity. So, at a speed faster than a modern space rocket u" We get 10 km/s =0,99999999944 .

It is therefore not surprising that it is impossible to notice an increase in mass with increasing speed at such relatively low speeds. But elementary particles in modern charged particle accelerators reach enormous speeds. If the speed of a particle is only 90 km/s less than the speed of light, then its mass increases 40 times.

Calculation of force F

Powerful electron accelerators are capable of accelerating these particles to speeds that are only 35-50 m/s less than the speed of light. In this case, the mass of the electron increases approximately 2000 times. In order for such an electron to be kept in a circular orbit, a force must act on it from the magnetic field that is 2000 times greater than one would expect without taking into account the dependence of mass on speed. It is no longer possible to use Newtonian mechanics to calculate the trajectories of fast particles.

Taking into account relation (2.6), the momentum of the body is equal to:

The basic law of relativistic dynamics is written in the same form:

However, the momentum of the body is determined here by formula (2.7), and not simply by the product.

Thus, mass, considered constant since Newton's time, actually depends on speed.

As the speed of movement increases, the mass of the body, which determines its inert properties, increases. At u®с body weight in accordance with equation (2.6) increases unlimitedly ( m®¥); therefore, the acceleration tends to zero and the speed practically stops increasing, no matter how long the force acts.

The need to use the relativistic equation of motion when calculating charged particle accelerators means that the theory of relativity in our time has become an engineering science.

Newton's laws of mechanics can be considered as a special case of relativistic mechanics, valid at speeds of motion of bodies much lower than the speed of light.

The relativistic equation of motion, which takes into account the dependence of mass on velocity, is used in the design of particle accelerators and other relativistic devices.

? 1 . Write down the formula for the dependence of body mass on the speed of its movement. 2 . Under what condition can the mass of a body be considered independent of speed?

mathematics formulas, linear algebra and geometry

§ 100. Expression of kinetic energy through the mass and speed of a body

In §§ 97 and 98 we saw that it is possible to create a store of potential energy by causing some force to do work, lifting a load or compressing a spring. In the same way, it is possible to create a reserve of kinetic energy as a result of the work of some force. Indeed, if a body, under the influence of an external force, receives acceleration and moves, then this force does work, and the body acquires speed, i.e., acquires kinetic energy. For example, the pressure force of powder gases in the barrel of a gun, pushing out a bullet, does work, due to which a reserve of kinetic energy of the bullet is created. Conversely, if work is done as a result of the movement of the bullet (for example, the bullet rises up or, hitting an obstacle, causes destruction), then the kinetic energy of the bullet decreases.

Let us trace the transition of work into kinetic energy using an example when only one force acts on a body (in the case of many forces, this is the resultant of all forces acting on the body). Let us assume that a constant force begins to act on a body of mass , which was at rest; under the influence of a force, the body will move uniformly accelerated with acceleration . Having traveled a distance in the direction of the force, the body will acquire a speed associated with the distance traveled by the formula (§ 22). From here we find the work of force:

.

In the same way, if a force directed against its movement begins to act on a body moving at a speed, then it will slow down its movement and stop, having done work against the acting force, also equal to , before stopping. This means that the kinetic energy of a moving body is equal to half the product of its mass and the square of its speed:

Since a change in kinetic energy, like a change in potential energy, is equal to the work (positive or negative) produced by this change, kinetic energy is also measured in units of work, i.e., joules.

100.1. A body of mass moves with speed due to inertia. A force begins to act on the body along the direction of motion of the body, as a result of which after some time the speed of the body becomes equal to . Show that the increment in the kinetic energy of a body is equal to the work done by the force for the case when the speed: a) increases; b) decreases; c) changes sign.

100.2. What is the most work spent on: giving a stationary train a speed of 5 m/s or accelerating it from a speed of 5 m/s to a speed of 10 m/s?

How to find the mass of a car in physics

How to find mass knowing speed

You will need

• - pen;
• - paper for notes.

Instructions

The simplest case is the movement of one body with a given uniform speed. The distance that the body has traveled is known. Find the travel time: t = S/v, hour, where S is the distance, v is the average speed of the body.

The second example is for the oncoming movement of bodies. A car moves from point A to point B at a speed of 50 km/h. A moped simultaneously drove towards him from point B at a speed of 30 km/h. The distance between points A and B is 100 km. You need to find the time after which they will meet.

Label the meeting point with the letter K. Let the distance AK traveled by the car be x km. Then the motorcyclist’s path will be 100 km. From the conditions of the problem it follows that the travel time for a car and a moped is the same. Make up the equation: x/v = (S-x)/v’, where v, v’ are the speeds of the car and moped. Substituting the data, solve the equation: x = 62.5 km. Now find the time: t = 62.5/50 = 1.25 hours or 1 hour 15 minutes. Third example - the same conditions are given, but the car left 20 minutes later than the moped. Determine how long the car will travel before meeting the moped. Create an equation similar to the previous one. But in this case, the travel time of a moped will be 20 minutes longer than that of a car. To equalize the parts, subtract one third of an hour from the right side of the expression: x/v = (S-x)/v’-1/3. Find x – 56.25. Calculate the time: t = 56.25/50 = 1.125 hours or 1 hour 7 minutes 30 seconds.

The fourth example is a problem involving the movement of bodies in one direction. A car and a moped are moving from point A at the same speeds. It is known that the car left half an hour later. How long will it take him to catch up with the moped?

In this case, the distance traveled by the vehicles will be the same. Let the car travel time be x hours, then the moped travel time will be x+0.5 hours. You have the equation: vx = v’(x+0.5). Solve the equation by plugging in the speed and find x - 0.75 hours or 45 minutes.

Fifth example – a car and a moped are moving at the same speeds in the same direction, but the moped left point B, located 10 km from point A, half an hour earlier. Calculate how long after the start the car will catch up with the moped.

The distance traveled by the car is 10 km more. Add this difference to the motorcyclist’s path and equalize the parts of the expression: vx = v’(x+0.5)-10. Substituting the speed values ​​and solving it, you will get the answer: t = 1.25 hours or 1 hour 15 minutes.

Elastic force acceleration

• what is the speed of the time machine

How to find mass?

Many of us at school asked the question: “How to find body mass”? Now we will try to answer this question.

Finding mass through its volume

Let's say you have a two hundred liter barrel at your disposal. You intend to completely fill it with diesel fuel, which you use to heat your small boiler room. How to find the mass of this barrel filled with diesel fuel? Let's try to solve this seemingly simplest problem together with you.

Solving the problem of how to find the mass of a substance through its volume is quite easy. To do this, apply the formula for the specific density of a substance

where p is the specific density of the substance;

m—its mass;

v - occupied volume.

The measures of mass will be grams, kilograms and tons. Volume measures: cubic centimeters, decimeters and meters. Specific density will be calculated in kg/dm³, kg/m³, g/cm³, t/m³.

Thus, in accordance with the conditions of the problem, we have at our disposal a barrel with a volume of two hundred liters. This means that its volume is 2 m³.

But you want to know how to find mass. From the above formula it is derived as follows:

First we need to find the value p - the specific density of diesel fuel. You can find this value using the reference book.

In the book we find that p = 860.0 kg/m³.

Then we substitute the obtained values ​​into the formula:

m = 860*2 = 1720.0 (kg)

Thus, the answer to the question of how to find the mass was found. One ton and seven hundred and twenty kilograms is the weight of two hundred liters of summer diesel fuel. Then you can make an approximate calculation of the total weight of the barrel and the capacity of the rack for the barrel of solarium in the same way.

Finding mass through density and volume

Very often in practical tasks in physics you can find quantities such as mass, density and volume. In order to solve the problem of how to find the mass of a body, you need to know its volume and density.

Items you will need:

1) Roulette.

2) Calculator (computer).

3) Capacity for measurement.

4) Ruler.

It is known that objects with the same volume, but made of different materials, will have different masses (for example, metal and wood). The masses of bodies that are made of a certain material (without voids) are directly proportional to the volume of the objects in question. Otherwise, the constant is the ratio of the mass to the volume of an object. This indicator is called “substance density”. We will denote it by the letter d.

Now you need to solve the problem of how to find the mass in accordance with the formula d = m/V, where

m is the mass of the object (in kilograms),

V is its volume (in cubic meters).

Thus, the density of a substance is the mass per unit volume.

If you need to find the density of the material from which an object is made, you should use the density table, which can be found in a standard physics textbook.

The volume of an object is calculated using the formula V = h * S, where

V – volume (m³),

H – object height (m),

S – area of ​​the base of the object (m²).

If you cannot clearly measure the geometric parameters of the body, then you should resort to the laws of Archimedes. To do this, you will need a vessel that has a scale used to measure the volume of liquids and lower the object into water, that is, into a vessel that has divisions on it. The volume by which the contents of the vessel will be increased is the volume of the body that is immersed in it.

Knowing the volume V and density d of an object, you can easily find its mass using the formula m = d * V. Before calculating the mass, you need to bring all measurement units into a single system, for example, into the SI system, which is an international measuring system.

In accordance with the above formulas, the following conclusion can be drawn: to find the required amount of mass with a known volume and known density, it is necessary to multiply the density value of the material from which the body is made by the volume of the body.

Calculation of body mass and volume

In order to determine the density of a substance, it is necessary to divide the mass of a body by its volume:

Body weight can be determined using scales. How to find the volume of a body?

If the body has the shape of a rectangular parallelepiped (Fig. 24), then its volume is found according to the formula

If it has some other shape, then its volume can be found using a method that was discovered by the ancient Greek scientist Archimedes in the 3rd century. BC e.

Archimedes was born in Syracuse on the island of Sicily. His father, the astronomer Phidias, was a relative of Hiero, who became in 270 BC. e. king of the city in which they lived.

Not all of Archimedes' works have reached us. Many of his discoveries became known thanks to later authors, whose surviving works describe his inventions. So, for example, the Roman architect Vitruvius (1st century BC) in one of his writings told the following story: “As for Archimedes, of all his many and varied discoveries, the discovery that I will talk about seems to me to have been made with boundless wit. During his reign in Syracuse, after the successful completion of all his activities, Hiero vowed to donate a golden crown to the immortal gods in some temple. He agreed with the master on a high price for the work and gave him the required amount of gold by weight. On the appointed day, the master brought his work to the king, who found it perfectly executed; After weighing, the weight of the crown turned out to correspond to the issued weight of gold.

After this, a denunciation was made that part of the gold had been taken from the crown and the same amount of silver had been mixed in instead. Hiero was angry that he had been tricked, and, not finding a way to expose this theft, asked Archimedes to think carefully about it. He, immersed in thoughts on this issue, somehow accidentally came to the bathhouse and there, plunging into the bathtub, he noticed that the same amount of water was flowing out of it as the volume of his body immersed in the bathtub. Having realized the value of this fact, he, without hesitation, jumped out of the bath with joy, went home naked and in a loud voice informed everyone that he had found what he was looking for. He ran and shouted the same thing in Greek: “Eureka, Eureka! (Found, found!)."

Then, writes Vitruvius, Archimedes took a vessel filled to the top with water and dropped into it a gold bar equal in weight to the crown. Having measured the volume of displaced water, he again filled the vessel with water and lowered the crown into it. The volume of water displaced by the crown turned out to be greater than the volume of water displaced by the gold bar. The larger volume of the crown meant that it contained a substance less dense than gold. Therefore, the experiment carried out by Archimedes showed that part of the gold was stolen.

So, to determine the volume of a body that has an irregular shape, it is enough to measure the volume of water displaced by this body. If you have a measuring cylinder (beaker), this is easy to do.

In cases where the mass and density of a body are known, its volume can be found using the formula following from formula (10.1):

This shows that to determine the volume of a body, the mass of this body must be divided by its density.

If, on the contrary, the volume of a body is known, then, knowing what substance it consists of, one can find its mass:

To determine the mass of a body, the density of the body must be multiplied by its volume.

1. What methods of determining volume do you know? 2. What do you know about Archimedes? 3. How can you find the mass of a body based on its density and volume? Experimental task. Take a piece of soap that has the shape of a rectangular parallelepiped, on which its mass is indicated. After taking the necessary measurements, determine the density of the soap.

In chemistry you can’t do without a lot of substances. After all, this is one of the most important parameters of a chemical element. We will tell you in this article how to find the mass of a substance in various ways.

First of all, you need to find the desired element using the periodic table, which you can download on the Internet or buy. The fractional numbers under the sign of an element are its atomic mass. It needs to be multiplied by the index. The index shows how many molecules of an element are contained in a given substance.

1. When you have a complex substance, you need to multiply the atomic mass of each element of the substance by its index. Now you need to add up the atomic masses you obtained. This mass is measured in units of gram/mol (g/mol). We will show how to find the molar mass of a substance using the example of calculating the molecular mass of sulfuric acid and water:

   H2SO4 = (H)*2 + (S) + (O)*4 = 1*2 + 32 + 16*4 = 98g/mol;

   H2O = (H)*2 + (O) = 1*2 + 16 = 18g/mol.

   The molar mass of simple substances that consist of one element is calculated in the same way.

2. You can calculate molecular weight using an existing table of molecular weights, which can be downloaded online or purchased at a bookstore
3. You can calculate molar mass using formulas and equate it to molecular mass. In this case, the units of measurement must be changed from “g/mol” to “amu”.

   When, for example, you know the volume, pressure, mass and temperature on the Kelvin scale (if Celsius, then you need to convert), then you can find out how to find the molecular mass of a substance using the Mendeleev-Claiperon equation:

   M = (m*R*T)/(P*V),

   where R is the universal gas constant; M is the molecular (molar mass), a.m.u.

4. You can calculate the molar mass using the formula:

   where n is the amount of substance; m is the mass of a given substance. Here you need to express the amount of substance using volume (n = V/VM) or Avogadro's number (n = N/NA).

5. If the volume of a gas is given, then its molecular weight can be found by taking a sealed container with a known volume and pumping out the air from it. Now you need to weigh the cylinder on the scales. Next, pump gas into it and weigh it again. The difference in mass of an empty cylinder and a cylinder with gas is the mass of the gas we need.
6. When you need to carry out the cryoscopy process, you need to calculate the molecular weight using the formula:

   M = P1*Ek*(1000/P2*Δtk),

   where P1 is the mass of the dissolved substance, g; P2 is the mass of the solvent, g; Ek is the cryoscopic constant of the solvent, which can be found from the corresponding table. This constant is different for different liquids; Δtk is the temperature difference, which is measured using a thermometer.

Now you know how to find the mass of a substance, be it simple or complex, in any state of aggregation.

In chemistry and physics, we often come across problems in which it is necessary to calculate the mass of a substance, knowing its volume. How to find mass through volume. A table of densities will help you with this, since in order to find the mass, you need to know both the density and volume of the substance.

If the problem statement does not indicate density, you can look at the table, which contains such data about each substance. Ideally, of course, you need to learn such a table, but you can also refer to a chemistry textbook.

The rule states that the volume of a substance multiplied by its density equals the mass of that substance. From this rule, the formula for mass through volume is derived. It looks like this: m = V*p. Where m is mass, V is volume, and p is density. Knowing the number that is equal to the volume, you can look up the number that will be equal to the density and multiply the data. This way you can get a lot.

Example calculation

For example, a volume of 5 ml is given. The volume of a substance is calculated in units such as liters and milliliters. The substance whose mass needs to be found is gelatin. Looking at the table, you can see that its density is 1.3 g/ml. Now use the formula. Volume V is 5 ml. It is necessary to multiply 5 ml. by 1.3 g/ml. That is: 5 * 1.3 = 6.5 grams. So m - mass is 6.5 grams. Why gram: when multiplying volume by density, we have units such as milligrams. We reduce them, leaving grams, which indicate mass.

You can use another method. It is necessary to know or have at hand the periodic table. This method involves using the molar mass of the substance (in the table). You need to know the formula, which states that the mass of a substance is equal to the product of volume and molar mass. That is, m = V*M, where V is the volume of a given substance, and M is its molar mass.

In actual problems in physics and mathematics, quantities such as volume, mass and density. Knowing the density and volume of a body or substance, it is absolutely possible to detect it mass .

You will need

• – computer or calculator;
• – roulette;
• – measuring container;
• - ruler.

Instructions

1. As you know, objects that have the same volume, but are made of different materials, will have different masses (wood and metal, glass and plastic). The masses of bodies made of the same substance (without voids) are directly proportional to the volume of the objects in question. On the contrary, a continuous quantity is the ratio of the mass of an object to its volume. This quantity is called the “density of a substance.” In the future we will denote it by the letter d.

2. Based on the definition, d=m/V, where m is the mass of an object (kg), V is its volume (m3). As can be seen from the formula, the density of a substance is the mass per unit of its volume.

3. You can find out the density of the substance from which an object is made from the density table in the appendix to a physics textbook or on the website http://www.kristallikov.net/page15.html, where the densities of virtually all existing substances are given.

5. If it is not possible to accurately measure the geometric dimensions of a body, use Archimedes' law. To do this, take a vessel that has a scale (or divisions) for measuring the volume of liquid, lower the object into water (into the vessel itself, equipped with divisions). The volume that increases the contents of the vessel is the volume of the body immersed in it.

6. If the density d and volume V of an object are known, it is always possible to determine its mass using the formula: m=V*d. Before calculating mass, convert all units of measurement into one system, say, the SI international system of measurement.

7. The result from the above formulas is the following: in order to obtain the desired value of mass, knowing the density and volume, you need to multiply the value of the volume of the body by the value of the density of the substance from which it is made.

Mass body traditionally determined experimentally. To do this, take a load, put it on the scales and get the measurement result. But when solving physical problems given in textbooks, measuring mass for objective reasons is unrealistic, but there is certain data about the body. Knowing these data, it is possible to determine the mass body implicitly by calculation.

Instructions

1. In school courses in physics, chemistry, and astronomy, one can encounter the representation of mass. By weight body find reciprocal quantities - volume, density, force. Mass is a quantitative indicator of a substance; therefore, in chemistry problems, the number of a substance is found based on mass. Mass depends on the properties of the substance of which the body is composed, as well as on the number of this substance. There are several main ways to calculate mass. They are chosen depending on what other physical quantities are specified in the problem. Let's look at each case separately.

2. The most commonly known method for finding mass body is its calculation based on volume and density. True, in a number of problems, before determining the mass, it is necessary to calculate the volume itself, guided by other geometric calculations body. Let's say, for a cylinder with a known base area and height, made of a substance with a known density, the mass will be equal to: m=?*V=?*S*h, where Vcyl.=S*h, ? – density, S – area of ​​the base of the cylinder, h – height of the cylinder. If the volume is indicated directly in the problem, to find the mass it is quite primitive to multiply it by the density: m=?*V

3. Another branch of physics where it is necessary to calculate mass is dynamics. Traditionally, it studies the interaction between body mi, the action of external forces on body, the state of bodies in uniform motion. Any body with force F receives acceleration when interacting with another body. At the same time, it has a certain mass m. Mass is related to force by the following relation: F=m*a, where a is the acceleration of a given body; m - mass body From here you can find out the mass body:m=F/a

4. In chemistry textbooks we come across representations of the number of a substance and molar mass. Through these two quantities it is also possible to express the mass of a substance. Since the number of a substance is a physical quantity proportional to the number of particles that make up the substance, and the molar mass is the mass of one mole of a substance, the mass of a given number of this substance can be calculated as follows: mв = Mв * nв, where Mв is the molar mass, nв - number of substance

Video on the topic

Helpful advice
An example of a problem for finding the mass of a body. It is possible that a small steel ball of radius R = 5 cm is given. Determine the mass of the ball if it is known that p iron = 7.8 mg/m^3. Initially, find the volume of the ball. It is equal to: V = 4? R ^ 2 = 4 * 3.14 * 25 = 314 cm ^ 3 The mass is calculated as follows: m = p * V = 7.8 * 314 = 24.492 g

Density is the ratio of mass to the volume it occupies - for solids, and the ratio of molar mass to molar volume - for gases. In its most general form, volume (or molar volume) will be the ratio of mass (or molar mass) to its density. Density vestima. What to do? First determine the mass, then calculate the volume, then make the necessary corrections.

Instructions

1. The volume of a gas is equal to the ratio of the product of the number of a substance multiplied by its molar mass to the already known density. In other words, even knowing the density, you need to know the molar mass of the gas and the number of the substance, that is, how many moles of gas you have. In the thesis, knowing how many moles of gas you have, you can calculate its volume, even without knowing the density - according to Avogadro’s law, one mole of any gas occupies a volume of 22.4 liters. If you definitely calculate the volume through density, then you will need to find out the mass of gas in an as yet unknown volume.

2. The volume of a solid body can be determined, even without knowing the density, by easily measuring it, and in the case of a difficult and very irregular shape, the volume is determined, say, by the volume of liquid displaced by the solid body. However, if you need to calculate volume specifically through density, then the volume of a solid body is the ratio of the mass of the body to its density, and mass is usually determined by simple weighing. If weighing the body for some reason (say, it is too huge or moving) is unthinkable, then you will have to resort to rather difficult indirect calculations. For example, for a moving body, mass is the ratio of twice the kinetic energy to the square of its speed, or the ratio of the force applied to the body to its acceleration. For a very large body at rest, one will have to resort to calculations in relation to the mass of the Earth, using the gravitational continuum and the moment of rotation. Or - through the calculation of the specific heat capacity of the substance; in any case, using only density to calculate volume will be unsatisfactory.

3. Having calculated the mass of a solid, you can calculate the volume by simply dividing the mass by the density.

Note!
1. The above methods are more or less applicable only in the case of homogeneity of the substance of which the solid body consists2. The above methods are more or less applicable in a relatively narrow temperature range - from minus 25 to plus 25 degrees Celsius. When the state of aggregation of a substance changes, the density can change abruptly; in this case, the formulas and calculation methods will be completely different.

Mass as a physical quantity is a parameter that characterizes the force of a body’s influence on gravity. To calculate body weight in physics it is required to know two of its quantities: the density of the body material and its volume.

Instructions

1. Let a certain body be given with volume V and density of its substance p. Then it's mass calculated like this: m = p*V. For clarity, an example is given: Let an aluminum block with a volume of 5 cubic meters be given. meters. The density of aluminum is 2700 kg/cubic. meter. In this case, the mass of the block will be: m = 2700/5 = 540 kg.

Note!
The concept of mass is often confused with another, no less rare, physical quantity - weight. Is weight measured in n/m? and characterizes the force that acts on the fulcrum. Mass, by its nature, does not have any point of support, and, as noted, affects only the gravity of the Earth.

When solving some physical problems, it is necessary to detect density body. Occasionally, the density of a physical body needs to be determined in practice, say, in order to find out whether it will sink or not. By the way, the human body can also be classified as a physical body. Moreover, the concept of the “density” of the human body has long ago come into use. Thus, a “firmly built” person is traditionally called “dense,” and one who has the opposite body constitution is called “loose.”

You will need

• calculator, scales, ruler, measuring cup, table of density of substances.

Instructions

1. In order to detect the density of a physical body, determine what substance or material it consists of. After this, take the table of density of substances and find the corresponding substance in it. So, let's say, if an object is made of aluminum, its density will be 2.7 g/cm?.

2. If the body consists of several substances, then find the density of all of them in the corresponding tables. In order to detect the density of a body in its entirety, determine the contribution of the entire substance to the formation of the density of the object. To do this, determine the volume or mass of the entire homogeneous part, and then calculate the mass and volume of each body.

3. Let, say, the body consists of 2 parts with masses m1 and m2, respectively. The density of the entire part is ?1 and ?2. In order to find the average density of a body, find the total volume: V = V1 + V2 = m1 * ?1 + m2 * ?2, and then divide by the total mass of the body (m = m1 + m2): ? = V / m = (m1 * ?1 + m2 * ?2) / (m1 + m2), where: V – total volume of the body; V1 and V2 – volume of the first and 2nd parts of the body, respectively; m – total body mass ;m1 and m2 are the mass of the first and second parts of the body, respectively;? – average density of the body; ?1 and ?2 – density of the first and 2nd parts of the body, respectively.

4. If the volumes (V1 and V2) of the entire part of the body, as well as their densities, are known, to calculate the density of the body, use a similar formula:? = V / m = (V1 + V2) / (m1 + m2) = (V1 + V2) / (V1 / ?1 + V2 / ?2). The parameter designations are the same as in the previous formula.

5. If the material (substance) of which the body is composed is unknown or has a variable density (say, wood, the density of which depends on humidity), in order to determine its density, determine its volume and divide by mass. That is, use the formula:? = V / m. To do this, you will finally have to calculate or measure the volume and mass of the body, but this method will give the most accurate result. If the body has the shape of a primitive geometric figure, calculate its volume using the appropriate stereometry formulas. Determine the volume of difficult bodies through the volume of liquid displaced by them. Detect your body weight with weighing support.

Tip 6: How to detect mass if volume and density are known

The mass of a body is its most important physical factor. In modern physical science, there is a distinction between the concept of “mass”: gravitational mass (as the degree of influence of a body on earth’s gravity) and inertial mass (what force will be needed to bring the body out of a state of inertia). In any case, discover mass very easy if you are famous density and body volume.

Instructions

1. In the event that the body has known indicators such as its volume (V) and density(p), then to calculate body weight you will need to use the formula: m = p*V.

2. For clarity, it is permissible to give an example. Required to be discovered mass concrete slab, whose volume is 15 m?. Solution: to find the mass of a concrete slab, you only need to know it density. In order to find out this information, you need to use a table of densities of different substances.

3. According to this table density concrete is 2300 kg/m?. Then in order to discover mass concrete slab, you will need to perform a primitive algebraic operation: m = 15 * 2300 = 34500 kg, or 34.5 tons. Result: the mass of the concrete slab is 34.5 tons

4. The traditional method of measuring mass occurs using one of the oldest instruments in society - supported by scales. This occurs due to the comparison of body weight with the help of the reference mass of the load - weights.

Note!
When carrying out the calculation using the above formula, you need to understand that in this way the rest mass of a given body is known. A fascinating fact is that many elementary particles have an oscillating mass, which depends on the speed of their movement. If an elementary particle moves with the speed of a body, then this particle is massless (say, a photon). If the speed of a particle is lower than the speed of light, then such a particle is called bulky.

Helpful advice
When measuring mass, it is never possible to forget in which system the final result will be given. This means that in the SI system mass is measured in kilograms, while in the CGS system mass is measured in grams. Mass is also measured in tons, centners, carats, pounds, ounces, poods, and many other units depending on the country and culture. In our country, for example, mass has long been measured in poods, berks, and zolotniks.

You have a two hundred liter barrel. You plan to fill it entirely with diesel fuel, which you use to heat your mini-boiler room. How much will it weigh filled with diesel fuel? Now let's calculate.

You will need

• – table of specific densities of substances;
• – knowledge of making simple mathematical calculations.

Instructions

1. In order to determine the mass of a substance by its volume, use the formula for the specific density of the substance. p = m/vhere p is the specific density of the substance; m is its mass; v is the occupied volume. We will calculate mass in grams, kilograms and tons. Volumes in cubic centimeters, decimeters and measures. And the specific density, respectively, in g/cm3, kg/dm3, kg/m3, t/m3.

2. It turns out that according to the conditions of the problem, you have a two-hundred-liter barrel. This means: a barrel with a capacity of 2 m3. It is called a two-hundred-liter barrel because water, with its specific density equal to one, contains 200 liters in such a barrel. You are concerned about the mass. Consequently, bring it to the first place in the presented formula.m = p*vOn the right side of the formula, the value p is unfamiliar - the specific density of diesel fuel. Find it in the directory. It’s even easier to search the Internet for “specific gravity of diesel fuel.”

3. We discovered: the density of summer diesel fuel at t = +200 C is 860 kg/m3. Substitute the values ​​into the formula: m = 860*2 = 1720 (kg) 1 ton and 720 kg - this is how much 200 liters of summer diesel fuel weigh. Having hung the barrel in advance, you can calculate the total weight and estimate the capacity of the rack for the barrel of solarium.

4. In rural areas, it is useful to calculate in advance the mass of firewood needed by cubic capacity in order to determine the carrying capacity of the transport on which this firewood will be delivered. For example, you need at least 15 cubic meters for the winter. meters of birch firewood. Look in reference books for the density of birch firewood. This is: 650 kg/m3. Calculate the mass by substituting the values ​​into the same specific density formula. m = 650 * 15 = 9750 (kg) Now, based on the load capacity and capacity of the body, you can decide on the type of vehicle and the number of trips.

Video on the topic

Note!
Older people are more familiar with the concept of specific gravity. The specific density of a substance is the same as specific gravity.

There are situations when you need to calculate mass liquids contained in some container. This can be during a training session in the laboratory, or while solving a household problem, say, when repairing or painting.

Instructions

1. The easiest way is to resort to weighing. First, weigh the container together with the liquid, then pour the liquid into another container of suitable size and weigh the empty container. And after that, all that remains is to subtract the smaller value from the larger value, and you will get the result. Of course, this method can only be used when dealing with non-viscous liquids, which, after overflowing, do not actually remain on the walls and bottom of the first container. That is, some number will still remain, but it will be so small that it can be neglected; this will not affect the accuracy of the calculations.

2. What if the liquid is viscous, say glycerin? How then to determine it mass? In this case, you need to know its density (?) and occupied volume (V). And then everything is more elementary. Mass (M) is calculated using the formula M = ?V. Of course, before calculating, you need to convert the factors into an integral system of units.

3. Density liquids can be found in a physical or chemical reference book. But it’s cooler to use a measuring device - a density meter (densitometer). And the volume can be calculated by knowing the shape and overall dimensions of the container (if it has the correct geometric shape). Let's say, if the same glycerin is in a cylindrical barrel with a base diameter d and height h, then the volume of the barrel is calculated by the formula: ?d^2h/4.

4. Let's imagine you are given such a task. During a laboratory experiment, a liquid of mass m, located in a calorimeter container and having a heat capacity c, was heated from the initial temperature t1 to the final temperature t2. A quantity of heat equal to Q was expended on this heating. What is the mass of this liquids ?

5. All quantities, in addition to m, are known; heat losses during the experiment can be neglected. There is certainly nothing difficult in the calculation. You just need to remember the formula that combines the number of heat, mass liquids, its heat capacity and temperature difference. It is as follows: Q = mc(t2-t1). Consequently, the mass liquids calculated by the formula: m = Q/c(t2-t1). By substituting the quantities you know into the formula, you can easily calculate mass liquids m.

The value of Planck's continuous, denoted by the letter h, was determined experimentally in laboratory conditions with an accuracy of ten decimal places. By its definition, it is also possible to perform a skill in a physical classroom, but the accuracy will be much less.

You will need

• – photocell with external photoelectric effect;
• – light source with monochromator;
• – continuously adjustable 12 V power supply;
• – voltmeter;
• – microammeter;
• – lamp 12 V, 0.1 A;
• – a calculator that works with numbers represented in exponential form.

Instructions

1. Use a photocell with an external photoeffect for the skill. An element with an internal photoelectric effect (i.e., not a vacuum, but a semiconductor) will not work. Test it for suitability for carrying out the skill, for which connect it to the microammeter easily, observing the polarity. Point the light at it - the arrow should deviate. If this does not happen, use a different type of photocell.

2. Without changing the polarity of connecting either the photocell or the microammeter, break the circuit and connect an adjustable power source to its break, the output voltage of which can be smoothly changed from 0 to 12 V (with two knobs for bold and precise adjustment). Attention: this source should be turned on not in direct, but in reverse polarity, so that with its voltage it does not increase, but reduces the current through the element. Connect a voltmeter in parallel to it - this time in the polarity corresponding to the markings on the source. This may not be done if the unit has a built-in voltmeter. Also connect a load in parallel with the output, say a 12V, 0.1A lamp, in case the source's internal resistance is high. The lamp light should not fall on the photocell.

3. Set the source voltage to zero. Direct a stream of light from a source with a monochromator into the photocell, setting the wavelength to about 650 nanometers. Smoothly increasing the voltage of the power source, ensure that the current through the microammeter becomes zero. Leave the regulator in this location. Record the voltmeter and monochromator scale readings.

4. Set the monochromator to a wavelength of about 450 nanometers. Slightly increase the output voltage of the power supply so that the current through the photocell returns to zero. Record the new voltmeter and monochromator scale readings.

5. Calculate the frequency of light in hertz for the first and second skills. To do this, divide the speed of light in vacuum, equal to 299,792,458 m/s, by the wavelength, converted in advance from nanometers to meters. For simplicity, assume the refractive index of air to be 1.

6. Subtract the larger voltage from the smaller one. Multiply the total by the charge of the electron, equal to 1.602176565(35)·10^(?19) coulombs (C), and then divide by the total of subtracting the higher frequency from the lower. The result is a continuous Planck expressed in joules times a second (J s). If it is close to the official value of 6.62606957(29)·10^(-34) J·s, the skill can be considered positive.

Video on the topic

Note!
Use caution when working with electrical equipment.

Class: 7

Lesson objectives.

1. Educational: systematize the knowledge that students have about the concepts: “density”, “mass”, “volume”, expand the scope of knowledge about these concepts, develop the ability to apply the studied material to solve practical problems.

2. Developmental: formation of logical thinking, continue to develop the skill of solving physical problems.

3. Educational: instilling in students friendly communication and mutual assistance.

Lesson type: combined.

Equipment: 15 sets of tables 1 and 2.

During the classes

1. Organizational stage.

2. Updating knowledge.

Teacher activities	Student activities
- What is density?	A physical quantity that shows how much of a substance is contained in a unit volume.
- What does it mean that the density of iron is 6800 kg/m 3? How can you find the density of a substance? What does body weight depend on?	This means that the mass of 1 m 3 of iron is 6800 kg. To find the density of a substance, you need to divide the mass by the volume.
- How to find body mass?	The mass of a body depends on its volume and the density of the substance of which the body consists.
- How to find the volume of a body if its mass and the substance of which the body consists are known?	To find the mass of a body, gently multiply its density by its volume. To find the volume of a body, you need to divide its mass by its density.

3. Problem solving
1. How will we grade work in class based on the principle of addition or subtraction?	Based on the principle of addition.
2. Table 1(Appendix No. 1). There are loads in the warehouse: chalk, cork, birch, ice, steel. Each cargo is packed in 2 m3 containers. Five vehicles were called in to transport these goods. Your task is to distribute the cargo among the vehicles.	Find the mass of the loads.
- What needs to be done to distribute cargo among vehicles?
- How to find the mass of a substance if its density and volume are known?	kg/m 3
- In what units is the density of matter measured?	In kilograms
- In what units will the mass be calculated?	In tons and kilograms
- In what units is the carrying capacity of vehicles expressed?	In tons, and for a Muscovite in kilograms
- In what units should the mass of cargo be obtained? Solve this problem and distribute the cargo among the cars. The teacher checks the correctness of the completed task with the first student to solve it and appoints him as his assistant. In the cards (Appendix No. 3) of the students, records are made of the number of points scored.	Students solve problems and distribute loads.
3. table 2(Appendix No. 2). There are five different liquids that have the same mass. These liquids need to be poured into five different vessels. What needs to be done to pour liquids into vessels?	Find the volume of liquids.
- How to find the volume if the mass of the substance and its density are known?
- In what units will the calculated volume be obtained?	in m3.
- In what units is the volume of the vessels given?	In liters and milliliters
- In what units should the volume of liquids be obtained?	In liters and milliliters
Solve this problem and distribute the liquid among the vessels. The teacher checks the correctness of the completed task with the first student to solve it and appoints him as his assistant. The number of points scored is recorded on student cards.	Students solve the problem.
4. Reflection.
- What physical concepts did you use to complete the tasks? Compare the number of points you gave yourself with the number of points the evaluators gave you. What conclusion can you draw for yourself? Are you ready for the test?	Mass, density, volume.

Homework: repeat 18-22.