Mass weight m, suspended from a spring, oscillates with a period T and amplitude What will happen with the period of oscillations, the maximum potential energy of the spring and the frequency of oscillations, if the weight of the load is reduced at a constant amplitude?

For each quantity, determine the corresponding change pattern:

1) increased;

2) decreased;

3) has not changed.

Write down the selected numbers for each physical quantity in the table. The numbers in the answer may be repeated.

Decision.

The period of oscillation is related to the mass of the load and the stiffness of the spring k ratio With a decrease in mass, the period of oscillations will decrease. The frequency is inversely proportional to the period, which means that the frequency will increase.

With the maximum potential energy of the spring, things are a little more complicated. To answer the question, what will happen to it, it is essential that the spring is oriented vertically (for a horizontal spring pendulum, with a constant amplitude, this value will naturally remain unchanged). Indeed, when a load is suspended from a vertical spring, it immediately stretches slightly to balance the force of gravity acting on the load. Let us define this initial tension: It is this state that is the equilibrium position for a vertical spring pendulum, oscillations occur around it, the load rises and falls from this position by the magnitude of the amplitude. When moving down from the equilibrium position, the spring continues to stretch, which means that the potential energy of the spring continues to increase. When moving upwards from a position of equilibrium, the deformation of the spring is first reduced, and if the spring begins to compress. The maximum potential energy of the spring corresponds to the state when it is maximally stretched, which means, in our case, this is the position when the load has dropped as far down as possible. Thus, the maximum potential energy of the spring is

It can be seen from this formula that for a vertical spring pendulum with a constant amplitude and a decrease in the mass of the load, the maximum potential energy of the spring will decrease.

Answer: 221.

Answer: 221

a guest 19.12.2012 02:31

i have a question, how to explain the oscillation frequency increases or not. thanks

a guest

Good day!

The solution says that the frequency is inversely proportional to the period:. After you figured out what happened to the period, how the frequency changed is already clear.

a guest 10.01.2013 18:51

Hello! Why will the maximum potential energy of the spring decrease? After all, the amplitude remains the same, and the maximum potential energy depends on the tensile force, that is, on the amplitude? that is, the maximum potential energy should remain the same ...

a guest

Good day!

Read the solution carefully, it says it all.

Nikita Losik 07.05.2013 10:33

You have the wrong answer, because the amplitude does not change, the potential energy does not change either, you have muddied up the solution too much.

http://sverh-zadacha.ucoz.ru/ege/2010prob/v3/2010B-3.htm

The very first task

Alexei

Good day!

Not too clever. The potential energy of a spring is a quantity. Here --- deformation. It is determined not only by the amplitude, but also by the initial stretching.

Please note that in such a pendulum, a spring in general can be stretched all the way, that is, its energy will not vanish at all, and the pendulum will oscillate as usual.

a guest 24.05.2013 09:50

Good afternoon! An indication at the end of a problem of a constant amplitude leads to incorrect decisions. This indication in itself is already an erroneous statement of the question. Therefore, the "broken spears" of predecessors are also.

Alexei

Good day!

Amplitude is a deviation from the equilibrium position. A change in mass leads to a change in the equilibrium position. No discrepancies should arise

Grigory Kovalchuk 22.11.2015 10:35

Hello, and due to what has the vibration frequency increased?

Irina Safiulina

By changing the weight of the cargo. For convenience, see the period formula for a mathematical pendulum. The period with decreasing mass decreases. Frequency is a quantity inversely proportional to the period, therefore, with a decrease in mass, the frequency increases.

a guest 23.02.2016 08:37

Gentlemen, in the MODEL of a vertical spring pendulum there is no dependence of the maximum energy of the system (what is potential, what is kinetic, and, as a consequence, their sum) on the mass of the load. The initial stretch compensates for the effect of gravity, and then gravity is not considered. Yes, mass enters into the equation of motion of the pendulum, it characterizes the oscillation frequency, but the energy of the system depends on the amplitude and stiffness of the spring, E \u003d (kA ^ 2) / 2. Potential energy E \u003d (kx ^ 2) / 2 is also independent of mass. Into the formula kinetic energy mass is included, but mathematics says that it also does not affect - when integrating the coordinates and squaring it, the masses in the formula for kinetic energy cancel out.

Of course, the spring itself gets an initial stretch. But its introduction (stretching) into the model of free undamped oscillations without reservations leads to violation of the law of conservation of energy. The potential E at the lower point is then equal to k (A + x0) ^ 2/2, and at the upper point k (A-x0) ^ 2/2, and the model requires to recognize them equal to each other and equal to the total energy. Then to comply with the Z.S.E. it is necessary to enter the potential energy of the load mgh into the model, but is this not in the school curriculum?

The question is formulated illiterately (by the way, it is not the only one, in the section of pendulums there is also an emphasis on the inequality of potential energies at the upper and lower points), the answer to it is offered incorrectly (within the framework of the school curriculum)

Anton

Let the axis be directed downwards and its origin is at the spring suspension point. Then the potential energy of the system is

Study of the law of conservation of mechanical energy under the action of gravity and elasticity forces on a body

I. Preparatory part.

1) Carefully watch the videos and while watching, write down the basic information on the topic "The Law of Conservation of Energy" in a notebook for practical work.

2) To carry out practical work No. 2, it is necessary to repeat the summary of the lesson No. 3 "Laws of conservation of energy" and read and analyze the following text.

Potential energy

A stone, falling from a certain height to the Earth, leaves a dent on the Earth's surface. During the fall, he does work to overcome the air resistance, and after touching the ground, he does the work to overcome the resistance force of the soil, since it has energy. If you pump air into a jar closed with a cork, then at a certain air pressure the cork will fly out of the jar, while the air will do the work to overcome the friction of the cork against the neck of the jar, due to the fact that the air has energy. Thus, the body can do work if it has energy.

When work is done, the state of the body changes and its energy changes. The change in energy is equal to the perfect work (therefore, both the unit of measurement of energy and the unit of work are J).

Potential energycalled the energy of interaction of bodies or body parts, depending on their relative position.

Since bodies interact with the Earth, then they have the potential energy of interaction with the Earth.

If the body mass mfalls from a height to the height , then the work of gravity location on is equal to: or (Figure 1).

fig. one

In the resulting formula characterizes initial position (state) body, characterizes end position (state) body. The quantity - potential energy of the body in the initial state ; magnitude - potential energy of the body in the final state .

You can write , or , or ... Thus, the work of gravity is equal to the change in the potential energy of the body... The "-" sign means that when the body moves downward and, accordingly, when the force of gravity performs positive work the potential energy of the body will decrease ... If the body rises up, then the work of gravity is negative, and body potential energy increases .

If the body is at a certain height h relative to the surface of the Earth, then its potential energy, in this state, is . The value of potential energy depends on the level relative to which it is measured. The level at which the potential energy is zero is called zero level.

In contrast to kinetic energy, bodies at rest have potential energy. Since potential energy is the energy of interaction, it does not refer to one body, but to a system of interacting bodies. In this case, this system consists of the Earth and the body raised above it.

Elastically deformed bodies have potential energy. Suppose the left end of the spring is secured and a weight is attached to the right end of the spring. If the spring is compressed by displacing its right end by, then an elastic force appears in the spring, directed to the right (Fig. 2).

fig. 2

If we now leave the spring to itself, then its right end will move, the elongation of the spring will be equal, and the elastic force.

The work of the elastic force is equal to the change in the potential energy of the spring.

- potential energy of the spring in the initial state

- potential energy of the spring in the final state

When the spring is stretched and compressed, the elastic force performs negative work, the potential energy of the spring increases , and when the spring moves to the equilibrium position, the elastic force performs positive work, and potential energy decreases .

If the spring is deformed and its turns are displaced relative to the equilibrium position by a distance x, then the potential energy of the spring in this state is .

Potential Energy Theorem:A physical body (or a system of bodies) will always strive to occupy a position in which the potential energy is equal to 0 or is minimal relative to other positions of this body.

Kinetic energy

Moving bodies can also do work. For example, a moving piston compresses the gas in the cylinder, a moving projectile pierces a target, etc. therefore, moving bodies have energy. The energy possessed by a moving body is called kinetic energy.Kinetic energy depends on body weight and speed:

This follows from the transformation of the work formula.

Job ... Force ... Substituting this expression into the work formula, we get .

Because

then or where

- kinetic energy of the body in the first state

- kinetic energy of the body in the second state

Thus, the work of force is equal to the change in the kinetic energy of the body , or ... This statement - kinetic energy theorem... if the force does positive work, then kinetic energy increases , if the work of the force is negative, then kinetic energy of the body decreases .

Mechanical energy

Total mechanical energy E body - a physical quantity equal to the sum of its potential and kinetic energy: .

The law of conservation of mechanical energy:The total mechanical energy of a closed system of bodies between which conservative forces act (gravitational forces or elastic forces) is conserved.

where and - potential energy and kinetic energy of a body in state 1 or at time 1,

where and - potential energy and kinetic energy of the body in state 2 or at time 2.

Conservative strength -force, the work of which when moving a material point depends only on the initial and final position of the point in space.

In real systems, frictional forces act that are not conservative, therefore, in such systems, the total mechanical energy is not conserved, it turns into internal energy.

3) For successful completion of the test tasks, study the examples of step-by-step solution of problems using the law of conservation of energy below.

Example 1:A body weighing 10 kg freely falls from a height of 20 m from a state of rest. What is the kinetic energy at the moment of impact on the Earth? At what point on the trajectory is the kinetic energy three times the potential? Neglect air resistance.

Step number	Algorithm	Performance
1.		A body weighing 10 kg freely falls from a height of 20 m from a state of rest. What is the kinetic energy at the moment of impact on the Earth? At what point on the trajectory is the kinetic energy three times the potential? Neglect air resistance.
2.	)	Given: t \u003d 10 kg 20 m
3.
4.
5.	Under the word "Solution" 1) we draw a schematic drawing explaining the situation in the problem,	Condition 0 t Condition 2 Condition 1
2) then we write down the energy conservation law in general form
3) further reasoning	Let's take the Earth as the reference body, then: Considering that , , we get: or (1) Let us write the law of conservation of energy for a point of the trajectory, where : Considering that and , we get: from where (2)
6.		(1) (2)
7.	Write down the answer	Answer: , .

Example 2: ... At what height will the kinetic energy of the stone be equal to its potential energy? Neglect air resistance.

Step number	Algorithm	Performance
1.	Read the text of the problem carefully	A stone is thrown vertically upward with speed ... At what height will the kinetic energy of the stone be equal to its potential energy? Neglect air resistance.
2.	Write down in "Given" the letter designation and numerical value known from the text physical quantities... You need to know the acceleration of gravity (in some problems it is allowed to round up to the value )	Given: 0
3.	Under the horizontal line, write down the letter designation of the unknown (required) value, the sign "\u003d" and "?"
4.	Check if all values \u200b\u200bare in SI units. If not, then translate and write the result in the column "in SI"	In this problem, all quantities are initially conditionally in the SI system, and, accordingly, we skip the column "in SI".
5.	Under the word "Solution" 4) we depict a schematic drawing explaining the situation in the task,	Condition 0 t Condition 2 , Condition 1
5) then we write down the law of conservation of energy in general form	According to the law of conservation of energy, since the system is closed ... You can also write .
6) further reasoning	Since the stone is thrown from the Earth, then and then . Considering that , then we get . Consequently,
6.	Substitute numerical values, along with units of measure, carry out calculation and work with units of measure
7.	Write down the answer	Answer:

If a body is attached to a stretched spring, then the spring will act on it with a certain force, under the action of which the body will begin to shift. Hence the work will be done

The force with which the spring acts on the body is not constant, so to calculate the work we will use graphically... Plot the dependence of the elastic force F \u003d kx on the coordinate, which is a straight line

The area of \u200b\u200bthe highlighted triangle below the graph is equal to the maximum work that the spring can do. It is clear that it is equal to:

In order to assign a potential energy equal to the maximum work (1) to the spring, it is necessary to show that this work does not depend on the trajectory of the body. To prove this statement, it is enough to consider the work on a small section of displacement Δr when moving along an arbitrary trajectory

In this case, this work is completely determined by the change in the deformation of the spring x, therefore it does not depend on the trajectory of the body.

Thus, the elastic forces obeying Hooke's law are potential, and the potential energy of a deformed spring is determined by the formula

The zero level of potential energy, calculated by formula (2), corresponds to an undeformed spring.
Let's calculate what minimum work should be done in order to stretch a spring with stiffness k by an amount x

To deform a spring, an external force must be applied to it. Obviously, this work will be minimal if the external applied force at any point is equal to the elastic force acting from the side of the spring. Therefore, the work of this force will be equal to: A \u003d kx2 / 2, that is, an increase in the potential energy of the spring.

19. Potential energy of gravitational attraction

All bodies with mass are attracted to each other with a force that obeys the law of universal gravitation of I. Newton. Therefore, attracting bodies have the energy of interaction

Let us show that the work of gravitational forces does not depend on the shape of the trajectory, that is, gravitational forces are also potential. To do this, consider the motion of a small body of mass m interacting with another massive body of mass M, which we will assume to be motionless
As follows from Newton's law, the force F acting between bodies is directed along the line connecting these bodies. Therefore, when a body m moves along an arc of a circle centered at the point where the body M is, the work of the gravitational force is zero, since the vectors of forces and displacements remain mutually perpendicular all the time. When moving along a segment directed to the center of the body M, the displacement vectors and forces are parallel, therefore, in this case, when the bodies approach, the work of the gravitational force is positive, and when the bodies move away, it is negative. Further, we note that in radial motion, the work of the force of attraction depends only on the initial and final distances between the bodies. So, when moving along the segments (Fig. 162) DE and D1E1, the perfect work is equal, since the laws of change of forces from distance on both segments are the same. Finally, an arbitrary trajectory of the body m can be divided into a set of arc and radial segments (for example, the broken line ABCDE).

When moving along arcs, the work is zero, when moving along radial segments, the work does not depend on the position of this segment, therefore, the work of the gravitational force depends only on the initial and final distances between the bodies, which was required to be proved.
Note that when proving potentiality, we used only the fact that gravitational forces are central, that is, directed along a straight line connecting the bodies, and did not mention a specific form of the dependence of the force on distance. Therefore, all central forces are potential.
We have proved the potentiality of the force of gravitational interaction between two point bodies. But for gravitational interactions, the principle of superposition is valid: the force acting on the body from the side of the system of point bodies is equal to the sum of the forces of pair interactions, each of which is potential, therefore, their sum is also potential. Indeed, if the work of each force of pair interaction does not depend on the trajectory, then their sum also does not depend on the shape of the trajectory. Thus, all gravitational forces are potential.
It remains for us to obtain a specific expression for the potential energy of gravitational interaction.
To calculate the work of the force of attraction between two point bodies, it is enough to calculate this work when moving along a radial segment when changing the distance from r1 to r2

Once again, we will use the graphical method, for which we will plot the dependence of the attractive force F \u003d GMm / r2 on the distance r between the bodies. Then the area under the graph of this dependence within the specified limits will be equal to the required work

Calculating this area is not a very difficult task, however, it requires certain mathematical knowledge and skills. Without going into the details of this calculation, we present the final result: for a given dependence of force on distance, the area under the graph, or the work of the force of attraction, is determined by the formula

А12 \u003d GMm (1 / r2 - 1 / r1).

Since we have proved that gravitational forces are potential, this work is equal to a decrease in the potential energy of interaction, that is

А12 \u003d GMm (1 / r2 - 1 / r1) \u003d −ΔU \u003d - (U2 - U1).

From this expression, you can determine the expression for the potential energy of the gravitational interaction:

U (r) \u003d −GMm / r. (one)

With this definition, the potential energy is negative and tends to zero at an infinite distance between the bodies: U (∞) \u003d 0. Formula (1) determines the work that the force of gravitational attraction will do with increasing distance from r to infinity, and since with such a motion the vectors forces and displacements are directed in opposite directions, then this work is negative. With the opposite motion, with the approach of bodies from an infinite distance to a distance r, the work of the force of attraction will be positive. This work can be calculated by determining the potential energy:

We emphasize that potential energy is a characteristic of the interaction of at least two bodies. One cannot say that the interaction energy "belongs" to one of the bodies, or how to "divide this energy between the bodies." Therefore, when we talk about a change in potential energy, we mean a change in the energy of a system of interacting bodies. However, in some cases it is still permissible to talk about a change in the potential energy of one body. So, when describing the motion of a small, in comparison with the Earth, body in the gravity field of the Earth, we talk about the force acting on the body from the Earth, as a rule, without mentioning or taking into account equal strengthacting from the body to the Earth. The fact is that with the enormous mass of the Earth, the change in its speed is vanishingly small. Therefore, a change in the potential energy of interaction leads to a noticeable change in the kinetic energy of the body and an infinitely small change in the kinetic energy of the Earth. In such a situation, it is permissible to speak of the potential energy of a body near the Earth's surface, that is, to "attribute" all the energy of gravitational interaction to a small body. In general, we can talk about the potential energy of an individual body if the other interacting bodies are motionless.

We have repeatedly emphasized that the point at which the potential energy is taken to be zero is chosen arbitrarily. In this case, such a point turned out to be an infinitely distant point. In a sense, this unusual conclusion can be recognized as reasonable: indeed, interaction disappears at an infinite distance - potential energy also disappears. From this point of view, the sign of potential energy also looks logical. Indeed, in order to spread two attracting bodies, external forces must do positive work, therefore, in such a process, the potential energy of the system must increase: here it grows, increases and ... becomes equal to zero!
If the attracting bodies touch, then the force of attraction cannot do positive work, but if the bodies are spaced apart, then such work can be done when the bodies come together. Therefore, it is often said that attracting bodies have negative energy, and the energy of repulsive bodies is positive. This statement is true only if the zero level of potential energy is chosen at infinity. So, if two bodies are connected by a spring, then with an increase in the distance between the bodies, the force of attraction will act between them, however, the energy of their interaction is positive. Do not forget that the zero level of potential energy corresponds to the state of an undeformed spring, not infinity) ..