Periodic oscillations are called harmonic , if the fluctuating value changes over time according to the law of cosine or sine:
Here 
- cyclic oscillation frequency, A is the maximum deviation of the oscillating quantity from the equilibrium position ( oscillation amplitude
),
φ( t)
= ω t+
φ 0
– oscillation phase
,
φ 0
– initial phase
.
The graph of harmonic oscillations is shown in Figure 1.
Picture 1– Graph of harmonic oscillations
With harmonic oscillations, the total energy of the system does not change with time. It can be shown that the total energy of a mechanical oscillatory system with harmonic vibrations is equal to:
.
Harmoniously oscillating quantity s(t) obeys the differential equation:
,
(1)
which is called differential equation of harmonic oscillations.
A mathematical pendulum is a material point suspended on an inextensible weightless thread, oscillating in one vertical plane under the action of gravity.
Codeban period
physical pendulum.
A physical pendulum is a rigid body fixed on a fixed horizontal axis (suspension axis) that does not pass through the center of gravity and oscillates about this axis under the action of gravity. Unlike a mathematical pendulum, the mass of such a body cannot be considered as a point mass.
At small deflection angles α (Fig. 7.4), the physical pendulum also performs harmonic oscillations. We will assume that the weight of the physical pendulum is applied to its center of gravity at point C. The force that returns the pendulum to the equilibrium position, in this case, will be the component of gravity - the force F.
To derive the law of motion of mathematical and physical pendulums, we use the basic equation for the dynamics of rotational motion
Moment of force: cannot be determined explicitly. Taking into account all the quantities included in the original differential equation of oscillations of a physical pendulum, it has the form:
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Solution to this equation 
Let us determine the length l of the mathematical pendulum, at which the period of its oscillations is equal to the period of oscillations of the physical pendulum, i.e. or
. From this relation, we determine
This formula determines the reduced length of a physical pendulum, i.e. the length of such a mathematical pendulum, the period of oscillation of which is equal to the period of oscillation of a given physical pendulum.
Spring pendulum
This is a weight attached to a spring, the mass of which can be neglected.
As long as the spring is not deformed, the elastic force on the body does not act. In a spring pendulum, oscillations are performed under the action of an elastic force.
Question 36 Energy of harmonic vibrations
With harmonic oscillations, the total energy of the system does not change with time. It can be shown that the total energy of a mechanical oscillatory system for harmonic vibrations is equal to.
Figure 1 the velocity and acceleration vectors of the ball are shown. Which direction is shown in Fig. 2 , has the resultant vector of all forces applied to the ball? B) 2
On the image the density of the probability of detecting a particle at different distances from the walls of the well is given. What does the value of the probability density at point A () indicate? C) the particle cannot be detected in the middle of the potential well
On the image are given Graphs of dependence of blackbody emissivity on wavelength for different temperatures. Which of the curves corresponds to the lowest temperature? E) 5
On the image shows the profile of the wave at a certain point in time. What is its wavelength? B) 0.4m
The figure shows the lines of force of an electrostatic field. Field strength highest at point:E) 1
On the image shown graph of fluctuations of a material point, the equation of which has the form: . What is the initial phase?B)
On the image the section of the conductor with current I is shown. The electric current in the conductor is directed perpendicular to the plane of the figure from us. Which of the directions indicated in the figure at point A corresponds to the direction of the magnetic induction vector? C) 3
How much will change wavelength of X-rays during Compton scattering through an angle of 90 0 ? Take a Compton wavelength of 2.4 pm. E) will not change
How much will change wavelength of X-rays at Compton scattering at an angle of 60 0 ? Take Compton wavelength 2.4 pm.B) 1.2 pm
How much will change optical the length of the path, if a glass plate 2.5 microns thick is placed in the path of a light beam traveling in vacuum? Refractive index of glass 1.5.A) 1.25 µm
How much will change period oscillations of a mathematical pendulum with an increase in its length by 4 times? A) increases by 2 times
How much Will the period of oscillation of a physical pendulum change with an increase in its mass by 4 times? Will not change
How much will change phase during one complete oscillation?
How much different phases of charge oscillations on the capacitor plates and current strength in the oscillatory circuit? A) p / 2 rad
On the gathering lens a beam of parallel rays falls, as shown in the figure. What number in the figure indicates the focus of the lens? D) 4
A ray of light falls on a glass plate with a refractive index of 1.5. Find the angle of incidence of the beam if the angle of reflection is 30 0 .C) 45 0
A rod 10 cm long carries a charge of 1 μC. What is the linear charge density on the rod? E) 10 -5 C/m
A constant torque acts on the body. Which of the following quantities change linearly with time? B) angular velocity
A force of 10 N acts on a body of mass 1 kg. Find the acceleration of the body: E) 10m/s 2
On the body with a mass of 1 kg, a force F = 3H acts for 2 seconds. Find the kinetic energy of the body after the action of the force. V 0 \u003d 0m / s. 18J
On the thin lens a beam of light falls. Select the path of the beam after its refraction by the lens. A) 1
Monochromatic light with a wavelength of 220 nm falls on a zinc plate. The maximum kinetic energy of photoelectrons is: (work function A=6.4 10 -19 J, m e =9.1 10 -31 kg.)C) 2.63 10-19 J.
For what is the energy of the photon consumed during the external photoelectric effect? D) on the work function of the electron and the communication of kinetic energy to it
Falls on the crack normally monochromatic light. The second dark diffraction band is observed at an angle =0.01. How many wavelengths of incident light is the width of the slit? B) 200
On the gap a normally parallel beam of monochromatic light with a wavelength of . At what angle will the third diffraction minimum of light be observed? D) 30 0
A parallel beam of light from a monochromatic source with a length of 0.6 μm normally falls on a slit 0.1 mm wide. The width of the central maximum in the diffraction pattern projected using a lens located directly behind the slit onto a screen spaced from the lens at a distance L = 1 m is equal to: C) 1.2 cm
Normally monochromatic light with a wavelength of 0.6 μm falls on a slit 0.1 mm wide. Determine the sine of the angle corresponding to the second maximum. D) 0.012
A normally parallel beam of monochromatic light with a wavelength of 500 nm falls on a slit 2 μm wide. At what angle will the second diffraction minimum of light be observed? A) 30 0
For a slot wide a = 0.005 mm Normally monochromatic light falls. Deviation angle of rays corresponding to the fifth dark diffraction line, j=300. Determine the wavelength of the incident light.C) 0.5 µm
For a slot wide a= 2 μm falls normally parallel beam of monochromatic light (=500 nm). Over what angle will the second-order diffraction minimum of light be observed? C) 30 0
For a slot wide A normally parallel beam of monochromatic light is incident with a wavelength of . At what angle will the third diffraction minimum of light be observed? D) 30 0
On the screen an interference pattern was obtained from two coherent sources emitting light with a wavelength of 0.65 μm. The distance between the fourth and fifth interference maxima on the screen is 1 cm. What is the distance from the sources to the screen if the distance between the sources is 0.13 mm? A) 2 m
the observer was passed by a car with the siren on. As the car approached, the observer heard a higher tone of sound, and as it moved away, a lower tone of sound. What effect will be observed if the siren is stationary, and an observer passes by it?D) when approaching, the tone will increase, when removed, it will decrease
name thermodynamic parameters. B) temperature, pressure, volume
Find the speed of the body at time t=1c.С) 4 m/s
oscillatory motion- periodic or almost periodic movement of a body, the coordinate, velocity and acceleration of which at regular intervals take approximately the same values.
Mechanical oscillations occur when, when a body is taken out of equilibrium, a force appears that tends to bring the body back.
Displacement x - deviation of the body from the equilibrium position.
Amplitude A - the module of the maximum displacement of the body.
Oscillation period T - time of one oscillation:
Oscillation frequency
The number of oscillations made by the body per unit time: During oscillations, the speed and acceleration change periodically. In the equilibrium position, the speed is maximum, the acceleration is zero. At the points of maximum displacement, the acceleration reaches its maximum, and the velocity vanishes.
GRAPH OF HARMONIC OSCILLATIONS
Harmonic oscillations occurring according to the law of sine or cosine are called:
where x(t) is the displacement of the system at time t, A is the amplitude, ω is the cyclic oscillation frequency.
If the deviation of the body from the equilibrium position is plotted along the vertical axis, and time is plotted along the horizontal axis, then we get a graph of the oscillation x = x(t) - the dependence of the body's displacement on time. With free harmonic oscillations, it is a sinusoid or a cosine wave. The figure shows graphs of displacement x, velocity projections V x and acceleration a x versus time.
As can be seen from the graphs, at the maximum displacement x, the speed V of the oscillating body is zero, the acceleration a, and hence the force acting on the body, are maximum and directed opposite to the displacement. In the equilibrium position, the displacement and acceleration vanish, the speed is maximum. The acceleration projection always has the opposite sign of the displacement.
ENERGY OF VIBRATIONAL MOVEMENT
The total mechanical energy of an oscillating body is equal to the sum of its kinetic and potential energies and, in the absence of friction, remains constant:
At the moment when the displacement reaches its maximum x = A, the speed, and with it the kinetic energy, vanishes.
In this case, the total energy is equal to the potential energy:
The total mechanical energy of an oscillating body is proportional to the square of the amplitude of its oscillations.
When the system passes the equilibrium position, the displacement and potential energy are equal to zero: x \u003d 0, E p \u003d 0. Therefore, the total energy is equal to the kinetic:
The total mechanical energy of an oscillating body is proportional to the square of its velocity in the equilibrium position. Consequently:
MATHEMATICAL PENDULUM
1. Mathematical pendulum is a material point suspended on a weightless inextensible thread.
In the equilibrium position, the force of gravity is compensated by the tension of the thread. If the pendulum is deflected and released, then the forces and will cease to compensate each other, and there will be a resultant force directed to the equilibrium position. Newton's second law:
For small fluctuations, when the displacement x is much less than l, the material point will move almost along the horizontal x axis. Then from the triangle MAB we get:
Because sin a \u003d x / l, then the projection of the resulting force R on the x-axis is equal to
The minus sign indicates that the force R is always directed against the displacement x.
2. So, during oscillations of a mathematical pendulum, as well as during oscillations of a spring pendulum, the restoring force is proportional to the displacement and is directed in the opposite direction.
Let's compare the expressions for the restoring force of the mathematical and spring pendulums:
It can be seen that mg/l is an analogue of k. Replacing k with mg/l in the formula for the period of a spring pendulum
we get the formula for the period of a mathematical pendulum:
The period of small oscillations of a mathematical pendulum does not depend on the amplitude.
A mathematical pendulum is used to measure time, to determine the acceleration of free fall at a given location on the earth's surface.
Free oscillations of a mathematical pendulum at small deflection angles are harmonic. They occur due to the resultant force of gravity and the tension of the thread, as well as the inertia of the load. The resultant of these forces is the restoring force.
Example. Determine the free fall acceleration on a planet where a pendulum 6.25 m long has a period of free oscillation of 3.14 s.
The period of oscillation of a mathematical pendulum depends on the length of the thread and the acceleration of free fall:
By squaring both sides of the equation, we get:
Answer: free fall acceleration is 25 m/s 2 .
Tasks and tests on the topic "Topic 4. "Mechanics. Vibrations and waves.
- Transverse and longitudinal waves. Wavelength
Lessons: 3 Assignments: 9 Tests: 1
- Sound waves. Sound speed - Mechanical oscillations and waves. Sound grade 9
The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.
So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) performs a harmonic oscillatory motion on a vertical screen (Fig. 1).
The displacement from the equilibrium position during harmonic vibrations is described by an equation (it is called the kinematic law of harmonic motion) of the form:
where x - displacement - a value characterizing the position of the oscillating point at time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given time; A - oscillation amplitude - the maximum displacement of the body from the equilibrium position; T - oscillation period - the time of one complete oscillation; those. the smallest period of time after which the values of physical quantities characterizing the oscillation are repeated; - initial phase;
The phase of the oscillation at time t. The oscillation phase is an argument of a periodic function, which, for a given oscillation amplitude, determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.
If at the initial moment of time the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law
If the oscillating point at is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law
The value V, the reciprocal of the period and equal to the number of complete oscillations performed in 1 s, is called the oscillation frequency:
If in time t the body makes N complete oscillations, then
the value 
, showing how many oscillations the body makes in s, is called cyclic (circular) frequency.
The kinematic law of harmonic motion can be written as:
Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).
Figure 2, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case .
Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:
where is the amplitude of the velocity projection on the x-axis.
This formula shows that during harmonic oscillations, the projection of the body velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by (Fig. 2, b).
To find out the dependence of acceleration, we find the time derivative of the velocity projection:
where is the amplitude of the acceleration projection on the x-axis.
For harmonic oscillations, the acceleration projection leads the phase shift by k (Fig. 2, c).
Physics test Harmonic oscillations for 9th grade students with answers. The test includes 10 multiple choice questions.
1. Choose the correct statement(s).
A. oscillations are called harmonic if they occur according to the sine law
B. oscillations are called harmonic if they occur according to the law of cosine
1) only A
2) only B
3) both A and B
4) neither A nor B
2. The figure shows the dependence of the coordinate of the center of a ball suspended on a spring from time to time. The oscillation amplitude is
1) 10cm
2) 20cm
3) -10 cm
4) -20cm
3. The figure shows a graph of oscillations of one of the points of the string. According to the graph, the oscillation amplitude is equal to
1) 1 10 -3 m
2) 2 10 -3 m
3) 3 10 -3 m
4) 4 10 -3 m
4. The figure shows the dependence of the coordinate of the center of a ball suspended on a spring from time to time. The oscillation period is
1) 2 s
2) 4 s
3) 6 s
4) 10 s
5. The figure shows a graph of oscillations of one of the points of the string. According to the graph, the period of these oscillations is equal to
1) 1 10 -3 s
2) 2 10 -3 s
3) 3 10 -3 s
4) 4 10 -3 s
6. The figure shows the dependence of the coordinate of the center of a ball suspended on a spring from time to time. The oscillation frequency is
1) 0.25 Hz
2) 0.5Hz
3) 2Hz
4) 4Hz
7. The figure shows a graph X, see oscillations of one of the points of the string. According to the graph, the frequency of these oscillations is equal to
1) 1000 Hz
2) 750Hz
3) 500Hz
4) 250Hz
8. The figure shows the dependence of the coordinate of the center of a ball suspended on a spring from time to time. What is the distance traveled by the ball in two complete oscillations?
1) 10cm
2) 20cm
3) 40cm
4) 80 cm
9. The figure shows the dependence of the coordinate of the center of a ball suspended on a spring from time to time. This dependency is
