Harmonic vibrations. Mechanical and electromagnetic vibrations The figure shows a graph of harmonic vibrations

The simplest type of oscillations are harmonic vibrations- oscillations in which the displacement of the oscillating point from the equilibrium position changes over time according to the law of sine or cosine.

Thus, with a uniform rotation of the ball in a circle, 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 is the displacement - a quantity 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 - amplitude of oscillations - maximum displacement of the body from the equilibrium position; T - period of oscillation - time of one complete oscillation; those. the shortest period of time after which the values ​​of physical quantities characterizing the oscillation are repeated; - initial phase;

Oscillation phase 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 inverse of the period and equal to the number of complete oscillations completed in 1 s, is called the oscillation frequency:

If during time t the body makes N complete oscillations, then

Size showing how many oscillations a 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 wave (or sine wave).

Figure 2, a shows a graph of the time dependence of the displacement of the oscillating point from the equilibrium position for the case.

Let's 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 onto the x-axis.

This formula shows that during harmonic oscillations, the projection of the body’s velocity onto the x-axis also changes according to a harmonic law with the same frequency, with a different amplitude and is ahead of the displacement in phase by (Fig. 2, b).

To clarify the dependence of acceleration, we find the time derivative of the velocity projection:

where is the amplitude of the acceleration projection onto the x-axis.

With harmonic oscillations, the acceleration projection is ahead of the phase displacement by k (Fig. 2, c).

Periodic oscillations are called harmonic , if the fluctuating quantity changes over time according to the law of cosine or sine:

Here
- cyclic oscillation frequency, A– maximum deviation of the fluctuating quantity from the equilibrium position ( vibration amplitude ), φ( t) = ω t+ φ 0 – oscillation phase , φ 0 – initial phase .

The graph of harmonic vibrations is presented in Figure 1.

Picture 1– Harmonic graph

With harmonic oscillations, the total energy of the system does not change over time. It can be shown that the total energy of a mechanical oscillatory system during harmonic oscillations is equal to:

.

Harmonically vibrating quantity s(t) obeys the differential equation:

, (1)

which is called differential equation of harmonic vibrations.

A mathematical pendulum is a material point suspended on an inextensible weightless thread, performing oscillatory motion in one vertical plane under the influence of gravity.

Code 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 which oscillates about this axis under the influence of gravity. Unlike a mathematical pendulum, the mass of such a body cannot be considered pointlike.

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 - force F.

To derive the law of motion of mathematical and physical pendulums, we use the basic equation of 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 has the form:

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 the 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 mass attached to a spring whose mass can be neglected.

While the spring is not deformed, the elastic force does not act on the body. In a spring pendulum, oscillations occur under the action of elastic force.

Question 36 Energy of harmonic vibrations

With harmonic oscillations, the total energy of the system does not change over time. It can be shown that the total energy of a mechanical oscillatory system during harmonic oscillations is equal.

In Figure 1 the vectors of the ball's speed and acceleration are depicted. Which direction shown in Fig. 2, has the vector of the resultant of all forces applied to the ball? B) 2

On the image given the probability density of detecting a particle on different distances from the walls of the pit. 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 blackbody emissivity versus wavelength for different temperatures. Which of the curves corresponds to the lowest temperature? E) 5

On the image shows the wave profile at a certain point in time. What is its wavelength?B) 0.4m

The figure shows the lines of force of the electrostatic field. The field strength is greatest at point: E) 1

On the image shown graph of oscillations of a material point, the equation of which has the form: . What is the initial phase?B)

On the image shows the cross section of a conductor with current I. Electricity in the conductor is directed perpendicular to the plane of the drawing 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 it change? wavelength of X-rays during Compton scattering at an angle of 90 0? Assume the Compton wavelength is 2.4 pm. E) will not change

How much will it change? wavelength of X-rays during Compton scattering at an angle of 60 0? Assume Compton wavelength 2.4 pm. B) 1.2 pm

How long will change optical what is the path length if a glass plate 2.5 microns thick is placed in the path of a light beam traveling in a vacuum? Refractive index of glass 1.5.A) 1.25 µm

How long will change period oscillations of a mathematical pendulum when its length increases by 4 times? A) increases by 2 times

How long will the period of oscillation of a physical pendulum change when its mass increases 4 times? Will not change

How much will it change? phase during one complete oscillation?

How long differ phase of charge oscillations on the capacitor plates and current strength in the oscillatory circuit? A) p/2 rad

On collecting 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 = 3 N is applied for 2 seconds. Find the kinetic energy of the body after the force is applied. V 0 =0m/s. 18J

On thin lens a ray of light falls. Select the path of the ray after its refraction by the lens.A) 1

Monochromatic light with a wavelength of 220 nm is incident on a zinc plate. The maximum kinetic energy of photoelectrons is equal to: (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 a photon spent during the external photoelectric effect? ​​D) on the work function of the electron and imparting kinetic energy to it

Falls onto the crack normal 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

To the slit width of a normally parallel beam of monochromatic light with wavelength . 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 is normally incident 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 located at a distance L = 1 m from the lens is: C) 1.2 cm

Normally monochromatic light with a wavelength of 0.6 μm is incident on a 0.1 mm wide slit. 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 is incident on a 2 µm wide slit. At what angle will the second diffraction minimum of light be observed? A) 30 0

For a gap width a=0.005 mm monochromatic light falls normally. The angle of deflection of the rays corresponding to the fifth dark diffraction line is j=300. Determine the wavelength of the incident light.C) 0.5 µm

For a gap width a= A normally parallel beam of monochromatic light ( =500 nm) is incident at 2 µm. At what angle will the second-order diffraction minimum of light be observed? C) 30 0

For a gap width A normally parallel beam of monochromatic light with wavelength λ is incident. 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 driven by a car with its siren turned on. When the car approached, the observer heard a higher pitch of sound, and when moving away, a lower pitch of sound. What effect will be observed if the siren is stationary and an observer drives past it?D) when approaching, the tone will increase, when moving away it will decrease

Name thermodynamic parameters. B) temperature, pressure, volume

Find the speed of the body at time t=1c.С) 4 m/s

Physics test Harmonic vibrations for 9th grade students with answers. The test includes 10 multiple-choice questions.

1. Select 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 cosine law

1) only A
2) only B
3) both A and B
4) neither A nor B

2. The figure shows the dependence of the coordinates of the center of a ball suspended on a spring on time. The amplitude of oscillations is equal to

1) 10 cm
2) 20 cm
3) -10 cm
4) -20 cm

3. The figure shows a graph of vibrations of one of the points on 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 coordinates of the center of a ball suspended on a spring on time. The period of oscillation is equal to

1) 2 s
2) 4 s
3) 6 s
4) 10 s

5. The figure shows a graph of vibrations of one of the points on 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 coordinates of the center of a ball suspended on a spring on time. The oscillation frequency is

1) 0.25 Hz
2) 0.5 Hz
3) 2 Hz
4) 4 Hz

7. The figure shows the graph X, cm vibrations of one of the points of the string. According to the graph, the frequency of these oscillations is equal to

1) 1000 Hz
2) 750 Hz
3) 500 Hz
4) 250 Hz

8. The figure shows the dependence of the coordinates of the center of a ball suspended on a spring on time. How far will the ball travel in two complete oscillations?

1) 10 cm
2) 20 cm
3) 40 cm
4) 80 cm

9. The figure shows the dependence of the coordinates of the center of a ball suspended on a spring on time. This dependency is

1. The figure shows a graph of the potential energy of a mathematical pendulum (relative to its equilibrium position) versus time. At the moment of time corresponding to point D on the graph, the total mechanical energy of the pendulum is equal to: 1) 4 J 2) 12 J 3) 16 J 4) 20 J 2. The figure shows a graph of the potential energy of a mathematical pendulum (relative to its equilibrium position) on time. At the moment of time, the kinetic energy of the pendulum is equal to: 1) 0 J 2) 10 J 3) 20 J 4) 40 J 3. The figure shows a graph of the potential energy of a mathematical pendulum (relative to its equilibrium position) versus time. At the moment of time, the kinetic energy of the pendulum is equal to: 1) 0 J 2) 8 J 3) 16 J 4) 32 J 4. How will the period of small oscillations of a mathematical pendulum change if the length of its thread is increased by 4 times? 1) will increase by 4 times 2) will increase by 2 times 3) will decrease by 4 times 4) will decrease by 2 times 5. The figure shows the dependence of the amplitude of steady-state oscillations of the pendulum on the frequency of the driving force (resonance curve). The amplitude of oscillation of this pendulum at resonance is 1) 1 cm 2) 2 cm 3) 8 cm 4) 10 cm 6. With free oscillations of a load on a string as a pendulum, its kinetic energy varies from 0 J to 50 J, the maximum value of potential energy is 50 J Within what limits does the total mechanical energy of the load change during such oscillations? 1) does not change and is equal to 0 J 2) changes from 0 J to 100 J 3) does not change and is equal to 50 J 4) does not change and is equal to 100 J 7. The load oscillates on a spring, moving along the axis. The figure shows a graph of the load coordinates versus time. In what parts of the graph does the elastic force of the spring applied to the load do positive work? 1) 2) 3) 4) and and and 8. The load oscillates on a spring, moving along the axis. The figure shows a graph of the load coordinates versus time. In what parts of the graph does the elastic force of the spring applied to the load do negative work? 1) 2) 3) 4) and and and 9. The load oscillates on a spring, moving along the axis. The figure shows a graph of the projection of the speed of the load onto this axis versus time. During the first 6 seconds of movement, the load traveled a distance of 1.5 m. What is the amplitude of the load’s oscillations? 1) 0.5 m 2) 0.75 m 3) 1 m 4) 1.5 m 10. A mathematical pendulum with an oscillation period T was tilted at a small angle from the equilibrium position and released without an initial speed (see figure). How long after this does the kinetic energy of the pendulum reach its minimum for the first time? Neglect air resistance. 1) 2) 3) 4) 11. A mathematical pendulum with an oscillation period T was deflected by a small angle from the equilibrium position and released with an initial speed equal to zero (see figure). How long after this does the potential energy of the pendulum reach its maximum again for the first time? Neglect air resistance. 1) 2) 3) 4) 12. A mathematical pendulum with an oscillation period T was deflected by a small angle from the equilibrium position and released with an initial speed equal to zero (see figure). How long after this does the kinetic energy of the pendulum reach its maximum for the second time? Neglect air resistance. 1) 2) 3) 4) 13. A mass of 50 g attached to a light spring oscillates freely. A graph of the x coordinate of this load versus time t is shown in the figure. The spring stiffness is 1) 3 N/m 2) 45 N/m 3) 180 N/m 4) 2400 N/m 14. How should the spring stiffness of the pendulum be changed in order to increase its oscillation frequency by 2 times? 1) decrease by 2 times 2) increase by 4 times 3) increase by 2 times 4) decrease by 4 times