Mechanical and electromagnetic vibrations. Harmonic fluctuations The figure shows a graph of fluctuations
A. 1 sec. B. 2 sec. V. 3 sec. G. 4 sec. D.
2. q= 10-2cos 20 t(Cl). What is the amplitude of charge oscillations?
A. 10-2 cl. B. cos 20 t Cl. V. 20t Cl. G. 20 cl. D. There is no correct answer among the answers A-D.
3. The oscillation period of the mathematical pendulum is 0.5 s. What is the cyclical frequency of the pendulum?
A. 0.5 Hz. B. 2 Hz. V. 4 π s-1. G.π s-1. D. There is no correct answer among the answers A-D.
4. Oh x= 0.4 sin 2 t
A. 0.4 m / s2. B. 0.2 m / s2. V. 0.1 m / s2. G. 0.8 m / s2. D. 1.6 m / s2.
5. Mass weight m suspended on a spring, performs harmonic oscillations with a cyclic frequency ω ω 2 vibrations of weight m 2=4m 1 on the same spring?
A. ω 2= ω l / 4. B. ω 2 = ω 1/2. V. ω 2= ω 1. G. ω 2=2 ω 1. D. ω 2=4 ω 1.
6. How will the oscillation frequency of a mathematical pendulum change if its length is increased by 4 times?
A. 1. B. 2. V. 3. G. 4. D. I = 0.
8. Which of the graphs (Fig. 4) expresses the dependence of the active resistance in the alternating current circuit on the frequency?
A. 1. B. 2. V. 3. G. 4. D. 5.
9. An active resistance of 10 ohms is included in an alternating current circuit with a frequency of 50 Hz. What is the amplitude of current fluctuations at an amplitude of voltage fluctuations at the terminals of the active resistance of 50 V?
A. 5 A. B. 0.2 A. V. 250 A. G. 0.1 A. D. There is no correct answer among the answers A-D.
10. How will the amplitude of fluctuations in the strength of the current flowing through the capacitor change if, with a constant amplitude of voltage fluctuations, the frequency of voltage fluctuations is increased by 2 times?
A. Will increase 2 times. B. Will decrease by 2 times. V. Will increase 4 times.
G. It will decrease by 4 times. D. Will not change.
11. The effective value of the voltage in the section of the alternating current circuit is 220 V. What is the amplitude of the voltage fluctuation in this section of the circuit?
A. 220 V. B. 440 V. V. 220 / https: //pandia.ru/text/79/060/images/image005_1.gif "width =" 25 height = 23 "height =" 23 "> V. D. There is no correct answer among the answers A-D.
12. Figure 5 shows a diagram of a lamp generator. Indicate the element of the generator circuit, due to the energy of which sustained electrical oscillations are maintained.
A. 1. B. 2. V. 3. G. 4. D. 5.
13. With harmonic vibrations along the axis Oh coordinate NS the body changes according to the law NS= 0.6 sin 3 t(m). What is the amplitude of velocity fluctuations?
A. 0.6 m / s. B. 0.2 m / s. V. 1.8 m / s. G. 5.4 m / s. D. There is no correct answer among the answers A-D.
14. With harmonic vibrations of a body on a spring, the maximum value of the kinetic energy is 20 J, the maximum value of the potential energy of the spring is 20 J. How does the total mechanical energy of the body and the spring change over time?
A. Ranges from 0 to 20 J. B. Ranges from 0 to 40 J.
V. Does not change and is equal to 20 J. G. Does not change and is equal to 40 J.
D. There is no correct answer among the answers A-D.
15. Which of the following oscillations are free: 1 - oscillations of a mathematical pendulum, 2 - oscillations of a piston in a cylinder of an automobile engine, 3 - oscillations of current in an induction generator, 4 - oscillations of amperage in a lamp generator, 5 - oscillations of amperage in an oscillatory circuit.
A. 4. B. 1, 5. V. 3, 4. G. 2, 3. D. There is no correct answer among the answers A-D.
MECHANICAL AND ELECTROMAGNETIC VIBRATIONS
Option 2
1.
Figure 1 shows a graph of time dependence t speed υ
a body performing harmonic vibrations along a straight line. What is the amplitude of the body's velocity fluctuations?
A. 10 m / s. B. 20 m / s. V. 3 m / s. G. 6 m / s.
D. There is no correct answer among the answers A-D.
2. Electrical vibrations in the oscillatory circuit are given by the equation I= 2 sin 10 t(A). What is the cyclic frequency of current fluctuations?
A. 2 s-1. B. 10t s-1. V. 10 s-1. G. sin 10 t s-1.
D. There is no correct answer among the answers A-D.
3. Oscillations of the current in the oscillating circuit occur with a cyclic frequency of 4π s-1. What is the period of current fluctuations?
A. 0.5 sec. B. 2 sec. V. 4π s ... G. 8π2 s. D. There is no correct answer among the answers A-D.
4. Oh acceleration changes according to the law Oh= 4 cos 2 t NS body?
A. 16 m. B. 8 m. V. 4 m. G. 2 m. D. 1m.
5. k 1, performs harmonic oscillations with a cyclic frequency ω 1. What is the cyclic frequency ω 2 vibrations of the same weight on a spring with a stiffness k 2=4k 1?
A. ω 2=4ω l. B. ω 2 =2ω 1. V. ω 2= ω 1/2. G. ω 2=ω 1/4. D. ω 2=ω 1.
6. How will the oscillation period of a mathematical pendulum change if its length is reduced by 4 times?
A. Will decrease by 2 times. B. It will decrease by 4 times. V. Will not change.
G. Will increase 2 times. D. Will increase 4 times.
7. The rotor of the alternator rotates in a uniform magnetic field. How will the amplitude of the induction EMF change with an increase in its rotation frequency by 2 times?
A. Will not change. B. Will increase 2 times. V. Will decrease by 2 times. G. Will increase 4 times.
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D. It will decrease by 4 times.
8.
A. 1. B. 2. V. 3. G. 4. D. 5.
9. How will the amplitude of fluctuations in the strength of the current flowing through the resistance change if, with a constant amplitude of voltage fluctuations, the frequency of fluctuations of the applied voltage is increased by 2 times?
A. Will increase 2 times. B. Will decrease by 2 times. V. Will not change.
G. Will increase 4 times. D. It will decrease by 4 times.
10. How will the amplitude of fluctuations in the strength of the current flowing through the coil, the active resistance of which is zero, change if, with a constant amplitude of voltage fluctuations, the frequency of fluctuations is increased by 2 times?
A. Will increase 2 times. B. Will decrease by 2 times. V. Will increase 4 times.
G. It will decrease by 4 times. D. Will not change.
11. The graph of the dependence of the voltage in the section of the alternating current circuit on time is shown in Figure 3. What is the effective value of the voltage?
A. 50 B. B. 50https: //pandia.ru/text/79/060/images/image005_1.gif "width =" 25 height = 23 "height =" 23 "> B. G. 0 B.
D. There is no correct answer among the answers A-D.
12. Figure 4 shows a diagram of a lamp generator. Indicate the element of the generator circuit in which electrical oscillations directly occur.
A. 1. B. 2. V. 3. G. 4. D. 5.
13. With harmonic vibrations along the axis Oh the speed of the body changes according to the law υ = 6 cos 3 t(m / s). What is the amplitude of acceleration oscillations?
A. 54 m / s2. B. 18 m / s2. V. 6 m / s2. G. 2 m / s2. D. There is no correct answer among the answers A-D.
14. With harmonic electrical oscillations in the oscillatory circuit, the maximum value of the energy of the electric field of the capacitor is 50 J, the maximum value of the energy of the magnetic field of the coil is 50 J. How does the total energy of the electromagnetic field of the circuit change over time?
A. Ranges from 0 to 50 J. B. Ranges from 0 to 100 J. V. Does not change and is equal to 100 J.
G. Does not change and is equal to 50 J. D. There is no correct answer among the answers A-D.
15. Which of the following oscillations are forced: 1 - oscillations of a mathematical pendulum; 2 - vibrations of the piston in the cylinder of an automobile engine; 3 - current fluctuations in the induction generator; 4 - fluctuations in the current in the lamp generator; 5 - current fluctuations in the oscillatory circuit.
A. 4. B. 1, 5. V. 3, 4. G. 2, 3. D. There is no correct answer among the answers A-D.
MECHANICAL AND ELECTROMAGNETIC VIBRATIONS
Option 3
1.
Figure 1 shows a graph of the dependence on the time of the charge of the capacitor with harmonic oscillations in the oscillatory circuit. What is the frequency of oscillations of the charge in the oscillatory circuit?
A. 10 Hz. B. 5 Hz. V. 3.3 Hz. G. 2.5 Hz.
D. There is no correct answer among the answers A-D.
2. Vibrations of the load along the axis Oh are given by the equation https://pandia.ru/text/79/060/images/image012_0.gif "width =" 45 "height =" 41 ">. V. 2t. G. https://pandia.ru/text/79/060/images/image014_0.jpg "align =" left "width =" 158 height = 165 "height =" 165 "> 4. With harmonic vibrations along the axis Oh the coordinate of the body changes according to the law x=0,9 sin 3t(m). What is the amplitude of acceleration oscillations?
A. 0.1 m / s2. B. 0.3 m / s2. V. 0.9 m / s2. G. 2.7 m / s2. D. 8.1 m / s2.
5. Mass weight T 1, suspended on a spring, performs harmonic oscillations with a period T 1. What is the period T 2 vibrations of weight m2 = 4m 1 on the same spring?
A. T 2= T 1/4. B. T 2= T 1/2. V. T 2= T 1. G. T 2 = 2T 1. D. T 2 = 4T 1.
6. How will the period of free electrical oscillations in the oscillatory circuit change if the inductance L to increase the coils by 4 times?
A. Will increase 4 times. B. Will increase 2 times. V. Will not change.
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G. Will decrease by 2 times. D. It will decrease by 4 times.
7. A rectangular wire frame rotates at a constant speed in a uniform magnetic field (Fig. 2). Which of the graphs shown in Figure 3 corresponds to the dependence of the EMF of induction in the frame on time?
A. 1. B. 2. V. 3. G. 4. D. I=0.
8. Which of the graphs (Fig. 4) expresses the dependence of the inductive reactance in the alternating current circuit on the frequency?
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A. 1. B. 2. V. 3. G. 4. D. 5.
9. An active resistance of 10 ohms is included in an alternating current circuit with a frequency of 50 Hz. What is the amplitude of voltage fluctuations across an active resistance of 10 ohms with an amplitude of current fluctuations in the circuit of 5 A?
A. 0.5 V. B. 50 B. V. 1 B. G. 250 V. D. There is no correct answer among the answers A-D.
10. How will the amplitude of the voltage fluctuations on the capacitor change if, with a constant amplitude of the current fluctuations, the frequency of the current change is reduced by 2 times?
A. Will not change. B. Will increase 2 times. V. Will increase 4 times.
G. Will decrease by 2 times. D. It will decrease by 4 times.
11. The effective value of the current in the alternating current circuit is 1 A. What is the amplitude of the current fluctuations in this circuit?
A. 1 A. B. https://pandia.ru/text/79/060/images/image017_0.gif "width =" 33 "height =" 23 "> A. D. There is no correct answer among the answers A-D.
12. Figure 5 shows a diagram of a lamp generator. Indicate the element of the generator circuit, with the help of which the feedback is carried out.
A. 1. B. 2. V. 3. G. 4. D. 5.
13. With electrical oscillations in the oscillatory circuit, the capacitor charge changes according to the law q=0,01 sin 10t(Cl). What is the amplitude of current fluctuations?
A. 0.01 A. B. 1 A. V. 0.1 A. G. 10−4 A. D. There is no correct answer among the answers A-D.
14. With harmonic vibrations of a body on a spring, the maximum value of its kinetic energy is 30 J. What is the maximum value of the potential energy of a compressed spring?
A. 0 J. B. 15 J. V. 30 J. G. 60 J.
D. There is no correct answer among the answers A-D.
15. Which of the following oscillations are self-oscillations: 1 - oscillations of a mathematical pendulum, 2 - oscillations of the piston in the cylinder of a car engine, 3 - oscillations of the current in the induction generator, 4 - oscillations of the amperage in the lamp generator, 5 - oscillations of the current in the oscillatory circuit?
A. 4. B. 1, 5. V. 3, 4. G. 2, 3. D. There is no correct answer among the answers A-D.
MECHANICAL AND ELECTROMAGNETIC VIBRATIONS
Option 4
1. Figure 1 shows a graph of the time dependence of the current through the coil of the oscillatory circuit. What is the period of current fluctuations?
A. 0.4 sec. B. 0.3 s. V. 0.2 sec. G. 0.1 sec. D. There is no correct answer among the answers A-D.
2.
The speed of a body performing oscillations is given by the equation 
(m / s). What is the initial phase of velocity fluctuations?
A. 5. B. 3 t+ π /3. V. 3t. G.π / 3.
D. There is no correct answer among the answers A-D.
3. Oscillations of the charge on the capacitor plates in the oscillatory circuit occur with a cyclic frequency of 4π s − 1. What is the period of oscillation of the charge on the capacitor plates?
A. 0.5 sec. B. 2 sec. V. 2π2 s. G.π s. D. There is no correct answer among the answers A-D.
4. With harmonic vibrations of the body along the axis Oh acceleration changes according to the law ax = 9 cos 3 t(m / s2). What is the amplitude of coordinate changes NS body?
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A. 1m. B. 3m. V. 9 m. G. 27 m. D. 81 m.
5. Load suspended on a spring with a stiffness k 1, performs harmonic oscillations with a period T 1. What is the period T 2 vibrations of the same body on a spring with stiffness k 2=4k 1?
A. T2= 4T 1. B. T2 = 2T 1. V. T2 = T 1.
G. T2 = T 1/
2.
D. T 2 =
T 1/4.
6. How will the period of free electrical oscillations in the oscillatory circuit change if the capacitance WITH increase the capacitor by 4 times?
A. It will decrease by 4 times. B. Will decrease by 2 times.
V. Will increase 4 times. G. Will increase 2 times. D. Will not change.
7. The rotor of an alternator rotates at a constant frequency in a uniform magnetic field. How will the EMF of induction change when the induction of the magnetic field is doubled?
A. Will increase 4 times. B. Will increase 2 times.
V. Will decrease by 2 times. G. It will decrease by 4 times. D. Will not change.
8. Which of the graphs (Fig. 2) expresses the dependence of the capacitance in the AC circuit on the frequency?
A. 1. B. 2. V. 3. G. 4. D. 5.
9. How will the amplitude of fluctuations in the strength of the current flowing through the active resistance change if, at a constant frequency of voltage fluctuations, the amplitude of fluctuations of the applied voltage is reduced by 2 times?
A. Will increase 2 times. B. Will decrease by 2 times. V. Will not change.
G. Will increase 4 times. D. It will decrease by 4 times.
10.
How will the amplitude of voltage fluctuations change on a coil, the active resistance of which is zero, if at a constant amplitude of the current strength, the frequency of the change in the current strength decreases by 2 times?
A. Will not change. B. Will increase 2 times. V. Will increase 4 times.
G. Will decrease by 2 times. D. It will decrease by 4 times.
11. The graph of the dependence of the current strength in the alternating current circuit on time is shown in Figure 3. What is the effective value of the current strength?
A. 0 A. B. https://pandia.ru/text/79/060/images/image025_1.gif "width =" 32 "height =" 23 "> A.
D. There is no correct answer among the answers A-D.
12. Figure 4 shows a diagram of a lamp generator. Indicate the elements of the generator circuit, with the help of which the energy supply from the constant voltage source is regulated.
A. 1. B. 2. V. 3. G. 4. D. 5.
13. With electrical oscillations in the oscillatory circuit, the current in a coil with an inductance of 1 H changes according to the law I=2 cos 100t(A). What is the amplitude of self-induction EMF oscillations?
A. 0.02 V. B. 2 B. V. 200 V. G. 2 × 104 V. D. There is no correct answer among the answers A-D.
14. With harmonic electrical vibrations in the oscillatory circuit, the maximum value of the energy of the electric field is 10 J. What is the maximum value of the energy of the magnetic field of the coil?
A. 0 J. B. 5 J. V. 10 J. G. 20 J. D. There is no correct answer among the answers A-D.
15. Which of the following vibrations are forced: 1 - vibrations of a load suspended on a spring, 2 - vibrations of a piston in a cylinder of an automobile engine, 3 - fluctuations in current in an induction generator, 4 - vibrations of a pendulum in hours, 5 - fluctuations in current in an oscillatory circuit ?
A. 2, 3. B. 1, 5. V. 4. G. 2, 4. D. There is no correct answer among the answers A-D.
Depending on the number of correct answers, a score is given on a five-point scale. Based on the experimental verification of the proposed tasks, the following scale is recommended for translating the results of knowledge testing with the help of tasks with a choice of answers into grades on a five-point system:
Number of correct answers: Score
Periodic oscillations are called harmonic , if the fluctuating quantity changes over time according to the cosine or sine law:
Here 
- cyclic vibration frequency, A- the maximum deviation of the fluctuating value from the equilibrium position ( vibration amplitude
),
φ( t)
= ω t+
φ 0
– oscillation phase
,
φ 0
– initial phase
.
The harmonic vibration graph is shown in Figure 1.
Picture 1- Graph 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 vibrational system with harmonic vibrations is equal to:
.
Harmonically fluctuating 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 that oscillates in one vertical plane under the action of gravity.
Codeburning 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 pointwise.
At small angles of deflection α (Fig. 7.4), the physical pendulum also performs harmonic oscillations. We will assume that the weight of a 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 power: cannot be defined explicitly. Taking into account all the quantities included in the original differential equation of oscillations of a physical pendulum has the form:
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The 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 oscillation period of which is equal to the oscillation period of the given physical pendulum.
Spring pendulum
It is a weight attached to a spring of negligible mass.
Until the spring is deformed, the elastic force does not act on the body. In a spring pendulum, oscillations occur 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 over time. It can be shown that the total energy of a mechanical vibrational system with harmonic vibrations is equal to.
1. The figure shows a graph of the dependence of the potential energy of a mathematical pendulum (relative to its equilibrium position) on time. At the moment in time corresponding to point D on the graph, the total mechanical energy of the pendulum is: 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) versus time. At the moment of time, the kinetic energy of the pendulum is: 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) it will increase 4 times 2) it will increase 2 times 3) it will decrease 4 times 4) it will decrease 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 oscillations of this pendulum at resonance is 1) 1 cm 2) 2 cm 3) 8 cm 4) 10 cm 6. With free oscillations of the load on the thread as a pendulum, its kinetic energy varies from 0 J to 50 J, the maximum potential energy is 50 J Within what limits does the total mechanical energy of the load change during such vibrations? 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 dependence of the coordinate of the cargo on time. In what parts of the graph does the spring force applied to the load perform positive work? 1) 2) 3) 4) and and and and 8. The load oscillates on the spring, moving along the axis. The figure shows a graph of the dependence of the coordinate of the cargo on time. In what parts of the graph does the spring force applied to the load perform negative work? 1) 2) 3) 4) and and and and 9. The load oscillates on the spring, moving along the axis. The figure shows a graph of the dependence of the projection of the cargo speed on this axis from time to time. During the first 6 s of movement, the load traveled a distance of 1.5 m. What is the amplitude of the load oscillations? 1) 0.5 m 2) 0.75 m 3) 1 m 4) 1.5 m 10. A mathematical pendulum with a period of oscillation T was deflected at a small angle from the equilibrium position and released without initial speed (see figure). How long after that will 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 a period of oscillations T was deflected at a small angle from the equilibrium position and released with an initial velocity equal to zero (see figure). How long after this does the potential energy of the pendulum reach its maximum for the first time? Neglect air resistance. 1) 2) 3) 4) 12. A mathematical pendulum with a period of oscillations 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 that will the kinetic energy of the pendulum reach its maximum for the second time? Neglect air resistance. 1) 2) 3) 4) 13. A 50 g weight attached to a light spring makes free vibrations. The graph of the dependence of the x coordinate of this load on the time t is shown in the figure. The stiffness of the spring is 1) 3 N / m 2) 45 N / m 3) 180 N / m 4) 2400 N / m 14. How should the stiffness of the pendulum spring be changed in order to increase its oscillation frequency by 2 times? 1) decrease 2 times 2) increase 4 times 3) increase 2 times 4) decrease 4 times
Oscillatory motion- periodic or almost periodic motion of a body, the coordinate, the speed and acceleration of which at regular intervals take on approximately the same values.
Mechanical vibrations occur when, when the body is removed from the equilibrium position, a force appears that tends to return the body back.
Displacement x - deviation of the body from the equilibrium position.
Amplitude A is the modulus of maximum displacement of the body.
The oscillation period T is the time of one oscillation:
Oscillation frequency
The number of vibrations made by the body per unit of time: During vibrations, 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, the speed vanishes.
HARMONIC VIBRATION GRAPH
Harmonic oscillations occurring according to the sine or cosine law are called:
where x (t) is the displacement of the system at the moment t, A is the amplitude, ω is the cyclic frequency of oscillations.
If we plot the deviation of the body from the equilibrium position along the vertical axis, and time along the horizontal axis, we get a graph of the oscillation x = x (t) - the dependence of the body’s displacement on time. With free harmonic oscillations, this is a sine or cosine wave. The figure shows the graphs of the dependence of the displacement x, projections of the velocity V x and acceleration a x on time.
As can be seen from the graphs, at the maximum displacement x, the velocity V of the oscillating body is zero, the acceleration a, and hence the force acting on the body, is maximum and directed opposite to the displacement. In the equilibrium position, displacement and acceleration are zero and the speed is maximum. The acceleration projection always has the opposite sign of the displacement.
VIBRATORY MOTION ENERGY
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 a maximum x = A, the velocity, and with it the kinetic energy, vanish.
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 = 0, E p = 0. Therefore, the total energy is equal to the kinetic one:
The total mechanical energy of an oscillating body is proportional to the square of its velocity in the equilibrium position. Hence:
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 force on the thread. If the pendulum is deflected and released, then the forces will cease to compensate each other, and a resultant force will arise, directed towards the equilibrium position. Newton's second law:
With 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 = x / l, then the projection of the resulting force R onto the x-axis is
The minus sign shows 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 us compare the expressions for the restoring force of the mathematical and spring pendulums:
It is seen that mg / l is analogous to k. Replacing, k by mg / l in the formula for the period of a spring pendulum
we get the formula for the period of the 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 gravity at a given place on the earth's surface.
Free vibrations of a mathematical pendulum at small angles of deflection are harmonic. They occur due to the resultant force of gravity and the tension force of the thread, as well as the inertia of the load. The resultant of these forces is a restoring force.
Example. Determine the acceleration due to gravity on a planet where a 6.25 m long pendulum has a free oscillation period of 3.14 s.
The period of oscillation of a mathematical pendulum depends on the length of the thread and the acceleration of gravity:
Squaring both sides of the equality, we get:
Answer: the acceleration of gravity is 25 m / s 2.
Problems and tests on the topic "Topic 4." Mechanics. Oscillations and waves "."
- Transverse and longitudinal waves. Wavelength
Lessons: 3 Assignments: 9 Tests: 1
- Sound waves. Sound speed - Mechanical vibrations and waves. Sound grade 9
Figure 1 the vectors of the speed and acceleration of the ball are shown. Which direction shown in fig. 2, has a vector of the resultant of all forces applied to the ball? B) 2
On the image the probability density of particle detection at various 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 the dependence of the emissivity of an absolutely black body on the wavelength for different temperatures. Which of the curves corresponds to the lowest temperature? E) 5
On the image depicts 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 the electrostatic field. The field strength is greatest at the 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 shows the cross-section of a conductor with current I. 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 it change X-ray wavelength for Compton scattering at an angle of 90 0 Accept Compton wavelength 2.4 pm E) will not change
How much will it change X-ray wavelength for Compton scattering at an angle of 60 0 Take a Compton wavelength of 2.4 pm B) 1.2 pm
How much will change optical the length of the path, if a glass plate with a thickness of 2.5 microns is placed in the path of a light beam traveling in a 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 it change phase during one complete oscillation?
How much differ phases of charge oscillations on the capacitor plates and the 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 is the number in the figure for 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 ray if the angle of reflection is 30 0 .C) 45 0
On a rod 10 cm long, there is 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 body weighing 1 kg is acted upon by a force of 10N. 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 = 0m / s. 18J
On thin lens a beam of light falls. Select the path of the beam after refraction by the lens. A) 1
Monochromatic light with a wavelength of 220 nm is incident on the 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 the energy of a photon is consumed with an external photoeffect? D) on the work function of the electron and the transfer of kinetic energy to it
Falls into the crack normally monochromatic light. The second dark diffraction fringe is observed at an angle of = 0.01. How many incident light wavelengths is the slit width? B) 200
On the slit the width of the incident is a normally parallel beam of monochromatic light with a 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 with a lens located directly behind the slit onto a screen spaced from the lens at a distance of L = 1 m is: C) 1.2 cm
Normal monochromatic light with a wavelength of 0.6 μm is incident 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 is incident on a slit 2 μm wide. At what angle will the second diffraction minimum of light be observed? A) 30 0
On a slot wide a = 0.005 mm normally monochromatic light falls. The deflection angle of the rays corresponding to the fifth dark diffraction line is j = 300. Determine the wavelength of the incident light. C) 0.5 μm
On a slot wide a = 2 µm, a normally parallel beam of monochromatic light (= 500 nm) is incident. Over what angle will the second-order diffraction minimum be observed? C) 30 0
On a slot wide a normally parallel beam of monochromatic light with a wavelength 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's car drove past with the siren on. When the vehicle approached, the observer heard a higher tone of sound, and when moving away a lower tone 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 it is removed, it will decrease
Name thermodynamic parameters B) temperature, pressure, volume
Find the speed of the body at the moment of time t = 1c. С) 4 m / s
