11TH GRADE · OSCILLATIONS AND WAVES

Mechanical vibrations

Period, frequency, phase, speed, acceleration and energy.

28 minutes25 theory cards

Open interactive lesson

LESSON MATERIAL

Basic Concepts

Hesitation

A time-repeated change in the state of a system near its equilibrium position.

The system goes through the same states repeatedly, and energy usually passes between two forms.

Harmonic oscillation

An oscillation in which a quantity changes according to the law of sine or cosine.

The restoring acceleration is directed against the displacement and is proportional to it; therefore the graph has a sinusoidal shape.

LESSON MATERIAL

Physical quantities

Oscillation period

T

Time of one complete oscillation.

Unit: second · $\mathrm{s}$

Oscillation frequency

\nu

The number of complete oscillations per unit of time.

Unit: hertz · $\mathrm{Gts}$

Number of oscillations

n

The number of complete oscillations completed.

Unit: dimensionless quantity

Cyclic frequency

\omega

The speed of change in the phase of oscillation.

Unit: radians per second · $\frac{\mathrm{rad}}{\mathrm{s}}$

Pendulum length

l

The distance from the suspension point to the center of mass of the load.

Unit: meter · $\mathrm{m}$

Cargo weight

m

The mass of a body oscillating.

Unit: kilogram · $\mathrm{kg}$

Spring stiffness

k

Spring deformation resistance coefficient.

Unit: newton per meter · $\frac{\mathrm{N}}{\mathrm{m}}$

Offset

x

Deviation from the equilibrium position.

Unit: meter · $\mathrm{m}$

Offset amplitude

x_m

Largest displacement modulus.

Unit: meter · $\mathrm{m}$

Initial phase

\varphi_0

Oscillation phase at the initial moment.

Unit: radian · $\mathrm{rad}$

Speed amplitude

v_m

The highest modulus of speed during oscillation.

Unit: meter per second · $\frac{\mathrm{m}}{\mathrm{s}}$

Acceleration amplitude

a_m

The largest acceleration module during oscillation.

Unit: meters per second squared · $\frac{\mathrm{m}}{\mathrm{s}^2}$

Energy of vibrations

E_0

Total mechanical energy of an ideal oscillator.

Unit: joule · $\mathrm{Dzh}$

LESSON MATERIAL

Lesson formulas

Period through observation time

T=\frac{t}{n}=\frac{1}{\nu}

The period is equal to the time divided by the number of oscillations and is the inverse of the frequency.

Oscillation frequency

\nu=\frac{n}{t}=\frac{1}{T}

Frequency is equal to the number of oscillations per time and the inverse of the period.

Hertz definition

1\,\mathrm{Gts}=1\,\mathrm{s}^{-1}

One hertz corresponds to one vibration per second.

Cyclic frequency

\omega=2\pi\nu=\frac{2\pi}{T}

The cyclic frequency is equal to the change in phase per unit time.

Period of a mathematical pendulum

T=2\pi\sqrt{\frac{l}{g}}

The period of small oscillations depends on the length of the pendulum and the acceleration of gravity.

Condition: The oscillations are small, air resistance and the mass of the thread are neglected.

Period of a spring pendulum

T=2\pi\sqrt{\frac{m}{k}}

The period depends on the mass of the load and the stiffness of the spring.

Equation of harmonic vibration

x=x_m\sin(\omega t+\varphi_0)

The displacement varies harmonically with phase.

Speed amplitude

v_m=x_m\omega

The maximum speed is equal to the product of the displacement amplitude and the cyclic frequency.

Acceleration amplitude

a_m=x_m\omega^2

The maximum acceleration is equal to the displacement amplitude times the square of the cyclic frequency.

Energy of a mechanical oscillator

E_0=\frac{mv_m^2}{2}=\frac{kx_m^2}{2}

The total energy is expressed in terms of the maximum velocity or displacement amplitude.

Related formulas

AmplitudeA, xₘ · m Spring stiffnessk · N/m PeriodT · with Frequencyν · Hz Cyclic frequencyω · s⁻¹ EnergyE, W · J

PRACTICE

Tasks on the topic

Oscillations and waves11 class · 30 problems