Гармоникалық тербелістердің теңдеулері мен графиктеріне есептер шығару. Физика, 11 сынып, презентация.

Гармоникалық тербелістердің теңдеулері мен графиктеріне есептер шығару

Сабақ мақсаты:

- Гармоникалық тербелістерді анықтау

- Есептерді шешу үшін гармоникалық тербелістер теңдеуін қолдану

- Гармоникалық тербелістер кестесін құру

Definition:

the acceleration is always proportional, but opposite to the displacement from the equilibrium position

Requirements:

An oscillating mass

An equilibrium position

A restoring force which acts to return the mass to its equilibrium

Properties:

It is identical to the projection of a uniform circular motion on an axis.

2. The curves (x-t, v-t and a-t) are sinusoidal with acceleration leading velocity by π/2 and velocity leads displacement by π/2.

3. The period of oscillation is independent of amplitude (isochronous).

Requirements for simple harmonic motion

Displacement vs time

amplitude

period (=T)

Displaced systems oscillate

around stable equil. points

Equil. point

Simple Harmonic Oscillator

Analogy with circular motion

s.h.m equations

Displacement

xm = amplitude (A)

Velocity

vm = maximum velocity

Acceleration

am = maximum acceleration

x = xm sin (ωt)

x = xm cos (ωt)

v = vm cos (ωt)

a = - am sin (ωt)

vm = ω x0

am = -ω2 x0

Graphs of s.h.m.

Simple harmonic motion

Equil. point

T

T= period = time for 1 complete oscillation

f = frequency = No of oscillations/time

= 1/T

Pure Sine-like curve

Energy of an oscillating mass

An oscillating spring-mass has a starting elastic potential energy (no kinetic energy):

When the mass start to oscillate there potential energy is changed into kinetic energy:

The total energy of the system is given by:

Graphs of Energy of SHM

Energy of an oscillating mass

If the mass is suspended there is also the gravitation potential energy and the conservation of energy is:

If friction is taken into account then its work must be added to the total energy of the system.

Ki + PEsi + PEgi = Kf + PEsf + PEgf

Wf = (Kf + PEsf + Pegf) – (Ki + PEsi + PEgi)

Velocity – Period - Frequency

The velocity of an oscillating mass-spring at any given position can be calculated using different methods:

Using energy conservation



Using newton’s 2nd law and differentiation:

F = -kx

ma = -kx