Mechanical oscillations

Acceleration of elastic force

$$a = -\frac{k\cdot x}{m}$$

a - acceleration
k - stiffness
x - elongation (shortening) of the object
m - mass

Find It is known that:

Calculate 'a'

Elastic force

$$F = -k\cdot x$$

F - force
k - stiffness
x - elongation (shortening) of the object

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Calculate 'F'

Equation of motion of mathematical pendulum

$$a = -\frac{g\cdot x}{l}$$

a - acceleration
g - free fall acceleration
x - declination
l - pendulum length

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Calculate 'a'

Equation of free oscillations

$$a = -\omega^{2}\cdot x$$

a - acceleration
ω - angular (cyclic, radian) frequency
x - declination

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Calculate 'a'

Equation of motion of spring pendulum

$$\omega^{2} = \frac{k}{m}$$

ω - angular (cyclic, radian) frequency
k - stiffness
m - mass

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Calculate 'ω'

Equation of motion of mathematical pendulum

$$\omega^{2} = \frac{g}{l}$$

ω - angular (cyclic, radian) frequency
g - free fall acceleration
l - pendulum length

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Calculate 'ω'

Free oscillations: declination

$$x = x_{m}\cdot cos(\omega\cdot t)$$

x - declination
x_m - maximum declination
ω - angular (cyclic, radian) frequency
t - time

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Calculate 'x'

Frequency and period of oscillations

$$\nu = \frac{1}{T}$$

ν - frequency
T - period

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Calculate 'ν'

Cyclic frequency of oscillations

$$\omega = \frac{2\cdot \pi}{T}$$

ω - angular (cyclic, radian) frequency
T - period

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Calculate 'ω'

Cyclic frequency of oscillations

$$\omega = 2\cdot \pi\cdot \nu$$

ω - angular (cyclic, radian) frequency
ν - frequency

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Calculate 'ω'

Phase of harmonic oscillations

$$\phi = \omega\cdot t$$

φ - phase
ω - angular (cyclic, radian) frequency
t - time

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Calculate 'φ'

Phase of harmonic oscillations

$$\phi = \frac{2\cdot \pi\cdot t}{T}$$

φ - phase
t - time
T - period

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Calculate 'φ'

Phase of harmonic oscillations

$$\phi = 2\cdot \pi\cdot \nu\cdot t$$

φ - phase
ν - frequency
t - time

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Calculate 'φ'

Harmonic oscillations: declination

$$x = x_{m}\cdot cos(\omega\cdot t+\phi)$$

x - declination
x_m - maximum declination
ω - angular (cyclic, radian) frequency
t - time
φ - phase

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Calculate 'x'

Oscillation period of spring pendulum

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

T - period
m - mass
k - stiffness

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Calculate 'T'

Oscillation period of mathematical pendulum

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

T - period
l - pendulum length
g - free fall acceleration

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Calculate 'T'

Harmonic oscillations: body speed

$$v = v_{m}\cdot cos(\omega\cdot t+\frac{\pi}{2})$$

v - speed (velocity)
v_max - maximum speed
ω - angular (cyclic, radian) frequency
t - time

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Calculate 'v'

Harmonic oscillations: body speed

$$v = v_{m}\cdot sin(\omega\cdot t)$$

v - speed (velocity)
v_max - maximum speed
ω - angular (cyclic, radian) frequency
t - time

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Calculate 'v'

Harmonic oscillations: body acceleration

$$a = a_{m}\cdot cos(\omega\cdot t+\pi)$$

a - acceleration
a_m - maximum acceleration
ω - angular (cyclic, radian) frequency
t - time

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Calculate 'a'

Harmonic oscillations: body acceleration

$$a = -\omega^{2}\cdot x\cdot cos(\omega\cdot t)$$

a - acceleration
ω - angular (cyclic, radian) frequency
x - declination
t - time

Find It is known that:

Calculate 'a'

Harmonic oscillations: body speed

$$v = -\omega\cdot x\cdot sin(\omega\cdot t)$$

v - speed (velocity)
ω - angular (cyclic, radian) frequency
x - declination
t - time

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Calculate 'v'

Harmonic oscillations: body maximum speed

$$v_{m} = \omega\cdot x_{m}$$

v_max - maximum speed
ω - angular (cyclic, radian) frequency
x_m - maximum declination

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Calculate 'v_m'

Harmonic oscillations: body maximum acceleration

$$a_{m} = \omega\cdot v_{m}$$

a_m - maximum acceleration
ω - angular (cyclic, radian) frequency
v_max - maximum speed

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Calculate 'a_m'

Harmonic oscillations: body maximum acceleration

$$a_{m} = \omega^{2}\cdot x_{m}$$

a_m - maximum acceleration
ω - angular (cyclic, radian) frequency
x_m - maximum declination

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Calculate 'a_m'

Harmonic oscillations: body kinetic energy

$$E_{k} = \frac{m\cdot v^{2}}{2}$$

E_k - kinetic energy
m - mass
v - speed (velocity)

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Calculate 'E_k'

Harmonic oscillations: body potential energy

$$E_{p} = \frac{k\cdot x^{2}}{2}$$

E_p - potential energy
k - stiffness
x - declination

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Calculate 'E_p'

Harmonic oscillations: body total energy

$$E = E_{_k}+E_{_p}$$

E - energy
E_k - kinetic energy
E_p - potential energy

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Calculate 'E'

Harmonic oscillations: body total energy

$$E = (\frac{m\cdot v^{2}}{2})+(\frac{k\cdot x^{2}}{2})$$

E - energy
m - mass
v - speed (velocity)
k - stiffness
x - declination

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Calculate 'E'

Resonance - oscillation amplitude

$$x = \frac{F}{\omega\cdot \mu}$$

x - declination
F - force
ω - angular (cyclic, radian) frequency
μ - coefficient of friction

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Calculate 'x'