Oscillations Formula

Question 1

Frequency Conversion

Answer 1

1 hertz = 1Hz = 1 oscillation per second = 1s^-1

Question 2

Period of a complete oscillation

Answer 2

T = 1/frequency

Question 3

Simple Harmonic Motion Displacement

Answer 3

x = xm cos(ωt + Φ)

• xm is amplitude

Question 4

angular frequency (SHM)

Answer 4

ω = (2π)/T
ω = 2πf

Question 5

velocity (SHM)

Answer 5

v = -ω xm sin(ωt + Φ)

Question 6

accelaration (SHM)

Answer 6

a = -ω^2 xm cos(ωt + Φ)

Question 7

Linear oscillator (angular frequency)

Answer 7

ω = sqrt(k/m)

note: under the influence of Hooke’s Law [F=-kx]

Question 8

Linear Oscillator (period)

Answer 8

T = 2π × sqrt(m/k)

note: under the influence of Hooke’s Law [F=-kx]

Question 9

Kinetic Energy (SHM)

Answer 9

K = 1/2 mv^2

Question 10

Potential Energy (SHM)

Answer 10

U = 1/2 kx^2

Question 11

Mechanical Energy (no friction)

Answer 11

E = K + U

Question 12

period of a torsion pendulum

Answer 12

T = 2π × sqrt(I/k)

Question 13

period of simple pendulum

Answer 13

T = 2π × sqrt(L/g)

Question 14

period of physical pendulum

Answer 14

T = 2π × sqrt(I/mgh)

Question 15

damping force

Answer 15

F = -bv

• b is a damping constant
  v is the velocity of the oscillator

Question 16

Displacement of the oscillator (damped)

Answer 16

x(t) = xm e^(-bt/2m) cos (ω’t + Φ)

Question 17

Angular frequency (damped oscillator)

Answer 17

ω’ = sqrt[ (k/m) - (b^2/4m^2)]

Question 18

small damping constant

Answer 18

b &laquo_space;sqrt(km)
then ω’ is approx ω

Question 19

Mechanical Energy of the Oscillator (damped)

Answer 19

E(t) = 1/2 kx^2 e^(-bt/m)

Question 20

forced oscillations and resonance

Answer 20

angular frequence (ω(d)) = natural angular frequency (ω)