18.1-18.2 simple harmonic motion

the equilibrium position

the position at which an object on a string eventually comes to rest

oscillating about equilibrium

an object moving from maximum height at one side to maximum height on the other side and back and so on

examples of oscillating motion

an object on a spring moving up and down repeatedly; a pendulum moving to and fro repeatedly; a ball bearing rolling from side to side; a small boat rocking from side to side

the motion of an oscillating object

moves repeatedly one way then in the opposite direction through its equilibrium position

the amplitude of oscillation

the maximum displacement of the oscillating object from equilibrium

free vibrations

when the amplitude of oscillations is constant and no frictional forces are present

the time period, T, of oscillating motion

the time for one complete cycle of oscillation

the frequency of oscillations

the number of cycles per second made by an oscillating object

the unit of frequency

hertz, Hz

equation for the time period, T

T = 1/f

equation for the angular frequency of oscillating motion

ω = 2π/T = 2πf

the unit of angular frequency, ω

radian per second (rad s^-1)

the time period of oscillating motion for 2 children on adjacent identical swings

the same for both, because the swings are identical, + their phase difference stays the same as they oscillate

equation for the phase difference, in radians, for 2 objects oscillating at the same frequency

phase difference = 2πt/ΔT, where ΔT is the time between successive instant when the 2 objects are at maximum displacement in the same direction

the variation of speed for an oscillating object

speeds up as it returns to equilibrium, and slows down as it moves away from equilibrium

what is the variation of velocity with time given by

the gradient of the displacement-time graph

when is the magnitude of velocity of an oscillating object greatest

when the gradient of the displacement-time graph is greatest = at zero displacement when the object passes through equilibrium

when is velocity of an oscillating object zero

when the gradient of the displacement-time graph is zero = at maximum displacement in either direction

what is the variation of acceleration with time given by

the gradient of the velocity-time graph

when is the acceleration of an oscillating object greatest

when the gradient of the velocity-time graph is greatest = when the velocity is zero = at maximum displacement in the opposite direction

when is the acceleration of an oscillating object zero

when the gradient of the velocity-time graph is zero = when the displacement is zero + velocity is a maximum

simple harmonic motion definition

oscillating motion in which the acceleration is proportional to the displacement, and always in the opposite direction to the displacement --> a ∝ -X

equation for the definition of simple harmonic motion

a = -ω^2X, where X = displacement, and ω = angular frequency