Simple Harmonic Motion: Lecture 21 from Introductory Physics I, Study notes of Physics

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PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 21

Simple Harmonic Motion ! , f, T determined by mass and spring constant A, " determined by initial conditions: x (0), v (0) Last Lecture f =

T

! = 2 " f =

T

x = A cos(! t " #)

v = "! A sin(! t " #)

a = "!

2

A (cos! t " #)

k m

Example 13. 4 a An object undergoing simple harmonic motion follows the expression,

x ( t ) = 4 + 2 cos[! ( t " 3 )]

The amplitude of the motion is: a) 1 cm b) 2 cm c) 3 cm d) 4 cm e) -4 cm Where x will be in cm if t is in seconds

Example 13. 4 b An object undergoing simple harmonic motion follows the expression,

x ( t ) = 4 + 2 cos[! ( t " 3 )]

The period of the motion is: a) 1/3 s b) 1 /2 s c) 1 s d) 2 s e) 2 /! s Here, x will be in cm if t is in seconds

Example 13. 4 d An object undergoing simple harmonic motion follows the expression,

x ( t ) = 4 + 2 cos[! ( t " 3 )]

The angular frequency of the motion is: a) 1/3 rad/s b) 1 /2 rad/s c) 1 rad/s d) 2 rad/s e)! rad/s Here, x will be in cm if t is in seconds

Example 13. 4 e An object undergoing simple harmonic motion follows the expression,

x ( t ) = 4 + 2 cos[! ( t " 3 )]

The object will pass through the equilibrium position at the times, t = _____ seconds a) …, -2, -1, 0, 1, 2 … b) …, -1.5, -0.5, 0.5, 1.5, 2.5, … c) …, -1.5, -1, -0.5, 0, 0.5, 1.0, 1.5, … d) …, -4, -2, 0, 2, 4, … e) …, -2.5, -0.5, 1.5, 3.5, Here, x will be in cm if t is in seconds

Simple Pendulum F =! mg sin " sin " = x x 2

• L 2

x L F #! mg L x

g L

max

cos(! t # $)

Simple pendulum Frequency independent of mass and amplitude! (for small amplitudes)

g L

Example 13. 5 A man enters a tall tower, needing to know its height h. He notes that a long pendulum extends from the roof almost to the ground and that its period is 15.5 s. (a) How tall is the tower? (b) If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s 2 , what is the period of the pendulum there? a) 59.7 m b) 37.6 s

Damped Oscillations In real systems, friction slows motion

Longitudinal (Compression) Waves Elements move parallel to wave motion. Example - Sound waves

Transverse Waves Elements move perpendicular to wave motion. Examples - strings, light waves

Snapshot of a Transverse Wave wavelength x y = A cos 2! x "

$

% & ' ( )

Snapshot of Longitudinal Wave

y could refer to pressure or density

y = A cos 2!

x