Simple Harmonic Motion

Simple harmonic motionSimple harmonic motion (SHM)(SHM) a type a type of wavelike motion that describes the of wavelike motion that describes the behavior of manybehavior of many physical phenomena:physical phenomena:– a penduluma pendulum– a bob attached to a springa bob attached to a spring– low amplitude waves in air (sound), water, the low amplitude waves in air (sound), water, the

groundground– the electromagnetic field of laser lightthe electromagnetic field of laser light– vibration of a plucked guitar stringvibration of a plucked guitar string– the electric current of most AC power suppliesthe electric current of most AC power supplies

This wavelike motion is repetitive.This wavelike motion is repetitive.

It is caused by a It is caused by a restoring forcerestoring force that acts in the opposite direction that acts in the opposite direction of the displacement.of the displacement.

If we stretch a If we stretch a spring with a spring with a mass on the mass on the end and let it end and let it go, the mass go, the mass will oscillate will oscillate back and back and forth.forth.

Under small Under small displacements, the displacements, the simple pendulum simple pendulum behaves as a harmonic behaves as a harmonic oscillator.oscillator.

the the restoring forcerestoring force is a is a component of the bob’s component of the bob’s weight.weight.

L

mg

Fg,x

Fg,y

Ft

The The period (period (T)T) is the amount of time it takes is the amount of time it takes a wave to go through 1 cycle.a wave to go through 1 cycle.

Frequency (Frequency (ff ) ) is the number of cycles per is the number of cycles per second.second.

T

t

unit of a frequency =unit of a frequency = hertzhertz (Hz)(Hz)

Heinrich Hertz Heinrich Hertz (1847-(1847-1894)1894), discovered radio , discovered radio waves.waves.

f = 1 / Tf = 1 / T

T = 1 / fT = 1 / f

The maximum displacement from some The maximum displacement from some equilibrium (mid point) position.equilibrium (mid point) position.

t

The period of a mass-spring system The period of a mass-spring system depends on the mass of the object and depends on the mass of the object and the spring constant.the spring constant.

T = 2T = 2ππ √(m/k) √(m/k)

Sample ProblemSample ProblemThe body of a 1275 kg car is supported on a frame by The body of a 1275 kg car is supported on a frame by fourfour springs. Two people riding in the car have a combined springs. Two people riding in the car have a combined mass of 153 kg. When driven on a pothole in the road, the mass of 153 kg. When driven on a pothole in the road, the frame vibrates with a period of 0.84 s. For the first few frame vibrates with a period of 0.84 s. For the first few seconds, the vibration approaches simple harmonic seconds, the vibration approaches simple harmonic motion. Find the spring constant of a motion. Find the spring constant of a singlesingle spring. spring.

k = ?

T = 2T = 2ππ √(m / k)√(m / k)

TT² = (4² = (4ππ²m) / k²m) / k

k = (4k = (4ππ²m) / T²²m) / T²

k = [4k = [4ππ² (357 kg)] / (0.84 s)² =² (357 kg)] / (0.84 s)² = 2.00 x 102.00 x 1044 N/m N/m

m = (1275 kg + 153 kg) / 4 =m = (1275 kg + 153 kg) / 4 = 357 kg357 kg

T= 0.84 s

The period of a simple pendulum The period of a simple pendulum depends on the string length and depends on the string length and gravity.gravity.

T = 2T = 2ππ √(L/g) √(L/g)