Simple Harmonic Motion

 Simple harmonic motion (SHM) a type of wavelike motion that describes the behavior of many physical phenomena: –a pendulum –a bob attached to a spring –low amplitude waves in air (sound), water, the ground –the electromagnetic field of laser light –vibration of a plucked guitar string –the electric current of most AC power supplies

 This wavelike motion is repetitive.  It is caused by a restoring force that acts in the opposite direction of the displacement.

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

 Under small displacements, the simple pendulum behaves as a harmonic oscillator.  the restoring force is a component of the bob’s weight. L mg F g,x F g,y FtFt

 The period (T) is the amount of time it takes a wave to go through 1 cycle.  Frequency (f ) is the number of cycles per second.

 unit of a frequency = hertz (Hz)  Heinrich Hertz ( ), discovered radio waves. f = 1 / T T = 1 / f

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

 The period of a mass-spring system depends on the mass of the object and the spring constant. T = 2π √(m/k)

Sample Problem The body of a 1275 kg car is supported on a frame by four springs. Two people riding in the car have a combined 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 seconds, the vibration approaches simple harmonic motion. Find the spring constant of a single spring. k = ? T = 2π √(m / k) T² = (4π²m) / k k = (4π²m) / T² k = [4π² (357 kg)] / (0.84 s)² = 2.00 x 10 4 N/m m = (1275 kg kg) / 4 = 357 kg T= 0.84 s

 The period of a simple pendulum depends on the string length and gravity. T = 2π √(L/g)