Vibrations and Waves Chapter 11

Most object oscillate (vibrate) because solids are elastic and they will vibrate when given an impulse Tuning forks, violin strings, even a spider’s web The spider uses the vibrations to detect prey

11-1 Simple Harmonic Motion An object that vibrates or oscillates moves back and forth over the same path in the same time period This is said to be periodic motion Equilibrium position the position where the net force equals zero.

Spring Constant (Spring stiffness constant) The magnitude of the restoring force, F of a spring is directly proportional to the displacement, x F=-kx Negative because the restoring force is always in the opposite direction of the displacement k, spring stiffness constant, depends upon the spring

The force is not a constant force and therefore the acceleration is not constant either. The linear equations developed earlier cannot be used. However, acceleration at a certain point can be found by using Newton’s 2 nd law

Displacement, x, (m) is measured from the equilibrium position Amplitude, A, (m) is the maximum displacement (should be equal on either side of equilibrium) Cycle a complete vibration Period, T, (s) the time to complete one cycle Frequency, f, (Hz) the number of complete cycles per second

Periods and frequencies are reciprocals of each other

Any vibrating system for which F=-kx is said to exhibit simple harmonic motion (SHM) Such a system is often called a simple harmonic oscillator (SHO)

11-2 Energy in the SHO Work has to be done to stretch the spring and W=½kx 2, this equals the gain in PE (sec. 6-4) So, PE=½kx 2 The total energy must include KE E=½mv 2 + ½kx 2 V is velocity of mass X is displacement from equilibrium

At maximum amplitude At the extreme points v=0 and x = A E = ½kA 2 So the total mechanical energy of SHO is proportional to the square of the amplitude

At equilibrium The x value is zero so there is 0 PE and the velocity is maximum E = ½m 2 max

At any point Energy is part KE and part PE ½mv 2 + ½kx 2 = ½kA 2 Rearranging

11-3 The Period The period depends upon the stiffness of a spring, but also on the mass that is oscillating Oddly though, period doesn’t depend upon the amplitude So