Periodic Motion What is periodic motion? When a vibration or oscillation repeats itself over and over the motion is said to be periodic. Most objects vibrate briefly when given an impulse Electrical oscillations occur in TV’s and radio’s, atoms vibrate around a fixed spot

Terminology Oscillation or Vibration – a motion that repeats itself with no net displacement. Equilibrium Position – the point that the object oscillates around. Also known as the rest position. Displacement – how far the mass is from the equilibrium point (x) Maximum displacement – how far the mass moves from the equilibrium position. (xmax occurs at A) Amplitude (A) – the distance from the equilibrium point to the maximum displacement. Cycle – a complete to and fro motion.

Terminology Period (T) – the time needed to complete one cycle. (units – seconds) Frequency (f) – the number of cycles completed in one second. Units are Hertz. (Hz = 1/s = s-1) Formula f =1/T T=1/f

Simple Harmonic Motion Simple Harmonic Motion is any motion in which the restoring force is proportional to displacement. Examples: An acrobat swinging on a trapeze Child on a playground swing Pendulum of a clock or metronome Mass on the end of a spring Restoring force – the force that pushes or pulls the mass back to equilibrium.

FSpring = FElastic = – kx Hooke’s Law In 1678 Robert Hooke found that most mass-spring systems obey a simple relationship between force and displacement for small displacements. This is a restoring force. FSpring = FElastic = – kx The force, F, is negative because it is always a restoring force, pulling or pushing the opposite direction of the displacement. The Applied Force is in the opposite direction of the Spring Force.

Spring Constant “k” The value of the constant measures the ‘stiffness’ of the spring. The larger the value, the stiffer the spring Unit for the spring constant, k, is N/m.

Period of a Spring Period of a spring is given by the following equation: m = mass in kg k = spring constant in N/m What happens to the period as mass increases? What happens to the period as the spring constant increases? What is the frequency equation?

Elastic Potential Energy At maximum displacement the potential energy is at its maximum and kinetic is at its minimum. At the equilibrium position the potential is at its minimum and kinetic is at its maximum. US = USpring = ½ k x2 KE = ½ m v2

Mass on a Spring HORIZONTAL MOTION If the object is in MOTION then at the equilibrium position (x=0) the velocity is at the maximum. At the maximum displacement, spring force and acceleration reaches a maximum and the velocity is zero. Known as a turning point. The acceleration is in the opposite direction of the motion.

E = K + U E = 1/2mv2 + 1/2kx2 A E = 1/2kA2 E = 1/2mv02 A E = 1/2kA2 x0 Maximum displacement is A = x for the example below. A E = 1/2kA2 E = 1/2mv02 A E = 1/2kA2 x0

Conservation of energy review Remember that all energy is conserved The energy just changes from one form to an other. Initial Energy = Final Energy Ki + Ui = Kf +PUf The example is for a horizontal spring system. Solving for “vf” as for a spring released at “A” and finding the maximum velocity as the spring passes through the equilibrium position.

Finding the velocity in general depends on the original amplitude and the location in the cycle, x.

Potential Energy in a Spring Example A spring with a force constant of 5.2 N/m has a relaxed length of 2.45 m. When a mass is attached to the end of the spring and allowed to come to rest, the vertical length of the spring is 3.57 m. Calculate the elastic potential energy stored in the spring.

Spring Constant in a Spring Example A mass of 0.30 kg is attached to a spring and is set into vibration with a period of 0.24 s. What is the spring constant of the spring?

Period and Frequency in a Spring Example A spring of spring constant 30.0 N/m is attached to different masses, and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: 2.3 kg 15 g 1.9 kg

Simple Graph of SHM Cosine Graph This means the position is a function of time.

Graph of Unit Circle x0 q=0

x0 q=p/2

x0 q=p

x0 q=3p/2

Notice that the radius of the circle equals the amplitude of the spring. x0 q=2p

Amplitude is independent of the period!!! The maximum velocity is equal to the path length of the circle (2pr) divided by time.

Is a pendulum simple harmonic motion? Simple Pendulum Is a pendulum simple harmonic motion? Simple pendulum is a mass on the end of a string. The mass is called a “BOB”. Assume the mass is concentrated at a point. Neglect air resistance and friction.

The restoring force is a component of the bob’s weight (-mg sin q). If the restoring force is proportional to the displacement the pendulum’s motion is simple harmonic. There are two forces acting on the pendulum: The tension in the string The weight of the bob

Motion of a Pendulum Ftension Weight Ftension Weight Weighty Weightx

Restoring force As the pendulum is pulled back the x component of the weight gets larger and the y component gets smaller. Therefore the greater the displacement the larger the restoring force For small displacements the pendulums motion is simple harmonic.

Energy of a Pendulum Energy is conserved. At maximum displacement: Velocity is zero, acceleration is largest, Energy is all potential. At equilibrium: Velocity is the largest, acceleration is zero, Energy is all kinetic.

Amplitude, Period, and Frequency The time it takes for a pendulum to swing from one side to the other and back again is one period. The number of complete cycles in one second is the frequency.

Small Angle Notes Extra Notes

Figure 14.19 Figure 14.19

Figure 14.19B Figure 14.19B

Figure 14.19A Figure 14.19A

What is this length here? Answer is l cos q Figure 14.20 What is this length here? Answer is l (1 - cos q) l sin q

Trig of a Pendulum Restoring force = -mg sinq For small angles sin q = q Using x = L q gets: F = -(mg/L)x This is similar to Hooke’s Law with k = mg/L Using the equations derived for a spring : Note that the period of Pendulum does NOT depend on the mass of the bob!! q L FT mg sinq mg q mg cosq x

Figure 14.22

Period of a Pendulum Depends on the length and free fall acceleration. For small amplitudes the period DOES NOT depend on the amplitude.

Damped Harmonic Motion The amplitude of any real oscillating spring slowly decreases. This is damped harmonic motion Damping is due to friction and air Forced Vibrations/ Resonance When a system is set in motion then left alone it vibrates at its natural frequency (f0) When an outside force is constantly applied it creates forced vibrations The amplitude of the forced vibration depends on the difference between f and f0.

Now with x(t) find v(t) and a(t)