Two Chapter 7

• Simple Harmonic Motion - The type of vibratory or periodic motion where the restoring force is proportional to the displacement.

• The above formula is Hooke's Law. F is the restoring force, k is a constant, and x is the displacement from the equilibrium position.

E.G. A weight hung on a spring displays SHM.

• if at equilibrium, x = 0 and F = 0
• if above equilibrium, x > 0 and F is towards equilibrium.
• if below equilibrium, x < 0 and F is towards equilibrium.
• The magnitude of F depends upon the displacement from equilibrium.
• Any elastic system obeying Hooke's law may oscillate with SHM.

• It can be shown that a point on a rotator projected onto a screen will move up and down just as a weight on the end of a spring does.

• As the rotator turns, the shadow of the knob goes up and down in unison with the weight on the spring.
• Because of this motion in unison, we can develop an equation relating the period of the motion to the position and acceleration of the weight.

• The knob experiences a centripetal force given by F_c = ma_c.
• The weight only moves up and down, therefore, it moves at a velocity equal to the vertical component of V_o and has acceleration equal to the vertical component of a_c.
• The maximum, vertical velocity is at B, while the minimum vertical velocity is at A and C.
• The maximum vertical acceleration is at A and C, while the minimum vertical acceleration is at B.
• From the diagram, one can see we have similar triangles. Therefore, the ratios of corresponding sides must be equal.

• For a rotating rotator, a_c and R are constant. Therefore,

• Rearranging gives a = Cx
• Since F = ma, we can solve for a, substitute this into a = Cx, and then solve the result for F to get F = kx
  • Since for the weight being synchronized with the rotator, we can prove that F = kx applies, we can conclude that the projection of the knob and the weight display SHM.
  • a_c for the rotator is given by

• Solving for T gives:

• The above relationship is true for all SHM. T is the period of the motion, x is the displacement from the equilibrium position, and a is the acceleration of the object at the displacement x.
• Note that x can be replaced by amplitude (A), if a in the equation is the maximum acceleration.

• A mass hung on a spring obeys Hooke's Law, and so it experiences a force and acceleration.

• k is the spring constant, m is mass of the object hung from the spring.
• k is proportional to the stiffness of the spring. Therefore, stiff springs have a small period.
• Large mass causes large periods.
• Amplitude does not affect the period (Amplitude is not in the equation.).

• Note that Fs must be proportional to the arc length not x.
• x is approximately equal to the arc length for small angles q.
• Since pendulums obey Hooke's Law for small angles, they display Simple Harmonic Motion for small angles

• The above equation:
  • gives the period of a pendulum swinging through small angles
  • it is approximate
  • T is independent of the mass and amplitude
  • the equation can be used to determine the value of g.

• When the paper is pulled by the bouncing weight, the pen will draw a sine curve onto the paper like the one shown on the right.
• If we refer back to the rotator, then the synchronized rotator would turn through 360^o in a time T (the period).
• If the rotator turns through q in a time t, then we can set up a proportion

• A sine curve is defined by the equation

• Equation I is used when your calculator is set on degrees. Equation II is used when your calculator is set on radians.
• The variables in the equations are defined as:
  • t is time
  • T is period
  • A is amplitude (maximum displacement from equilibrium)
  • x is displacement of the weight from equilibrium at time t
• These equations can be used to describe Simple Harmonic Motion by using angles instead of time.
  • For a pendulum,
    • at maximum height (maximum x) q = 90^o or 270^o
    • at minimum height (minimum x) q = 0^o or 180^o or 360^o
• For both waves and Simple Harmonic Motion, we can use the angle q to specify the phase of the wave or Simple Harmonic Motion.
• Phase Angle - The angle used to describe the phase of a wave or Simple Harmonic Motion.
  • It is not the same as the angle the pendulum is hung at.
• In Phase - have the same

• Out of Phase - occurs when the above conditions are not met.
• It is customary to describe phase with phase angle.
  • In phase - means the same phase angle
  • Out of phase - means different phase angle
• Damped Harmonic Motion - The reduction in amplitude as a result of the dissipation of energy.
  • Usually kinetic energy is converted to heat.
  • Below is a graphical representation of damped Harmonic Motion.

• Damping does not affect the period unless the rate of damping is large.
• The equations for Simple Harmonic Motion still apply because amplitude is not found in them (Note: that equations I and II above would not apply.)

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November 5, 2013