Simple Harmonic Motion

1
Simple Harmonic Motion

• Holt Physics
• Pages 438 - 451

2
Distinguish simple harmonic motion from other
forms of periodic motion.

• Periodic motion is motion in which a body moves
  repeatedly over the same path in equal time
  intervals.
• Examples uniform circular motion and simple
  harmonic motion.

3
Contd

• Simple Harmonic Motion (SHM) is a special type of
  periodic motion in which an object moves back and
  forth, along a straight line or arc.
• Examples pendulum, swings, vibrating spring,
  piston in an engine.
• In SHM, we ignore the effects of friction.
• Friction damps or slows down the motion of the
  particles. If we included the affect of friction
  then its called damped harmonic motion.

4
Contd

• For instance a person oscillating on a bungee
  cord would experience damped harmonic motion.
  Over time the amplitude of the oscillation
  changes due to the energy lost to friction.
• http//departments.weber.edu/physics/amiri/directo
  r/DCRfiles/Energy/bungee4s.dcr

5
State the conditions necessary for simple
harmonic motion.

• A spring wants to stay at its equilibrium or
  resting position.
• However, if a distorting force pulls down on the
  spring (when hanging an object from the spring,
  the distorting force is the weight of the
  object), the spring stretches to a point below
  the equilibrium position.
• The spring then creates a restoring force, which
  tries to bring the spring back to the equilibrium
  position.

6
Contd

• The distorting force and the restoring force are
  equal in magnitude and opposite in direction.
• FNET and the acceleration are always directed
  toward the equilibrium position.

7
Contd

• Applet showing the forces, displacement, and
  velocity of an object oscillating on a spring.
• http//www.ngsir.netfirms.com/englishhtm/SpringSHM
  .htm

8
Displacement Velocity Acceleration
9
Contd

• at equilibrium
• speed or velocity is at a maximum
• displacement (x) is zero
• acceleration is zero
• FNET is zero (Restoring Force Distorting Force)
  
• object continues to move due to inertia

10
Contd

• at endpoints
• speed or velocity is zero
• displacement (x) is at a maximum equal to the
  amplitude
• acceleration is at a maximum
• restoring force is at a maximum
• FNET is at a maximum

11
State Hookes law and apply it to the solution of
problems.

• Hookes Law relates the distorting force and the
  restoring force of a spring to the displacement
  from equilibrium.

12
Contd

• F magnitude of the distorting or restoring
  force in Newtons
• k spring constant or force constant (stiffness
  of a spring) in Newtons per meter (N/m)
• x displacement from equilibrium in meters

13
Calculate the frequency and period of any simple
harmonic motion.

• T period (time required for a complete
  vibration) in seconds
• f frequency in vibrations / second or Hertz

14
Relate uniform circular motion to simple harmonic
motion.

• The reference circle relates uniform circular
  motion to SHM.
• The shadow of an object moving in uniform
  circular motion acts like SHM.
• The speed of an object moving in uniform circular
  motion may be constant but the shadow wont move
  at a constant speed.
• The speed at the endpoints is zero and a maximum
  in the middle.
• The shadow only shows one component of the motion.

15
Contd

• Applet showing the forces, displacement, and
  velocity of an object oscillating on a spring and
  an object in uniform circular motion.
• http//www.ngsir.netfirms.com/englishhtm/SpringSHM
  .htm

16
Identify the positions of and calculate the
maximum velocity and maximum accelerations of a
particle in simple harmonic motion.

• The acceleration is a maximum at the endpoints
  and zero at the midpoint.
• The acceleration is directly proportional to the
  displacement, x.
• The radius of the reference circle is equal to
  the amplitude.
• The force and acceleration are always directed
  toward the midpoint.

17

• Fmax Force (N)
• m mass (kg)
• A - Amplitude (m)
• T period (seconds)

18
Contd

• This equation shows that the spring force
  according to Hookes Law is equal to the maximum
  force the spring experiences. K cancels on both
  sides.

19
Contd

• k spring constant (N/m)
• m mass (kg)
• T period (s)

20
Contd

• Vmax maximum velocity (m/s)
• A amplitude or maximum displacement (m)
• T period (s)

21
Contd

• Vmax maximum velocity (m/s)
• A amplitude or maximum displacement (m)
• T period (s)

22
Contd

• a acceleration (m/s2)
• v velocity (m/s)
• A amplitude (m)

23
Contd

• a acceleration (m/s2)
• A Amplitude(m)
• T period (s)

24
Contd

• T period (s)
• m mass (kg)
• k spring constant (N/m)

25
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26
Relate the motion of a simple pendulum to simple
harmonic motion.

• A pendulum is a type of SHM.
• A simple pendulum is a small, dense mass
  suspended by a cord of negligible mass.
• The period of the pendulum is directly
  proportional to the square root of the length and
  inversely proportional to the square root of the
  acceleration due to gravity.

27
Contd

• Applet of displacement, velocity, acceleration,
  force, etc acting on a pendulum.
• http//www.walter-fendt.de/ph11e/pendulum.htm

28
Contd

• T period (s)
• l length (m)
• g acceleration due to gravity (m/s2)