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.
• https//ngsir.netfirms.com/englishhtm/SpringSHM.ht
  m

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 (magnitude of Restoring Force
  magnitude of Distorting Force which is the
  weight)
• 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
• F -kx (hooks law)
• 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 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
• The distorting force is equal in magnitude but
  opposite in direction to the restoring force.

13
The spring shown to the right has an unstretched
length of 3 cm. When a 2 kg object is hung from
the spring, it comes to rest at the 7 cm mark.
What is the spring constant of this spring? 490
N/m What direction is the restoring force? upward
14
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

15

• A particle is moving in simple harmonic motion
  with a frequency of 10 Hz.
• What is its period?
• 0.1 sec
• How many complete oscillations does it make in
  one minute?
• 600 oscilations

16
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.

17
Contd

• Applet showing the forces, displacement, and
  velocity of an object oscillating on a spring and
  an object in uniform circular motion.
• http//www.physics.uoguelph.ca/tutorials/shm/phase
  0.html

18
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.

19
The mass on the end of a spring (which stretches
linearly) is in equilibrium as shown. It is
pulled down so that the pointer is opposite the
11 cm mark and then released. What is the
amplitude of the vibration? 4 cm What two places
will the restoring force be greatest? 11 cm and 3
cm Where will the restoring force be least? 7 cm
20

• Where is the speed greatest?
• 7 cm
• What two places is the speed least?
• 3 cm and 11 cm
• Where is the magnitude of the displacement
  greatest?
• 3 cm and 11 cm
• Where is the displacement least?
• 7 cm
• Where is the magnitude of the acceleration
  greatest?
• 3 cm and 11 cm
• Where is the acceleration least?
• 7 cm
• Where is the elastic potential energy the
  greatest?
• Where is the kinetic energy the greatest?

21
Contd

• Remember that in uniform circular motion, the
  velocity is calculated using

In SHM, the maximum velocity would be equal to
the velocity of the object in uniform circular
motion. The radius of the circle correlates to
the Amplitude (A) in SHM.
22
Contd

• Remember that in uniform circular motion, the
  centripetal acceleration is calculated using

In SHM, the maximum acceleration would be equal
to the acceleration of the object in uniform
circular motion. The radius of the circle
correlates to the Amplitude (A) in SHM.
23

• A mass hanging on a spring oscillates with an
  amplitude of 10 cm and a period of 2 seconds.
  What is the maximum speed of the object and where
  does it occur?
• 0.314 m/s at equilibrium
• What is the minimum speed of the object and where
  does it occur?
• 0 m/s at the end points.
• What is its maximum acceleration?
• 0.987 m/s2 at the end points

24

• An object moving is simple harmonic motion can be
  located using
• A is amplitude
• f is frequency
• x is displacement from equilibrium
• ? is angular velocity

25

• The mass on the end of a spring (which stretches
  linearly) is in equilibrium as shown.
• It is pulled down so that the pointer is
  opposite the 11 cm mark and then released. A
  spring vibrates in SHM according to the equation
  x 4 cospt.
• How many complete vibrations does it make in 10
  seconds?
• 5 vibrations

26

• The elastic potential energy content of the
  system is

So the maximum elastic potential energy is stored
at the end points of the oscillations where the
displacement is equal to the amplitude of the
vibration
At the end point, the object is not moving so
there is no kinetic energy. Therefore the total
energy content of the system is equal to
27

• A mass on a spring oscillates horizontally on a
  frictionless table with an amplitude of A. In
  terms of Eo (total mechanical energy of the
  system) when the mass is at A, Us ______ and K
  _________.
• Us Eo and K 0
• When the mass is at 0.5 A, then Us __________
  and K _________.
• Us 0.25 Eo and K 0.75Eo
• When the mass is at the equilibrium position,
  then Us _________ and K ________
• Us 0 and K Eo

28

• A 2 kg object is attached to a spring of force
  constant k 500 N / m. The spring is then
  stretched 3 cm from the equilibrium position and
  released. What is the maximum kinetic energy of
  this system?
• 0.225 J
• What is the maximum velocity it will attain?
• 0.47 m/s

29
Contd

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

30

• You want a mass that, when hung on the end of the
  spring, oscillates with a period of 3 seconds.
  If the spring constant is 5 N/m, the mass should
  be _______.
• 1.14 kg

31

• The period for a mass vibrating on very stiff
  springs (large values of k) will be (larger /
  smaller) compared to the same mass vibrating on a
  less stiff spring.
• Smaller
• If the value of k halves, the period will be
  ______ times as long.

32
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.

33
Contd

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

34

• A pendulum has a period of 2 seconds here on the
  surface of the earth. That pendulum is taken to
  the moon where the acceleration due to gravity is
  1/6 as much. What is the period of the pendulum
  on the moon?
• Squareroot of 6 times as much or 4.9 seconds.