Simple Pendulum

1
Simple Pendulum

• Take a simple pendulum

• Period (T) seconds
• The time to complete one cycle
• Frequency (f)- /sec 1/sec Hertz Hz cycles/sec
• Number of cycles per unit time

2
Pendulum

• Period (T)
• The time to complete one cycle
• Frequency (f)
• Number of cycles per unit time

3

• Pendulum bob with mass m
• Length l (from axis of rotation to center of mass
  of bob. see note below)
• Assume string massless
• Forces on bob
• Weight and Tension

Note A pendulum with a suspension material that
you cannot ignore the mass of, such a wood, metal
or large cord is called a physical pendulum and
the analysis would be different than what is
presented here.
4

• Oscillation occurs via
• Restoring force (as in spring/mass)
• Sum the forces

5

• Tension and radial component of weight cancel out

6

• Acceleration not directly proportional to
  displacement from equilibrium
• So, this motion is not simple harmonic motion

7

• But we can use small angle approximations

Check it out on your calculator ---- in RADIANS!
Sin? ?
8

• Angle is ratio of arc length to length of pendulum

9

• Again, assuming small angle,
• Arc length chord x

10

• This is precisely the same relationship for
  simple harmonic motion-
• Acceleration directly proportional to displacement

If angle is small a pendulum will execute simple
harmonic motion
11

• Derive an equivalent spring constant for a
  pendulum
• Recall the following for a spring mass system

• Recall for a pendulum

12

• Lets derive the period of a pendulum
• Recall the equivalent spring constant for a
  pendulum

• Recall the period of a mass spring system

• Now insert for k the above kp

• Which reduces to

13

• As already established experimentally
• Period of a simple pendulum at small angles
  (lt150) relies solely on length and location (g)

14

• Now, take a simple pendulum and take it to some
  other planet --- can you experimentally obtain
  the acceleration due to gravity?

• Measure the period and solve for g

15
Fig 10.30, p.353
Slide 15
16
Fig 10.p355a, p.355
Slide 16
17
(No Transcript)