Oscillations

1
Oscillations

• Phys 2101
• Gabriela González

2
A simple pendulum

• The horizontal force on a simple pendulum is
• F -mg sin ?
• For small oscillations, sin ? ?s/L, so
• F -(mg/L) s
• Another spring!
• SHM s sm cos(? t f)
• with ?2(mg/L/m)g/L
• and period T 2? (L/g) ½

A geophysicist is asked to measure how much the
acceleration of gravity changes at different
points in a mountain. He takes with him just a
mass, a string, a measuring tape, and a
stopwatch. How does he manage?
3
A real (physical) pendulum

• Restoring torque is ? -mgh ?
• Angular frequency is ?2mgh/I
• and period is T2? (I/mgh) ½
• Is a simple pendulum also a physical pendulum?
• Which pendulum has a longer period?
• a stick of length L and mass M
• a mass M at the end of a string of length L
• a disk with diameter L and mass M, hanging from
  its edge.

4
Simple Harmonic Motion

• Whats the frequency of oscillation of these
  systems?

k
k
k
k
m
m
k
m
5
The force law for SHM

• Displacement
• x(t) xm cos(? t f)
• Velocity
• v(t) dx(t)/dt -? xm sin(? t
  f)
• Acceleration
• a(t) dv(t)/dt -?2 xm cos(? t f)
  - ?2 x(t)

• Force
• F m a -m ?2 x -k x
• A spring! ??k/m

6
Simple Harmonic Motion Energy

• If F-kx like for a spring, potential energy
  -work done by the force is U½ k x2. Total
  mechanical energy is conserved
• E U K
• ½ k x2 ½ m v2
• ½ k xm2 ½ m vm2

7
Damped harmonic motion

• If there is a viscous damping force F - b v,
• then energy is not conserved, and oscillations
  are damped.

8
Driven harmonic motion resonance

• If a system with natural frequencies is driven by
  a harmonic force at one of those frequencies, the
  motion is AMPLIFIED, with the maximum amplitude
  only limited by existing damping forces.

Example Tacoma Narrows Bridge, WA,
1940 http//en.wikipedia.org/wiki/Tacoma_Narrows_B
ridge
9
Example

• A 2200 lb car drops 2in on its suspension when
  four 180 lb passengers step on the car. Assuming
  there are four suspension springs, what is the
  spring constant of each spring?
• The car with all four passengers over a rough
  washboard dirt road with corrugations 13ft
  apart which causes the car to bounce on its
  spring suspension. At what velocity will the car
  bounce up and down with the largest amplitude?