Oscillations

1
Oscillations

• Phys 2101
• Gabriela González

2
Oscillations

• Oscillations anything that swings back and
  forth.
• Harmonic oscillations described with
  trigonometric functions.
• Three kinds of oscillations
• free,
• damped and
• forced.

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Examples
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Simple Harmonic Motion Free oscillations

• Displacement
• x(t) xm cos(wm t f)

frequency (cps, Hz) fw/2p period (sec)
T1/f2p/w
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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

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Example

• A 5.00 kg block is attached to a spring with
  spring constant k2.45 N/cm.
• If the spring is then stretched 12.0 cm and
  released from rest,
• what is its period of oscillation?
• what is its maximum amplitude of oscillation?
• what is its maximum velocity?
• what is the maximum force the spring will exert
  on the body?

7
Example

• A 4.00 kg block hangs from a spring, extending it
  16.0cm from its unstretched position.
• What is the spring constant?
• The block is removed and a 0.5 kg body is hung
  from the same spring, letting it reach
  equilibrium. If the spring is then stretched 12.0
  cm further and released from rest,
• what is its period of oscillation?
• what is its maximum amplitude of oscillation?
• what is its maximum velocity?
• what is the maximum force the spring will exert
  on the body?
• What would be the period of oscillation of the
  same body hanging from the same spring in the
  Moon?

8
Example

• The graph shows the position function of a simple
  harmonic oscillator. What are the constants xm,
  ??? ? ??in the function x(t)xm cos(?t?) ?
• At what time after t0 does the particle cross
  the origin for the first time?
• What is its velocity and acceleration at that
  time?