Oscillations

1
Oscillations

• Phys 2101
• Gabriela González

2
Simple Harmonic Oscillator

• 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

3
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?

4
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?

5
Example

• A block is on a horizontal surface that is moving
  back and forth horizontally with simple harmonic
  motion of frequency 2 Hz. The coefficient of the
  static friction between block and surface is
  0.50. How great can the amplitude of the motion
  be if the block is not to slip along the surface?

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 (xm cos(?t f))2 ½ m (-?xm sin(?t
  f))2
• ½ k xm2 cos2(?t f) ½ m ?2 xm2 sin2(?t f)
• ½ k xm2 cos2(?t f) ½ m (k/m) xm2 sin2(?t
  f)
• ½ k xm2 cos2(?t f) ½ k xm2 sin2(?t f)
• ½ k xm2 (cos2(?t f) sin2(?t f))
• ½ k xm2

7
Simple Harmonic Motion Energy

• The block has a kinetic energy of 3J and the
  spring an elastic potential energy of 2 J when
  the block is at x2.0cm.
• What is the kinetic energy at x0?
• What is the potential energy at x-2.0cm?
• What is the potential energy at x-xm?

8
Torsion Pendulum

• Restoring torque (spring) ?-k?
• The angular displacement follows a simple
  harmonic motion
• ?(t)?m cos(? t f)
• with ?2 k/I (I rotational inertia).
• Period T 2? (I/k) ½

An engineer has an odd shaped object and needs to
find its rotational inertia about an axis through
the center of mass. She has a wire of unknown
properties, a rod, a measuring tape, a stopwatch,
and a scale. How can torsion pendulums help her?
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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?
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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.

11
Simple Harmonic Motion

• Whats the frequency of oscillation of these
  systems?

k
k
k
k
m
m
k
m