Lecture 21: Ideal Spring and Simple Harmonic Motion

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Lecture 21 Ideal Spring and Simple Harmonic
Motion

• New Material Textbook Chapters 10.1, 10.2 and
  10.3

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Ideal Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed position
  (for small x).
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant)

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Ideal Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant)

relaxed position
FX -kx gt 0
x
x ? 0
x0
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Ideal Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant)

relaxed position
FX - kx lt 0
x
x gt 0
x0
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Simple Harmonic Motion

• Consider the friction-free motion of an object
  attached
• to an ideal spring, i.e. a spring that behaves
  according to
• Hookes law.
• How does displacement, velocity and acceleration
• of the object vary with time ?
• Analogy
• Simple harmonic motion along x
• lt-gt x component of uniform
  circular motion

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What does moving along a circular path have to do
with moving back forth in a straight line
(oscillation about equilibrium) ??
x
8
8
q
R
7
7
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Velocity and Acceleration

• Using again the reference circle one finds for
  the velocity
• v - vT sin q - A w sin (w t)
• and for the acceleration
• a - ac cos q - A w2 cos (w t)
• 
  with w in rad/s

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Concept Question

• A mass on a spring oscillates back forth with
  simple harmonic motion of amplitude A. A plot of
  displacement (x) versus time (t) is shown below.
  At what points during its oscillation is the
  speed of the block biggest?
• 1. When x A or -A (i.e. maximum displacement)
• 2. When x 0 (i.e. zero displacement)
• 3. The speed of the mass is constant

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Concept Question

• A mass on a spring oscillates back forth with
  simple harmonic motion of amplitude A. A plot of
  displacement (x) versus time (t) is shown below.
  At what points during its oscillation is the
  magnitude of the acceleration of the block
  biggest?
• 1. When x A or -A (i.e. maximum displacement)
• 2. When x 0 (i.e. zero displacement)
• 3. The acceleration of the mass is constant

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Springs and Simple Harmonic Motion
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Simple Harmonic Motion
At t0 s, xA or At t0
s, x0 m
x(t) Acos(?t) v(t) -A?sin(?t) a(t)
-A?2cos(?t)
x(t) Asin(?t) v(t) A?cos(?t) a(t)
-A?2sin(?t)
OR
Period T (seconds per cycle) Frequency f
1/T (cycles per second) Angular frequency ?
2?f 2?/T For spring ?2 k/m
xmax A vmax A? amax A?2
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Elastic Potential Energy

• Work done by the (average) restoring force of the
  spring is
• W Fave s cos q ½ k (
  x0xf) (x0-xf)
• ½ k (x02 xf2)
  Epot,elastic,0- Epot,elastic,f
• The elastic potential energy has to be considered
  when
• Calculating the total mechanical energy of an
  object
• attached to a spring.