Chapter 12

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Chapter 12 part B

• The Energy of an Harmonic Oscillator
• The Pendulum

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Exercise 12.14

• 14. A 200-g block is attached to a horizontal
  spring and executes simple harmonic motion with a
  period of 0.250 s. The total energy of the system
  is 2.00 J. Find (a) the force constant of the
  spring and (b) the amplitude of the motion.

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Exercise 12.6

• 6. A particle moves along the x axis. It is
  initially at the position 0.270 m, moving with
  velocity 0.140 m/s and acceleration 0.320 m/s2.
  First, assume that it moves with constant
  acceleration for 4.50 s. Find (a) its position
  and (b) its velocity at the end of this time
  interval. Next, assume that it moves with simple
  harmonic motion for 4.50 s and that x 0 is its
  equilibrium position. Find (c) its position and
  (d) its velocity at the end of this time
  interval.

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Exercise 12.18
18. A 2.00-kg object is attached to a spring and
placed on a horizontal, smooth surface. A
horizontal force of 20.0 N is required to hold
the object at rest when it is pulled 0.200 m from
its equilibrium position (the origin of the x
axis). The object is now released from rest with
an initial position of xi 0.200 m, and it
subsequently undergoes simple harmonic
oscillations. Find (a) the force constant of the
spring, (b) the frequency of the oscillations,
and (c) the maximum speed of the object. Where
does this maximum speed occur? (d) Find the
maximum acceleration of the object. Where does it
occur? (e) Find the total energy of the
oscillating system. Find (f) the speed and (g)
the acceleration of the object when its position
is equal to one third of the maximum value.
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Exercise 12.11
11. A 0.500-kg object attached to a spring with a
force constant of 8.00 N/m vibrates in simple
harmonic motion with an amplitude of 10.0 cm.
Calculate (a) the maximum value of its speed and
acceleration, (b) the speed and acceleration when
the object is 6.00 cm from the equilibrium
position, and (c) the time interval required for
the object to move from x 0 to x 8.00 cm.
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Exercise 12.24
24. The angular position of a pendulum is
represented by the equation ? (0.032 0 rad) cos
?t, where ? is in radians and ? 4.43 rad/s.
Determine the period and length of the pendulum.
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Exercise 12.25

• 25. A particle of mass m slides without friction
  inside a hemispherical bowl of radius R. Show
  that if it starts from rest with a small
  displacement from equilibrium, the particle moves
  in simple harmonic motion with an angular
  frequency equal to that of a simple pendulum of
  length R (that is, ).

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Exercise 12.47

• 47. A pendulum of length L and mass M has a
  spring of force constant k connected to it at a
  distance h below its point of suspension. Find
  the frequency of vibration of the system for
  small values of the amplitude (small ?). Assume
  that the vertical suspension of length L is
  rigid, but ignore its mass.