when spring is stretched or compressed it has elastic potential energy
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When spring is stretched or compressed it has elastic potential energy.
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Welastic = 1/2 kx02 - 1/2 kxf
2
or Initial elastic potential energy minus Final elastic potential energy.
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PEelastic = 1/2 kx2 where k is the spring constant, and x is the distance the spring is compressed or stretched beyond its unstrained length. The unit is the joule (J).
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When external nonconservative forces do no net work on a system then total mechanical energy must be conserved.
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Ef = E0
Total mechanical energy = translational kinetic energy + rotational kinetic energy + gravitational potential energy + elastic potential energy.
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If there is no rotation, this becomes this equation:
E = 1/2 mv2 + mgh + 1/2 kx2
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Ex. 10 - A m = 0.200 kg object is vibrating on a horizontal, frictionless table. The spring (k = 545 N/m) is stretched initially to x0 = 0.0450 m and then released from rest. Determine the final translational speed vf
of the object when the final displacement of the spring is (a) xf = 0.0255 m and (b) xf = 0.
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Ex. 12 - A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. The ball, supported initially so that the spring is neither stretched nor compressed, is released form rest. How far does the ball fall before being momentarily stopped by the spring?
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A simple pendulum is a mass m suspended by a pivot P. When the object is pulled to one side and released, it will swing back and forth in a motion approximating simple harmonic motion.
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A series of substitutions finds that, for small angles,
2πf = √g/L
f is frequency, g is 9.80, and L is length.
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2πf = √g/LMass is algebraically eliminated, and it has no bearing on the frequency of a pendulum.
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Ex. 13 - Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s.
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A pendulum can be a real object, in which case it is called a physical pendulum.
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In reality, an object in simple harmonic motion will not vibrate forever. Friction, or some such force, will decrease the velocity and amplitude of the motion. This is called damped harmonic motion.
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A shock absorber introduces damping forces to reduce the vibrations of the ride. The smallest degree of damping that completely eliminates the oscillations is called critical damping.
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When damping exceeds the critical value, this is called overdamping. Underdamping is when the damping forces are less than what would eliminate the vibration.
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Automobile shocks are generally designed to produce underdamped motion.
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In driven harmonic motion, a driving force is applied to assist the object in its vibrational motion.
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If the frequency of this driving force is the same as the natural frequency of the system, the total mechanical energy of the system increases.
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Resonance is when a periodic force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion.
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Resonance usually occurs when the frequency of the driving force is equal to the natural frequency of the object.
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Formulas for frequency and period of an oscillating spring:
2•π•f = √ k/m2•π/ T = √ k/m
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