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Chapter 13

Oscillations About Equilibrium (Cont.)

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Outline Energy conservation in oscillatory

motion The simple pendulum

The period of a simple pendulum Driven oscillations and resonance

Energy conservation in oscillatory motion In an ideal system with no friction or

other nonconservative forces, the total energy (E) is conserved.

For a mass on a horizontal spring: Total energy E = K + U K = kinetic energy = (1/2)mv2

U = elastic potential energy = (1/2)kx2

The kinetic (K) and potential (U) energy change with time, but the total energy E is conserved.

It can be shown that E = (1/2)kA2, where A is the amplitude of oscillation.

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Energy vs. position in simple harmonic motion

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Example A 0.980-kg block slides on a

frictionless, horizontal surface with a speed of 1.32 m/s. The block encounters an unstretched spring with a force constant of 245 N/m.

(a) How far is the spring compressed before the block comes to rest?

(b) How long is the block in contact with the spring before it comes to rest?

Answers: (a) 0.0835 m (b) 0.0993 s

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The pendulum Simple pendulum: consists of

a mass m suspended by a light string or rod of length L.

Period of a pendulum: The period of a simple pendulum

depends on the length of the pendulum and the acceleration of gravity. It is independent of the mass and the amplitude.

T 2L

gL

m

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Example A pendulum is constructed from

a string 0.627 m long attached to a mass of 0.250 kg. The pendulum completes one oscillation every 1.59 s. Find the acceleration of gravity, g.

“Gravity maps” are valuable tools for geologists attempting to understand the underground properties of a given region, such as density of rocks.

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Driven oscillations and resonance

Natural frequency, f0: The frequency at which the system oscillates when it is not driven. In general, driving any system

at a frequency near its natural frequency results in large oscillations.

Resonance: This type of large response, due to frequency matching, is known as resonance.

Tacoma Narrows bridge, 1940

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Homework See online homework on

www.masteringphysics.com