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Chapter 13

Vibrations and Waves(Part 1)

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Outline

Hook’s Law and periodic motionElastic potential energyEquations of simple harmonic motionMotion of a pendulum

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Hooke’s LawFs = - k x

Fs is the spring forcek is the spring constant

It is a measure of the stiffness of the springA large k indicates a stiff spring and a small k indicates a soft spring

x is the displacement of the object from its equilibrium position

x = 0 at the equilibrium position

The negative sign indicates that the force is always directed opposite to the displacement

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Hooke’s Law Force

The force always acts toward the equilibrium position

It is called the restoring force

The direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position

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Hooke’s Law Applied to a Spring – Mass System

When x is positive (to the right), F is negative (to the left)When x = 0 (at equilibrium), F is 0When x is negative (to the left), F is positive (to the right)

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Motion of the Spring-Mass System

Assume the object is initially pulled to a distance Aand released from restAs the object moves toward the equilibrium position, F and a decrease, but v increasesAt x = 0, F and a are zero, but v is a maximumThe object’s momentum causes it to overshoot the equilibrium position

amF

xkFsrr

rr

=

−=

2

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Motion of the Spring-Mass System, cont

The force and acceleration start to increase in the opposite direction and velocity decreasesThe motion momentarily comes to a stop at x = - AIt then accelerates back toward the equilibrium positionThe motion continues indefinitely

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Simple Harmonic MotionMotion that occurs when the net force along the direction of motion obeys Hooke’s Law

The force is proportional to the displacement and always directed toward the equilibrium position

The motion of a spring mass system is an example of Simple Harmonic Motion

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Simple Harmonic Motion, cont.

Not all periodic motion over the same path can be considered Simple Harmonic motionTo be Simple Harmonic motion, the force needs to obey Hooke’s Law

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Periodic Motion:Amplitude

Amplitude, AThe amplitude is the maximum position of the object relative to the equilibrium positionIn the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A

-A A

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Periodic Motion: Period and Frequency

The period, T, is the time that it takes for the object to complete one complete cycle of motion

From x = A to x = - A and back to x = A

The frequency, ƒ, is the number of complete cycles or vibrations per unit time

ƒ = 1 / TFrequency is the reciprocal of the period

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Quick Quiz 13.1-2A block on the end of a spring is pulled to position x = A and released. Through what total distance does it travel in one fullcycle of its motion? (a) A/2; (b) A; (c) 2A ; (d) 4AWhich of the following pairs of vector quantities can’t both point in the same direction? (a) position and velocity (b) velocity and acceleration (c) position and acceleration

-A A

amF

xkFsrr

rr

=

−=

3

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Acceleration of an Object in Simple Harmonic Motion

Newton’s second law will relate force and accelerationThe force is given by Hooke’s LawF = - k x = m a

a = -kx / m

The acceleration is a function of positionAcceleration is not constant and therefore the uniformly accelerated motion equation cannot be applied

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Elastic Potential Energy

A compressed spring has potential energy

The compressed spring, when allowed to expand, can apply a force to an objectThe potential energy of the spring can be transformed into kinetic energy of the object

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Elastic Potential Energy, cont

The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy

PEs = ½kx2

The energy is stored only when the spring is stretched or compressedElastic potential energy can be added to the statements of Conservation of Energy and Work-Energy

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Energy in a Spring Mass System

A block sliding on a frictionless system collides with a light springThe block attaches to the springThe system oscillates in Simple Harmonic MotionThe spring force is conservative and the total energy of the system remains constant

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Velocity as a Function of Position

Conservation of Energy allows a calculation of the velocity of the object at any position in its motion

Speed is a maximum at x = 0Speed is zero at x = ±AThe ± indicates the object can be traveling in either direction

222

21

21

21 kxmvkA +=

)( 22 xAmk

−±=v

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Example: Problem #9

x0 =2.0 cmm = 100 ghmax = 60.0 cm

k - ?

4

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Simple Harmonic Motion and Uniform Circular Motion

A ball is attached to the rim of a turntable of radius AThe focus is on the shadow that the ball casts on the screenWhen the turntable rotates with a constant angular speed, the shadow moves in simple harmonic motion

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Period from Circular Motion

The ball rotates with constant speed v0, its period

Analogy with ball and spring system leads to

A

0

2vAT π

=

kmT π2=

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Period and Frequency from Circular Motion

Period

This gives the time required for an object of mass m attached to a spring of constant k to complete one cycle of its motion

Frequency

Units are cycles/second or Hertz, Hz

kmT π2=

mk

T π211ƒ ==

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Angular FrequencyThe angular frequency is related to the frequency

The frequency gives the number of cycles per secondThe angular frequency gives the number of radians per second

mk

== ƒ2πω

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Example: Problem #24

x0 =0.8 cmm = 320 kgM = 2x103 kg

f - ?mk

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T1ƒ

π==

x0