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Chapter 14 – Vibrations and Waves

Every swing follows the same path

This action is an example of vibrational motion

vibrational motion - mechanical oscillations around an

equilibrium point

14.1 Periodic Motion

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Each trip back and forth takes the same amount of time

This motion, which repeats in a regular cycle, is an example

of periodic motion

14.1 Periodic Motion

The simplest form of

periodic motion can be

represented by a mass

oscillating on the end of a

coil spring.

- mounted horizontally

- ignore mass of spring

- no friction

14.1 Periodic Motion

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- Any spring has a natural

length at which it exerts no

force on the mass, m. This

is the equilibrium position

- Moving the mass

compresses or stretches the

spring, and the spring then

exerts a force on the mass in

the direction of the equilibrium

position

- This is the restoring force

14.1 Periodic Motion

- At the equilibrium

position x = 0 and F = 0

- The further the mass is

moved (in either direction)

from the equilibrium

position, the greater the

restoring force, F

- The restoring force is

directly proportional to the

displacement from the

equilibrium position

14.1 Periodic Motion

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Hooke’s Law (restoring force of an ideal spring)

F = -kx

- The minus sign indicates the restoring force is always

opposite the direction of the displacement

- k is the “spring constant” (units of N/m)

- a stiffer spring has a larger value of k (more force is

required to stretch it)

- Note, the force changes as x changes, so the

acceleration of the mass is not constant

14.1 Periodic Motion

14.1 Periodic Motion

A spring stretches by 18 cm

when a bag of potatoes with a

mass of 5.71 kg is suspended

from its end.

a) Determine the spring

constant.

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14.1 Periodic Motion

A 57.1 kg cyclist sits on a

bicycle seat and compresses

the two springs that support it.

The spring constant equals

2.2 x 104 N/m for each spring.

How much is each spring

compressed? (Assume each

spring bears half the weight of

the cyclist)

14.1 Periodic Motion

- The spring has the potential to do work on the ball

- The work however, is NOT W = Fx, because F varies with

displacement

- We can use the average force:

F = 1__

2(0 + kx) =

__

2

1kx

W = Fx = __

2

1kx(x) =

__

2

1kx2

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14.1 Periodic Motion

Potential Energy in a Spring

The potential energy in a spring is equal to one-half

times the product of the spring constant and the

square of the displacement

__

2

1kx2PEsp =

14.1 Periodic Motion

A 0.5 kg block is used to compress a spring with

a spring constant of 80.0 N/m a distance of 2.0

cm (.02 m). After the spring is released, what is

the final speed of the block?

2.0 cm

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- First, the object is stretched

from the equilibrium position a

distance x = A

- The spring exerts a force to

pull towards equilibrium

position

- Because the mass has been accelerated, it passes by the

equilibrium position with considerable speed

- At the equilibrium position, F = 0, but the speed is a

maximum

14.1 Periodic Motion

- As its momentum carries it to

the left, the restoring force now

acts to slow (decelerate) the

mass, until is stops at x = -A

- The mass then begins to

move in the opposite

direction, until it reaches x = A

The cycle then repeats

(periodic motion)

14.1 Periodic Motion

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Terms for discussing periodic motion

Displacement – the distance of the mass from the

equilibrium point at any moment

Amplitude – the greatest distance from the equilibrium point

14.1 Periodic Motion

Cycle – a complete to-and-fro

motion (i.e. – from A to –A and back

to A)

Period (T) – the time required to

complete one cycle

Frequency (f) – the number of

cycles per second (measured in

Hertz, Hz)

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Frequency and Period are inversely related

If the frequency is 5 cycles per second (f = 5 Hz), what is

the period (seconds per cycle)?

1__

5s

f =1__

Tand T =

1__

f

14.1 Periodic Motion

Any vibrating system for which the

restoring force is directly proportional

to the negative of the displacement is

said to exhibit simple harmonic

motion.

14.1 Periodic Motion

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14.1 Periodic Motion

Period of a Mass-Spring System

Tm

k= 2π

14.1 Periodic Motion

A 1.0 kg mass attached to one end of a spring

completes one oscillation every 2.0 s. Find the spring

constant

What size mass will make the spring vibrate once

every 1.0 s?

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14.1 Periodic Motion

Simple harmonic motion can also be demonstrated with a simple pendulum

The net force on the pendulum is a restoring force

x = max

v = min

a = max

x = min

v= max

a = min

14.1 Periodic Motion

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14.1 Periodic Motion

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14.1 Periodic Motion

Period of a Pendulum

The period of a pendulum is equal to two pi times the

square root of the length of the pendulum divided by

the acceleration due to gravity

Tl

g= 2π

14.1 Periodic Motion

What is the period of a 99.4 cm long pendulum?

What is the period of a 3.98 m long pendulum?

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14.1 Periodic Motion

A desktop pendulum swings

back and forth once every

second.

How tall is this pendulum?

14.1 Periodic Motion

Resonance

The condition in which a time dependent force can

transmit large amounts of energy to an oscillating

object leading to a larger amplitude motion.

Resonance occurs when the frequency of the force

matches a natural frequency at which the object will

oscillate.