Adamped harmonic oscillator loses 6.0% of its mechanical energy per cycle. a. by what percentage does its frequency (equation 14-20) differ from its natural frequency? b. after how many periods will the amplitude have decreased to 1/2 of its original value?

1. The problem statement, all variables and given/known data

A damped harmonic oscillator loses 6.0% of it's mechanical energy per cycle. (a) By what percentage does it's frequency differ from the natural frequency f

0

=

(

1

2

π

)

√

k

m

? (b) After how many periods will the amplitude have decreased to  

1

e

of it's original value?

2. Relevant equations

natural frequency

f

0

=

(

1

2

π

)

√

k

m

damped frequency

f

′

=

1

2

π

√

k

m

−

b

2

4

m

2

displacement for lightly damped harmonic oscillator

x

=

A

e

(

−

b

2

m

)

t

c

o

s

ω

′

t

Total mechanical energy

E

=

1

2

k

A

2

=

1

2

m

v

2

m

a

x

And I know the mean half life,  

2

m

b

is the time until oscillations reach 1/e of original.

3. The attempt at a solution

I used the A^2 expression for E and the A decay term,  

A

e

(

−

b

2

m

)

t

,said it loses 6% of E when A^2 = .94A^2 (original) or in other words when

A

e

(

−

b

2

m

)

t

=  

√

0.94

A

so,  

e

(

−

b

2

m

)

t

=  

√

.94

−

b

2

m

t

=  

1

2

ln(.94)

t =  

− 1. The problem statement, all variables and given/known data

A damped harmonic oscillator loses 6.0% of it's mechanical energy per cycle. (a) By what percentage does it's frequency differ from the natural frequency f

0

=

(

1

2

π

)

√

k

m

? (b) After how many periods will the amplitude have decreased to  

1

e

of it's original value?

2. Relevant equations

natural frequency

f

0

=

(

1

2

π

)

√

k

m

damped frequency

f

′

=

1

2

π

√

k

m

−

b

2

4

m

2

displacement for lightly damped harmonic oscillator

x

=

A

e

(

−

b

2

m

)

t

c

o

s

ω

′

t

Total mechanical energy

E

=

1

2

k

A

2

=

1

2

m

v

2

m

a

x

And I know the mean half life,  

2

m

b

is the time until oscillations reach 1/e of original.

3. The attempt at a solution

I used the A^2 expression for E and the A decay term,  

A

e

(

−

b

2

m

)

t

,said it loses 6% of E when A^2 = .94A^2 (original) or in other words when

A

e

(

−

b

2

m

)

t

=  

√

0.94

A

so,  

e

(

−

b

2

m

)

t

=  

√

.94

−

b

2

m

t

=  

1

2

ln(.94)

t =  

−

m

b

ln(.94)  

But this is time and I need it to be one cycle so do I plug the period in for t?  

T = 1/f or 2∏ ω?

m

b

ln(.94)  

But this is time and I need it to be one cycle so do I plug the period in for t?  

T = 1/f or 2∏ ω?

the motion of a pendulum is a classic example of mechanical energy conservation. a pendulum consists of a mass (known as a bob ) attached by a string to a pivot point. as the pendulum moves it sweeps out a circular arc, moving back and forth in a periodic fashion.