Answer:
a. [tex]m' = \frac{m}{4}[/tex]
b. [tex]m' = \frac{m}{9}[/tex]
Explanation:
The frequency of a harmonic oscillator is given by the following formula:
[tex]\omega = \sqrt{\frac{k}{m}}[/tex] ----------------- equation (1)
a.
In order to double the frequency of this oscillator:
ω' = 2ω
m' = ?
Therefore,
[tex]\omega ' = 2\omega = \sqrt{\frac{k}{m'}}[/tex]
using equation (1):
[tex]2 \sqrt{\frac{k}{m}} = \sqrt{\frac{k}{m'}}\\\\ \frac{4}{m} = \frac{1}{m'}[/tex]
[tex]m' = \frac{m}{4}[/tex]
a.
In order to triple the frequency of this oscillator:
ω' = 3ω
m' = ?
Therefore,
[tex]\omega ' = 3\omega = \sqrt{\frac{k}{m'}}[/tex]
using equation (1):
[tex]3\sqrt{\frac{k}{m}} = \sqrt{\frac{k}{m'}}\\\\ \frac{9}{m} = \frac{1}{m'}[/tex]
[tex]m' = \frac{m}{9}[/tex]
A) To double the Frequency of a harmonic oscillator ;
Divide the mass by four i.e. m₁ = m / 4
B) To triple the frequency of a harmonic oscillator :
Divide the mass by nine (9) i.e. m₂ = m / 9
Given that The frequency of a harmonic oscillator is expressed as
[tex]w = \sqrt{\frac{k}{m} }[/tex] -- ( 1 )
A) Doubling the frequency
[tex]2w = \sqrt{\frac{k}{m_{1} } }[/tex] ---- ( 2 )
Applying equation ( 1 ) and ( 2 )
[tex]2\sqrt{\frac{k}{m} } = \sqrt{\frac{k}{m_{1} } }[/tex]
squaring both sides
( 4 / m ) = 1 / m₁
∴ m₁ ( new mass ) = m / 4
B) Tripling the frequency
3w = [tex]\sqrt{\frac{k}{m_{2} } }[/tex] ---- ( 3 )
applying equation ( 1 ) and ( 3 )
[tex]3 \sqrt{\frac{k}{m} } = \sqrt{\frac{k}{m_{2} } }[/tex]
squaring both sides
( 9 / m ) = 1 / m₂
∴ m₂ = m / 9
Hence we can conclude that To double the Frequency of a harmonic oscillator m₁ = m / 4 and To triple the frequency of a harmonic oscillator : m₂ = m / 9
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