Question

How should the mass of a harmonic oscillator be changed to double the frequency? Can the frequency be tripled by a suitable adjustment of the mass?

Answer 1

Answer:

a. $m' = \frac{m}{4}$

b. $m' = \frac{m}{9}$

Explanation:

The frequency of a harmonic oscillator is given by the following formula:

$\omega = \sqrt{\frac{k}{m}}$   ----------------- equation (1)

a.

In order to double the frequency of this oscillator:

ω' = 2ω

m' = ?

Therefore,

$\omega ' = 2\omega = \sqrt{\frac{k}{m'}}$

using equation (1):

$2 \sqrt{\frac{k}{m}} = \sqrt{\frac{k}{m'}}\\\\ \frac{4}{m} = \frac{1}{m'}$  

$m' = \frac{m}{4}$

a.

In order to triple the frequency of this oscillator:

ω' = 3ω

m' = ?

Therefore,

$\omega ' = 3\omega = \sqrt{\frac{k}{m'}}$

using equation (1):

$3\sqrt{\frac{k}{m}} = \sqrt{\frac{k}{m'}}\\\\ \frac{9}{m} = \frac{1}{m'}$  

$m' = \frac{m}{9}$