Question

To double the total energy of an oscillating mass-spring system, by what factor must the amplitude increase?

Answer 1

Given :

▪ For an oscillating mass-spring system,

• Initial total energy = E
• Final total energy = 2E

To Find :

▪ By what factor must the amplitude invrease.

Formula :

→ Formula of total energy associated with mass-spring system at any point is given by

$\bigstar\:\underline{\boxed{\bf{\red{E=\dfrac{1}{2}kA^2}}}}$

• E denotes total energy
• k denotes spring constant
• A denotes amplitude

Calculation :

→ From the above formula, we can say that

$\dashrightarrow\bf\:E\varpropto A^2\\ \\ \dashrightarrow\sf\:\dfrac{E_1}{E_2}=\dfrac{{A_1}^2}{{A_2}^2}\\ \\ \dashrightarrow\sf\:\dfrac{\cancel{E}}{2\times \cancel{E}}=\dfrac{A^2}{{A_2}^2}\\ \\ \dashrightarrow\sf\:\dfrac{1}{2}=\dfrac{A^2}{{A_2}^2}\\ \\ \dashrightarrow\sf\:{A_2}^2=2A^2\\ \\ \dashrightarrow\underline{\boxed{\bf{\gray{A_2=\sqrt{2}A}}}}\:\orange{\bigstar}$

→ Thus, amplitude will increase by factor √2.

Answer 2

Answer :

$\bigstar \;\bold{\bf{\blue{E=\frac{1}{2}\;kA^2 }}}$

where ,

• E denotes ' Total Energy '
• k denotes ' Spring Constant '
• A denotes ' Amplitude '

Here ,

• E ∝ A²

$\implies \bold{\frac{E_1}{E_2}=\frac{A_1^2}{A_2^2} }\\\\\bold{Here\;they\;say\;To\;double\;the\;total\;energy}\\\\\implies \bold{\frac{E}{2E} =\frac{A^2}{A_2^2} }\\\\\implies \bold{A_2^2=2A^2}\\\\\implies \bold{\bf{\orange{A_2=\sqrt{2}A}} }\;\bigstar$

So , to double the total energy Amplitude must increases to √2 A