Two identical point masses P and Q, suspended from two separate massless springs of spring constants and , respectively, oscillate vertically. If their maximum speeds are the same, the ratio of the amplitude of mass to the amplitude of mass is

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Solution

Q.
Two identical point masses P and Q, suspended from two separate massless springs of spring constants $k_{1}$ and $k_{2}$ , respectively, oscillate vertically. If their maximum speeds are the same, the ratio $(\frac{A_{Q}}{A_{P}})$ of the amplitude $A_{Q}$ of mass $Q$ to the amplitude $A_{P}$ of mass $P$ is [2025]

1 $\sqrt{\frac{k_{2}}{k_{1}}}$

2 $\sqrt{\frac{k_{1}}{k_{2}}}$

3 $\frac{k_{2}}{k_{1}}$

4 $\frac{k_{1}}{k_{2}}$

Ans.
(2)

For a spring-mass system, the angular frequency is given by

   $ω = \sqrt{\frac{k}{m}}$

For body $P : v_{1 (max)} = ω_{1} A_{P} = \sqrt{\frac{k_{1}}{m}} A_{P}$ ...(i)

For body $Q : v_{2 (max)} = ω_{2} A_{Q} = \sqrt{\frac{k_{2}}{m}} A_{Q}$ ...(ii)

Equate the equation (i) and (ii),

   $\sqrt{\frac{k_{1}}{m}} A_{P} = \sqrt{\frac{k_{2}}{m}} A_{Q}$

or $\sqrt{k_{1}} A_{P} = \sqrt{k_{2}} A_{Q}$

Ratio of the amplitudes of Q and P:

   $\frac{A_{Q}}{A_{P}} = \sqrt{\frac{k_{1}}{k_{2}}}$

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