System A consists of a mass m attached to a spring with a force constant k; system B has a mass 2m attached to a spring with a force constant k; system C has a mass 3m attached to a spring with a force constant 6k; and system D has a mass m attached to a spring with a force constant 4k.
Rank these systems in order of increasing period of oscillation. (Use only the symbols < and =, for example A < B = C.)

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AbsorbingMan AbsorbingMan · Answer 1 · 2021-04-19T08:40:44+02:00

Solution :

We know that the time period of oscillation of a spring mass system is given by :

[tex]$T = 2 \pi \sqrt{\frac{m}{k}}$[/tex] , where m is mass and k is the spring constant

∴ [tex]$T_A = 2 \pi \sqrt{\frac{m}{k}}$[/tex] .........(i)

[tex]$T_B = 2 \pi \sqrt{\frac{2m}{k}}$[/tex] ..........(ii)

[tex]$T_C = 2 \pi \sqrt{\frac{3m}{6k}} = 2 \pi \sqrt{\frac{m}{2k}}$[/tex] ..........(iii)

[tex]$T_D = 2 \pi \sqrt{\frac{m}{4k}}$[/tex] ...............(iv)

Comparing the equations (i), (ii), (iii) and (iv)

We get

[tex]$T_B > T_A > T_C > T_D$[/tex]

So in increasing order of time period, we get

[tex]$T_D < T_C < T_A < T_B$[/tex]