Answer to Question #80775 in Mechanics | Relativity for hamid

Question

does the time period of a spring mass system depends on medium? will it be different in air and water?

Answer 1

First, obtain the formula for period of a spring mass system in vacuum, zero friction surfaces: for any change in spring’s length Δx we can write according to Newton’s second law and Hooke’s law:
ma=kΔx ⇨m·ω^2 Δx=kΔx ⇨m·(2π/T)^2 Δx=kΔx ⇨T=2π√(m/k).
Now obtain the same formula for any medium. The medium will cause a resistive force F_R (drag or friction). Write this:
ma=kΔx+F_R ⇨m·ω^2 Δx=kΔx+F_R ⇨
⇨ m·(2π/T)^2 Δx=kΔx+F_R ⇨T=2π√(mΔx/(kΔx+F_R )).
In case of vacuum, no friction F_R=0 and
T=2π√(mΔx/(kΔx+F_R ))
becomes
T=2π√(m/k).