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प्रश्न
Describe the vertical oscillations of a spring.
उत्तर
Vertical oscillations of a spring: Let us consider a massless spring with stiffness constant or force constant k attached to a ceiling as shown in the figure. Let the length of the spring before loading mass m be L. If the block of mass m is attached to the other end of the spring, then the spring elongates by a length l. Let F1 be the restoring force due to the stretching of spring. Due to mass m, the gravitational force acts vertically downward. We can draw a free-body diagram for this system as shown in the figure. When the system is under equilibrium,
F1 + mg = 0 …........(1)
But the spring elongates by small displacement l, therefore,
`"F"_1 ∝ "l" ⇒ "F"_1 = -"kl"` ...................(2)
Substituting equation (2) in equation (1), we get
−kl + mg = 0
mg = kl (or) `"m"/"k" = "l"/"g"` ..................(3)
Suppose we apply a very small external force on the mass such that the mass further displaces downward by a displacement y, then it will oscillate up and down. Now, the restoring force due to this stretching of spring (total extension of spring is y + l) is
A massless spring with stiffness constant k
`"F"_2 ∝ ("y" + "l")`
F2 = −k(y + l) = −ky − kl .........(4)
Since, the mass moves up and down with acceleration `("d"^2"y")/"dt"^2`, by drawing the free body diagram for this case, we get
−ky − kl + mg = `"m"("d"^2"y")/"dt"^2` ......(5)
The net force acting on the mass due to this stretching is
F = F2 + mg
F = −ky − kl + mg .......(6)
The gravitational force opposes the restoring force. Substituting equation (3) in equation (6), we get
F = −ky − kl + kl = −ky
Applying Newton’s law, we get
`"m"("d"^2"y")/"dt"^2 = -"ky" ⇒ ("d"^2"y")/"dt"^2 = -"k"/"m""y"` .............(7)
The above equation is in the form of simple harmonic differential equation. Therefore, we get the time period as
T = `2π sqrt("m"/"k")` second .......(8)
The time period can be rewritten using equation (3)
T = `2π sqrt("m"/"k") = 2π sqrt("l"/"g")` ........(9)
The acceleration due to gravity g can be computed from the formula
g = `4π^2 ("l"/"T"^2) "ms"^(-2)` ...........(10)
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