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Spring Pendulum

A point mass suspended from a massless spring or placed on a frictionless horizontal plane attached with spring (Fig. 7) constitutes a linear harmonic spring pendulum
Fig. 7
Time period, 59797.png 59791.png
and frequency, 59785.png
Important Points
  • Time period of a spring pendulum depends on the mass suspended
    63911.png or 63905.png
    i.e., greater the mass greater will be the inertia and so lesser will be the frequency of oscillation and greater will be the time period.
  • The time period depends on the force constant k of the spring
    63899.png or 63893.png
  • Time of a spring pendulum is independent of acceleration due to gravity. That is why a clock based on spring pendulum will keep proper time every where on a hill or moon or in a satellite and time period of a spring pendulum will not change inside a liquid if damping effects are neglected.
  • If two masses of mass m1 and m2 are connected by a spring and made to oscillate on horizontal surface, the reduced mass mr is given by 63887.png
    Fig. 8
    So that 63875.png
  • If a spring pendulum oscillating in a vertical plane is made to oscillate on a horizontal surface, (or on inclined plane) time period will remain unchanged. However, equilibrium position for a spring in a horizontal plain is the position of natural length of spring as weight is balanced by reaction. While in case of vertical motion, equilibrium position will be L +y0 with ky0 = mg.
    Fig. 9
  • If the stretch in a vertically loaded spring is y0, then for equilibrium of mass m, ky0 = mg, i.e., 63959.png.
    So 63953.png
    Time period does not depend on gbecause along with g, yo will also change in such a way that 63946.pngremains constant.
  • The spring constant k is inversely proportional to the spring length.
    As 63939.png
    That means if the length of spring is halved, then its force constant becomes double.
  • A spring of length l is cut into two pieces of length l1 and l2 such that l1 =nl2.
    If the constant of a spring is k, then spring constant of first part,
    Spring constant of second part, k2 = (n + 1)k
    and ratio of spring constant, 63927.png


Important Facts and Formulae
  • The pendulum clock runs slow due to increase in its time period whereas it becomes fast due to decrease in time period.
  • If infinite spring with force constant k, 2k, 4k, 8k, …, respectively, are connected in series. The effective force constant of the spring will bek/2.
  • If y1 = a sin ωt and y2 = b cos ωt are two SHM then by the superimposition of these two SHM we get
    y = a sin ωt + b cos ωt = A sin (ωt + φ)
    This is also the equation of SHM
    where 62428.png and φ = tan–1(b/a)
  • If a particle performs SHM whose velocity is v1 at x1 distance from mean position and velocity v2 at distance x2, then

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